{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier
" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 
2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu
t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" 
-1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 
2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 26 "Discrete Fourier Transfor
m" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "We d
evelop the idea of a Fourier transform in order to analyze functions i
n creative ways:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 58 "temporal signals in the frequency domain (signal analysis
)" }}{PARA 0 "" 0 "" {TEXT -1 62 "spatial information in the momentum \+
domain (quantum mechanics)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 462 "For some functions the Fourier transform (FT) ca
n be calculated analytically, but we will be mostly concerned with dis
crete FTs for which fast algorithms (FFTs) exist. The DFT (or FFT) is \+
a numerical technique that can be applied to virtually any function. M
any digital measuring devices (oscilloscopes) have built-in hardware w
hich can carry out FFTs for signal analysis. In numerical analysis FFT
s are used to solve ordinary and partial differential equations." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "restart; with(LinearAlgebra)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(plots): with(intt
rans):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords \+
has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fx:
=(4/sqrt(5*Pi))/(1+x^2)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxG,$
*,\"\"%\"\"\"\"\"&!\"\"F)#F(\"\"#%#PiG#F*F,,&F(F(*$)%\"xGF,F(F(!\"#F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int(fx^2,x=-infinity..i
nfinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "fp:=fourier(fx,x,p);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#fpG,$*&#\"\"#\"\"&\"\"\"*(F)#F*F(%#PiGF,,&*(-%$expG6
#,$%\"pG!\"\"F*,&F4F*F*F*F*-%*HeavisideG6#F4F*F**(-F1F9F*,&F4F*F*F5F*-
F8F2F*F5F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int(fp^2,
p=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"
\"\"%#PiGF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The FT defined a
s part of the " }{TEXT 19 9 "inttrans " }{TEXT -1 82 "package misses a
 factor, and we redefine the answer so that the norm is preserved." }}
{PARA 0 "" 0 "" {TEXT -1 55 "The naive calculation of the FT for our f
unction fails:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "fpn:=1/sq
rt(2*Pi)*int(fx*exp(I*p*x),x=-infinity..infinity);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$fpnG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "
It can be fixed, however, by assuming that the variable " }{TEXT 19 1 
"p" }{TEXT -1 16 " is real-valued." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "assume(p,real);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "fpn:=1/sqrt(2*Pi)*int(fx*exp(I*p*x),x=-infinity..infi
nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fpnG,$*&#\"\"\"\"#5F(*,\"
\"##F(F+\"\"&F,-%'signumG6#%#p|irGF(,2*$)-%$expGF0F+F(F(*&F4F(F.F(!\"
\"F1F8F.F8*&F1F(F.F(F8*&F1F(F4F(F8*(F1F(F4F(F.F(F(F(F8F(F5F8F(F8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "fpn:=1/sqrt(2*Pi)*int(fx*cos
(p*x),x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fp
nG,$*&#\"\"\"\"\"&F(**\"\"##F(F+F)F,-%'signumG6#%#p|irGF(,*-%%sinhGF/F
(*&-%%coshGF/F(F-F(!\"\"*&F0F(F5F(F7*(F0F(F2F(F-F(F(F(F(F7" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "fpnI:=1/sqrt(2*Pi)*int(fx*sin(p*x),
x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fpnIG\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([fp/sqrt(2*Pi),
fpn],p=-5..5,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 878 280 
280 {PLOTDATA 2 "6&-%'CURVESG6$7in7$$!\"&\"\"!$\"3w&ysC6rob#!#>7$$!3YL
LLe%G?y%!#<$\"31I[[E44kIF-7$$!3OmmT&esBf%F1$\"3_Y\">N.#[#e$F-7$$!3ALL$
3s%3zVF1$\"3+\"33x))o]E%F-7$$!3_LL$e/$QkTF1$\"3e$*ecr?^v]F-7$$!3ommT5=
q]RF1$\"3`KgrWwgCgF-7$$!3ILL3_>f_PF1$\"3;j8@jDn]qF-7$$!3K++vo1YZNF1$\"
3_\\fB<HS#G)F-7$$!3;LL3-OJNLF1$\"3_j\\IFo2i(*F-7$$!3p***\\P*o%Q7$F1$\"
3O:*)3#>js9\"!#=7$$!3Kmmm\"RFj!HF1$\"3I;vFBv\"3N\"Ffn7$$!33LL$e4OZr#F1
$\"3WFtltK$eb\"Ffn7$$!3u*****\\n\\!*\\#F1$\"3v3!p!fEE==Ffn7$$!3%)*****
\\ixCG#F1$\"3*4'pgmr==@Ffn7$$!3#******\\KqP2#F1$\"3:Eo$)z\"\\QW#Ffn7$$
!39LL3-TC%)=F1$\"3qO?kKMprFFfn7$$!3[mmm\"4z)e;F1$\"3UP!*pZk+,KFfn7$$!3
Mmmmm`'zY\"F1$\"3#ftj*el;'f$Ffn7$$!3#****\\(=t)eC\"F1$\"3[iVd^QN'3%Ffn
7$$!3!ommmh5$\\5F1$\"3%p'z]&Rj'QXFfn7$$!3S$***\\(=[jL)Ffn$\"3seh.0N[Q]
Ffn7$$!3)f***\\iXg#G'Ffn$\"35fD7s=>%\\&Ffn7$$!3ndmmT&Q(RTFfn$\"3`xQL'o
C8\"fFfn7$$!3Ihm\"HdGe:$Ffn$\"39pp)>.`'ogFfn7$$!3%\\mmTg=><#Ffn$\"3Wyc
iI4K&>'Ffn7$$!3FK$3Fpy7k\"Ffn$\"3wk/)yYT\"[iFfn7$$!3g***\\7yQ16\"Ffn$
\"3)oN>9V>$)G'Ffn7$$!3iK$3_D)=`%)F-$\"3/d4&*oF>.jFfn7$$!3Epm\"zp))**z&
F-$\"3!f1k)z*>VJ'Ffn7$$!3#f+D19*yYJF-$\"3*GX**p'))[@jFfn7$$!3vDMLLe*e$
\\!#?$\"3c^-gI&yWK'Ffn7$$\"3+l;a)3RBE#F-$\"3[6?B:5'HK'Ffn7$$\"3bsmTgxE
=]F-$\"3+?+vFK&oJ'Ffn7$$\"37!o\"HKk>uxF-$\"3'>p1VO0kI'Ffn7$$\"3womT5D,
`5Ffn$\"3A2!f0*y&=H'Ffn7$$\"3Gq;zW#)>/;Ffn$\"3'3(\\X-qP^iFfn7$$\"3!=nm
\"zRQb@Ffn$\"3_QR:OL9(>'Ffn7$$\"3mOLL$e,]6$Ffn$\"3K4,0I+dugFfn7$$\"3_,
+](=>Y2%Ffn$\"3x<U%)*\\UD#fFfn7$$\"3summ\"zXu9'Ffn$\"3T%)4)e(zsAbFfn7$
$\"3#4+++]y))G)Ffn$\"3K'*p%R/_$\\]Ffn7$$\"3H++]i_QQ5F1$\"39h?9#GfSc%Ff
n7$$\"3b++D\"y%3T7F1$\"3]]`o<dC(4%Ffn7$$\"3+++]P![hY\"F1$\"35Qd)G+b+g$
Ffn7$$\"3iKLL$Qx$o;F1$\"3o%*G$G7t?=$Ffn7$$\"3Y+++v.I%)=F1$\"3/2drR:frF
Ffn7$$\"3?mm\"zpe*z?F1$\"3dmK1PAmLCFfn7$$\"3;,++D\\'QH#F1$\"39&e\">&eo
95#Ffn7$$\"3%HL$e9S8&\\#F1$\"32S!=.e`L#=Ffn7$$\"3s++D1#=bq#F1$\"3w4O*)
=YMm:Ffn7$$\"3\"HLL$3s?6HF1$\"3Mp_hw5#fM\"Ffn7$$\"3a***\\7`Wl7$F1$\"3#
[s+%H4#\\9\"Ffn7$$\"3enmmm*RRL$F1$\"3q)>R(H#)Rs(*F-7$$\"3%zmmTvJga$F1$
\"31)phS)*R;H)F-7$$\"3]MLe9tOcPF1$\"3-,[e3PoHqF-7$$\"31,++]Qk\\RF1$\"3
6B@.)*fpHgF-7$$\"3![LL3dg6<%F1$\"3k%)Rws[%y/&F-7$$\"3%ymmmw(GpVF1$\"3Y
4wKbW@*H%F-7$$\"3C++D\"oK0e%F1$\"3qz#3$y`Z<OF-7$$\"35,+v=5s#y%F1$\"3)3
lu*\\jLiIF-7$$\"\"&F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F\\_l-F$6$
7in7$F($\"3IQlZ76(ob#F-7$F/$\"3o!=$[E44kIF-7$F5$\"3M?1_L?[#e$F-7$F:$\"
3%e12x))o]E%F-7$F?$\"39wLcr?^v]F-7$FD$\"31?^rWwgCgF-7$FI$\"37w2@jDn]qF
-7$FN$\"3uYfB<HS#G)F-7$FS$\"3'3Z/t#o2i(*F-7$FX$\"3)y*))3#>js9\"Ffn7$Fh
n$\"3g'[xK_<3N\"Ffn7$F]o$\"3_&HdOFLeb\"Ffn7$Fbo$\"3f%**o!fEE==Ffn7$Fgo
$\"3Qnpgmr==@Ffn7$F\\p$\"3qEo$)z\"\\QW#Ffn7$Fap$\"3EP?kKMprFFfn7$Ffp$
\"3'=/*pZk+,KFfn7$F[q$\"3eUP'*el;'f$Ffn7$F`q$\"3qkVd^QN'3%Ffn7$Feq$\"3
skz]&Rj'QXFfn7$Fjq$\"3$)fh.0N[Q]Ffn7$F_r$\"3)obA@(=>%\\&Ffn7$Fdr$\"3Jv
QL'oC8\"fFfn7$Fir$\"3Eqp)>.`'ogFfnF]sFbs7$Fhs$\"3yb$>9V>$)G'Ffn7$F]t$
\"3$f&4&*oF>.jFfn7$Fbt$\"3zkS')z*>VJ'FfnFft7$F\\u$\"3B[-gI&yWK'Ffn7$Fb
u$\"3E4?B:5'HK'FfnFfuF[v7$Fav$\"371!f0*y&=H'Ffn7$Ffv$\"3'>(\\X-qP^iFfn
7$F[w$\"3SPR:OL9(>'Ffn7$F`w$\"3?3,0I+dugFfn7$Few$\"3m;U%)*\\UD#fFfnFiw
7$F_x$\"3W(*p%R/_$\\]Ffn7$Fdx$\"3#*e?9#GfSc%Ffn7$Fix$\"3%)[`o<dC(4%Ffn
7$F^y$\"3KNd)G+b+g$Ffn7$Fcy$\"3c$*G$G7t?=$Ffn7$Fhy$\"3g2drR:frFFfn7$F]
z$\"3ugK1PAmLCFfn7$Fbz$\"3q!e\">&eo95#Ffn7$Fgz$\"3.A!=.e`L#=Ffn7$F\\[l
$\"3E2O*)=YMm:Ffn7$Fa[l$\"35#G:m2@fM\"Ffn7$Ff[l$\"3L!y+%H4#\\9\"Ffn7$F
[\\l$\"3=3&R(H#)Rs(*F-7$F`\\l$\"3)phgS)*R;H)F-7$Fe\\l$\"3SxIe3PoHqF-7$
Fj\\l$\"3_7K.)*fpHgF-7$F_]l$\"3Q/Ows[%y/&F-7$Fd]l$\"3g'GH`X9#*H%F-7$Fi
]l$\"3RF%4$y`Z<OF-7$F^^l$\"3#ycs*\\jLiIF-7$Fc^lFa_l-Ff^l6&Fh^lF\\_lF\\
_lFi^l-%+AXESLABELSG6$Q#p|ir6\"Q!Fiil-%%VIEWG6$;F(Fc^l%(DEFAULTG" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The graph of the expressions
 is the same." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(f
p/sqrt(2*Pi) - fpn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"
\"&F&*(F'#F&\"\"#F*F),6*(-%$expG6#,$%#p|irG!\"\"F&-%*HeavisideG6#F1F&F
1F&F&*&F-F&F3F&F&*&-F.F5F&F1F&F2*(F8F&F1F&F3F&F&F8F&*&F8F&F3F&F2*&-%%s
inhGF5F&-%'signumGF5F&F&-%%coshGF5F2*&F@F&-%$absGF5F&F2*&F1F&F<F&F&F&F
&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(p=3,%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"&F&*(F'#F&\"\"#F*F),0*(
\"\"%F&-%$expG6#!\"$F&-%*HeavisideG6#\"\"$F&F&*&F*F&-F/F4F&!\"\"*(F*F&
F7F&F2F&F&*&-%%sinhGF4F&-%'signumGF4F&F&-%%coshGF4F8*&F?F&-%$absGF4F&F
8*&F5F&F;F&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simp
lify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 257 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 26 "Modify the above function " }{TEXT 19 2 "
fx" }{TEXT -1 186 " in such a way as to change the scale over which it
 varies by replacing the 1 in the denominator by some other constant. \+
Keep the function normalized by adjusting the factor in front of " }
{TEXT 19 2 "fx" }{TEXT -1 80 ". Calculate the FT, and observe the rela
tionship. Is it true that distributions " }{TEXT 19 2 "fx" }{TEXT -1 
106 " which have more concentration at smaller x-values have transform
s that emphasize more the large-p region?" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "fx:=(2/Pi)^(1/4)/sqr
t(a)*exp(-(x/a)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxG**\"\"##
\"\"\"\"\"%*&F(F(%#PiG!\"\"F'%\"aG#F,F&-%$expG6#,$*&%\"xGF&F-!\"#F,F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fourier(fx,x,p);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#**\"\"##\"\"\"\"\"%%#PiG#!\"\"F'%\"aG#
F*F$-%(fourierG6%-%$expG6#,$*&%\"xGF$F+!\"#F*F5%#p|irGF&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(a>0);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "fourier(fx,x,p);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#**)\"\"##\"\"\"\"\"%F')%#PiGF&F'%#a|irG#F'F%-%$expG6#,$*(F(!\"\"
%#p|irGF%F+F%F2F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The assumpti
on on " }{TEXT 19 1 "a" }{TEXT -1 144 " was required for the FT to be \+
meaningful. The example of a Gaussian is particularly illustrative, be
cause it is obvious how the distance scale " }{TEXT 19 1 "a" }{TEXT 
-1 49 " appears as the inverse momentum scale in the FT." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Now let us look at
 examples relevant for signal theory:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "fourier(sin(t),t,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*(%#PiG\"\"\",&-%&DiracG6#,&%\"wGF%F%!\"\"F,-F(6#,&F+F%F%F%F%F%^#F%
F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fourier(cos(3*t),t,w)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\",&-%&DiracG6#,&%\"wG
F%\"\"$F%F%-F(6#,&F+F%F,!\"\"F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
157 "We obtain a Dirac delta at both positive and negative frequency. \+
For signals made up of a few frequencies the Fourier representation wi
ll be very economical:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f
t:=sin(t)+2*cos(5*t)+1/2*sin(sqrt(5)*t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#ftG,(-%$sinG6#%\"tG\"\"\"*&\"\"#F*-%$cosG6#,$*&\"\"&F*F)F*F*F
*F**&#F*F,F*-F'6#*&F2F4F)F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "plot(ft,t=0..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 876 274 274 
{PLOTDATA 2 "6%-%'CURVESG6$7ez7$$\"\"!F)$\"\"#F)7$$\"3gmmTN@Ki8!#>$\"3
)zCB'eO@C?!#<7$$\"3ALL$3FWYs#F/$\"3gF]%)[h:R?F27$$\"3%)***\\iSmp3%F/$
\"3PF&\\*e_([/#F27$$\"3WmmmT&)G\\aF/$\"3mwn.G9YT?F27$$\"3m****\\7G$R<)
F/$\"3d#e$Ggo!y+#F27$$\"3GLLL3x&)*3\"!#=$\"3+ws-iduR>F27$$\"3$*****\\i
lyM;FJ$\"3H9WNOAt4<F27$$\"3emmm;arz@FJ$\"3#G0W(3ujv8F27$$\"3;L$e*)4bQl
#FJ$\"3c!osac)zC5F27$$\"3v***\\7y%*z7$FJ$\"3#\\wHVAi@V'FJ7$$\"3Lm;ajW8
-OFJ$\"37?A<DsmlDFJ7$$\"3[LL$e9ui2%FJ$!3?!)[v?*))Q4\"FJ7$$\"3\"pm\"H2Q
\\4YFJ$!3S@_*\\oZ)fYFJ7$$\"3z***\\(oMrU^FJ$!3G^D?/$eFN(FJ7$$\"3Cm\"z%*
HB$4aFJ$!3k3e+*GKWH)FJ7$$\"3oK$3-8Lfn&FJ$!3KG*3A!e%[$*)FJ7$$\"3X;HdX!Q
#4eFJ$!3C!Q%4vEmO\"*FJ7$$\"3B+v$4'HaUfFJ$!3!*pBN\")zRd#*FJ7$$\"3!H3-j(
y%e2'FJ$!3)*oIfN!3fH*FJ7$$\"3nmmm\"z_\"4iFJ$!3'oyR&pQV^#*FJ7$$\"3/nm;z
p!fu'FJ$!35\"oJpT[5B)FJ7$$\"3Unmmm6m#G(FJ$!3Wq(*4<fq1fFJ7$$\"3ynm;a`T>
yFJ$!3!))4l'GYc;CFJ7$$\"39ommT&phN)FJ$\"3uli#*))eK7?FJ7$$\"37M$3-js.*)
)FJ$\"3ey_b-+?bqFJ7$$\"3A,+v=ddC%*FJ$\"3_*>dD-G&Q7F27$$\"3?n;H2)y(e**F
J$\"3#3M`f%z6j<F27$$\"3KLLe*=)H\\5F2$\"3(e/h.\"RCUAF27$$\"3wmm;ac#))4
\"F2$\"3TFt$Hdjeh#F27$$\"3-++v=JN[6F2$\"3CUZQ(=eu*GF27$$\"3um;/^o6t6F2
$\"3%)4eLUp%y*HF27$$\"3CLLL$e!)y>\"F2$\"3g&\\oP\\>*oIF27$$\"3]m\"z%\\C
E57F2$\"3)o#zy$*R0$4$F27$$\"3)***\\i:VkA7F2$\"3c$RU$3aU4JF27$$\"3YL3x
\"=E]B\"F2$\"3,l*Qo4[z6$F27$$\"3smm\"z/3uC\"F2$\"3_[Pf&\\l&=JF27$$\"31
++vo3p)H\"F2$\"3X#)z1yu&o.$F27$$\"3ULLe*ot*\\8F2$\"3>,SyKP<BGF27$$\"3x
mmT5lD,9F2$\"3?#o=C4E$*[#F27$$\"3!****\\7LRDX\"F2$\"3/_*e`l!ya?F27$$\"
3!om;z4wb]\"F2$\"3L*Q+erNq_\"F27$$\"3[LLekGhe:F2$\"3w&[L-')*eK&*FJ7$$
\"3;++DJ'\\;h\"F2$\"3mD\\r]#R_q$FJ7$$\"3%om;zR'ok;F2$!3f9Bo$Gi_$=FJ7$$
\"34+++vIb<<F2$!3xYoSG5M=nFJ7$$\"3OLL3_(>/x\"F2$!3kZGG'pu`1\"F27$$\"3h
mm;HkGB=F2$!3I<Vy<')zR8F27$$\"33++D1J:w=F2$!3s'GjQ;J*y9F27$$\"3l;/,WIM
.>F2$!3%)pe$oJl^\\\"F27$$\"3VL3x\")H`I>F2$!3y2ew\"z(ot9F27$$\"3A]7`>Hs
d>F2$!3wflZ,PC:9F27$$\"3+n;HdG\"\\)>F2$!3Na$>$HuC@8F27$$\"37+D\"Gt#HR?
F2$!3$>\")e)>&Ha.\"F27$$\"3oLLL3En$4#F2$!3CV1Q!fpiR'FJ7$$\"35n;HK/dT@F
2$!3ov=AtV*fC#FJ7$$\"3_++Dc#o%*=#F2$\"3+,u[\">rgE#FJ7$$\"3%RL3-3mtB#F2
$\"3'Gt3\\hK3(oFJ7$$\"3#pmmT!RE&G#F2$\"3\\>Bqp]aH6F27$$\"3Z+]P4b=RBF2$
\"3qi:&)3zkt:F27$$\"3eLLe9r5$R#F2$\"3%p=&yY+%)G>F27$$\"3qm;z>(GqW#F2$
\"3uOiMF5po@F27$$\"3D+++D.&4]#F2$\"3yNvk:IIvAF27$$\"3Z+]7yg[9DF2$\"3Y>
Y0eH'*zAF27$$\"3C++DJ=-GDF2$\"3sedbg#RcF#F27$$\"3,+]P%ed:a#F2$\"3#*=Jd
%eZBE#F27$$\"3A++]PL4bDF2$\"30zY,y^9SAF27$$\"3A++vV[;#e#F2$\"3akSiRJWp
@F27$$\"3?+++]jB4EF2$\"3KCCHNp![1#F27$$\"3=++]i$zLm#F2$\"3-h(p,OE>w\"F
27$$\"3;+++vB_<FF2$\"3K-y/fD^`8F27$$\"3!)*******>*ppFF2$\"3QA8GitAy))F
J7$$\"3!******\\-w=#GF2$\"3rl3.u5BKQFJ7$$\"3)*******\\G0uGF2$!31\"HzD*
4uh7FJ7$$\"33+++v'Hi#HF2$!3%[AzSlt*fgFJ7$$\"3!p;H2B6O(HF2$!3_.]\\KWC*)
)*FJ7$$\"3pL$eky#*4-$F2$!39nf#ob_'*H\"F27$$\"3/+v=UVPoIF2$!3>Q#)*Hox6_
\"F27$$\"3&om;z*ev:JF2$!3ve!*\\qsgT;F27$$\"3>](=U_ER9$F2$!3IP)3g5LDm\"
F27$$\"3_L3_]r4sJF2$!3i^jF_M>X;F27$$\"3&o\"H#oxn-?$F2$!3G<<P$37,f\"F27
$$\"3>+]7.%Q%GKF2$!3w[GZ2Ee)\\\"F27$$\"3'o;HdlzZG$F2$!3[;z)erR]@\"F27$
$\"3_LLL347TLF2$!3?v)))4Xe+>)FJ7$$\"3WLLeR$\\))Q$F2$!3XGiEEOD.UFJ7$$\"
3#QLL3xxlV$F2$\"3[$=_M5/c1\"F/7$$\"3wLL3-iI%[$F2$\"3#>/d_4HaZ%FJ7$$\"3
nLLLLY.KNF2$\"3b)fN[OqQj)FJ7$$\"3;+DJXTb(e$F2$\"3SP#H&pHI'G\"F27$$\"3m
m;HdO2VOF2$\"3I%3Vm>\">4;F27$$\"37]7G8M$3n$F2$\"3%H;nf%HkB<F27$$\"3eL3
FpJf)p$F2$\"3BB<ceN$G!=F27$$\"3LDcEZIZ7PF2$\"3>S$)oP)4&G=F27$$\"31</ED
HNEPF2$\"3.&3UYp@Y%=F27$$\"3!)3_D.GBSPF2$\"3)z_<G965&=F27$$\"33++D\"o7
Tv$F2$\"39NJjZecZ=F27$$\"39L3xcoD.QF2$\"3k()zC^%odv\"F27$$\"3km;HK5S_Q
F2$\"3&oID]?p@a\"F27$$\"39+D\"y?X:!RF2$\"3_w=m-*)p:7F27$$\"3?LLL$Q*o]R
F2$\"3V)='*H4,&>zFJ7$$\"3R+D\"GL3Y+%F2$\"33^+s?&fgS#FJ7$$\"39n;H#GF&eS
F2$!3uU%pZ(\\DxOFJ7$$\"3_]7.dn[&3%F2$!3TH#zX#3Y8oFJ7$$\"3!R$3xJiW7TF2$
!3MlM:sVRY**FJ7$$\"3G</^1dSRTF2$!3q&>:G\\9EI\"F27$$\"3m++D\"=lj;%F2$!3
0d%pr?s.g\"F27$$\"3P+](=x3x@%F2$!3')H'z@P3C7#F27$$\"33++]iB0pUF2$!3/B
\\G@o'3c#F27$$\"3p+]7`fR?VF2$!3c\\qI@$o9*GF27$$\"3S++vV&R<P%F2$!3'4*yg
u9\"o4$F27$$\"3w3-QcC8&Q%F2$!3vKyK&3:%GJF27$$\"37</,p`_)R%F2$!3I,3t5jt
]JF27$$\"3]D1k\"G=>T%F2$!3uO)eozUP;$F27$$\"3'Q$3F%>6`U%F2$!3CHP#ycUu;$
F27$$\"3AU5!p5/(QWF2$!3=W!fG%R)=;$F27$$\"3e]7`>q4_WF2$!3q2L\\t_:ZJF27$
$\"3'*e9;K**[lWF2$!3J!yh?]%QBJF27$$\"3Kn;zWG))yWF2$!3cbXdV&Q24$F27$$\"
3y+DJ&\\aC`%F2$!3eN%o85cg(GF27$$\"3CML$e9Ege%F2$!39![]p=u@a#F27$$\"3hL
3FW;ANYF2$!3/t^zq.W`@F27$$\"3'QL3F9<Wo%F2$!3Q#z9!4A57<F27$$\"39Me9TEhL
ZF2$!3CBNC7#=jC\"F27$$\"3]LLeR\"3Gy%F2$!3/ls))paE`yFJ7$$\"3Lm\"H28se$[
F2$!3yVvx1BYiKFJ7$$\"3/+](=7O*))[F2$\"3M'y8?9S7:'F/7$$\"3vL3-8,+U\\F2$
\"3VavlV2R4NFJ7$$\"3emm;/T1&*\\F2$\"3-&Qfr(e$[A&FJ7$$\"3nTg_ZP%)3]F2$
\"3OttUc`8iaFJ7$$\"3(eT&)3RBE-&F2$\"3`U%R]Eo8h&FJ7$$\"3(4zWU..k.&F2$\"
3!RWj'f:9scFJ7$$\"31mTgxE=]]F2$\"3Cx=6-x]WcFJ7$$\"3C;HKk>ux]F2$\"3<%Hd
.B^kK&FJ7$$\"3Vm;/^7I0^F2$\"3%>zx*)Rfkm%FJ7$$\"3!o;zW#)>/;&F2$\"3)\\ja
&QB!=R#FJ7$$\"3=nm\"zRQb@&F2$!36XCvC(fGu*F/7$$\"3'3+D\"y#>NE&F2$!35+02
0\\EmXFJ7$$\"3nLLLe,]6`F2$!3^Pc56duG&)FJ7$$\"3Ym;aQ5[f`F2$!3ghM-'=M-E
\"F27$$\"3:++v=>Y2aF2$!3UslEx#e=l\"F27$$\"3wmT&Qe#GfaF2$!3\"G3!GT#))p-
#F27$$\"3PL$e*[K56bF2$!3`>aE()z=BBF27$$\"3)**\\iS\"R#Hc&F2$!3@O%QRH:a^
#F27$$\"3Znm;zXu9cF2$!3y@!\\\"oHO&e#F27$$\"3i+](oR!GocF2$!3Cl()3N`M=DF
27$$\"3yLLe9i\"=s&F2$!3'>*fB()yJ3BF27$$\"3%pm\"HK?NvdF2$!3'pVyo0RQ'>F2
7$$\"34+++]y))GeF2$!3E,==+9..:F27$$\"3!3+DJ?i7)eF2$!3G@(GIJi-l*FJ7$$\"
3k++DcljLfF2$!3'eso;Bxyr$FJ7$$\"3[+]P44,')fF2$\"3*\\S!zX[t>CFJ7$$\"3H+
+]i_QQgF2$\"3G'=Psbz'*R)FJ7$$\"3!3](=U,1*3'F2$\"31&oxP2m,P\"F27$$\"3U+
](=-N(RhF2$\"3>a!z%f)pJ#=F27$$\"3-+Dc,*4/>'F2$\"37'H2![$4W<#F27$$\"3b+
+D\"y%3TiF2$\"3S\"4VQ'\\`0CF27$$\"3/]7G)o<#piF2$\"3U03&=-NwZ#F27$$\"3S
+DJ&f]tH'F2$\"3C!z^Ccq$3DF27$$\"3/E\"G)[qT6jF2$\"3?_%=zK'=3DF27$$\"3z]
PM-N[DjF2$\"3?y5e;7r(\\#F27$$\"3`v$fe&*\\&RjF2$\"3S#z'=\"HkqZ#F27$$\"3
G+]P4kh`jF2$\"3_Mq&Q\"HTYCF27$$\"39+vVBA))4kF2$\"39o3QIruGAF27$$\"3+++
]P![hY'F2$\"3Y&ok3$))=w=F27$$\"3rK$eRP0n^'F2$\"3)3+;koO'p9F27$$\"3KmmT
5FEnlF2$\"3mZ-RN`#o+\"F27$$\"3#***\\(o/?yh'F2$\"3/nRxrg*z=&FJ7$$\"3iKL
L$Qx$omF2$\"3s;R)*\\'[')y$F/7$$\"3:***\\78eBs'F2$!3>A#[&RW_@VFJ7$$\"3c
mm;z)Qjx'F2$!3V_P%\\`hCB)FJ7$$\"3'RL$3F'>.$oF2$!3C34%>Pul5\"F27$$\"3Y+
++v.I%)oF2$!39+iAmwVh7F27$$\"3Z$ek`md(3pF2$!3o5(*y8f%fG\"F27$$\"3Ym\"H
d&\\@LpF2$!3s@Vf`,K\"G\"F27$$\"3M]P4YAndpF2$!3/1hv&flwC\"F27$$\"3ML$ek
`H@)pF2$!3%pOiz#)Hb=\"F27$$\"3A+v=<T/JqF2$!3A&y]jrl:!)*FJ7$$\"3?mm\"zp
e*zqF2$!39:IQ[ZFznFJ7$$\"3&**\\(oa_VLrF2$!37>wqu(Gue#FJ7$$\"3oL$e9\"=
\"p=(F2$\"33)\\vc%HdKAFJ7$$\"3Vn\"H#o$)QSsF2$\"3I!RcTk7\"HtFJ7$$\"3;,+
+D\\'QH(F2$\"3qMK=4D3L7F27$$\"3*Q$eR(>#=WtF2$\"3Ca>,N\\Ri;F27$$\"3gm;z
p%*\\%R(F2$\"3(>:=M-3L-#F27$$\"3A+v=Un\"[W(F2$\"3!R/+9&R[#H#F27$$\"3%H
L$e9S8&\\(F2$\"3aapAXyX_CF27$$\"3C+DcEIG3vF2$\"3]VG!>'=zuCF27$$\"3km;a
Q?V@vF2$\"3?-)y)Q,\"))[#F27$$\"30L3_]5eMvF2$\"3-^&4(4qW%\\#F27$$\"3W**
**\\i+tZvF2$\"3CYdL$ot;\\#F27$$\"39L$ek3GSd(F2$\"3mTXbKj'4Y#F27$$\"3%o
m;/6E.g(F2$\"3!\\p=Td'>(R#F27$$\"3AMLLe@#Hl(F2$\"3eW_/&p'Rv@F27$$\"3s+
+D1#=bq(F2$\"3zw3INneT=F27$$\"3yL3xc/%pv(F2$\"3M@KJdU<H9F27$$\"3#om\"H
2FO3yF2$\"3u/\\)=HdRf*FJ7$$\"3')*\\7y&\\yfyF2$\"3ss)4+W'RPYFJ7$$\"3\"H
LL$3s?6zF2$!3cd;*y3lHW#F/7$$\"3!*)\\i!R:/lzF2$!3gv`\"oC8b\"\\FJ7$$\"3y
l;zpe()=!)F2$!3'=\\v]D.Qz)FJ7$$\"3mK3_+-rs!)F2$!3IFa4Rrce6F27$$\"3a***
\\7`Wl7)F2$!3)[FxD)GX28F27$$\"3[$3x1'*oC:)F2$!36yz)p2J'G8F27$$\"3mlT5!
R$Ry\")F2$!3d\"*>p*p@gJ\"F27$$\"3h\\7`>yJ/#)F2$!3?=Wfh8kp7F27$$\"3cL$e
*[ACI#)F2$!3w='3@ar+>\"F27$$\"3Y,D\"y5\"4#G)F2$!3)4QG[t?XO*FJ7$$\"3enm
mm*RRL)F2$!3-A@%4*)4bq&FJ7$$\"3on;a8H'pQ)F2$!3V&4qs6R</\"FJ7$$\"3wnmTg
e)*R%)F2$\"31g&yXgfdC%FJ7$$\"3%ym\"H2)3I\\)F2$\"3'>p7!)e3<!)*FJ7$$\"3%
zmmTvJga)F2$\"3HrR^;7/D:F27$$\"3eM3FWch)f)F2$\"3FFY&\\gN$=?F27$$\"3A,]
PM&*>^')F2$\"3U@*o;,S:V#F27$$\"3'y;zWU$y.()F2$\"3FaN*H0jnt#F27$$\"3]ML
e9tOc()F2$\"3]'>l$\\sA8HF27$$\"3Qxo/)4Z%o()F2$\"3!='=RXa5MHF27$$\"3]=/
^\")o_!y)F2$\"3eP&erf@u%HF27$$\"3gfR(\\m1Ez)F2$\"3E\")[[X<7`HF27$$\"3q
+vV[ko/))F2$\"3/!p]wky6&HF27$$\"3p%ek`,Y)G))F2$\"3[v&3Kt(QCHF27$$\"3oo
;H#e0I&))F2$\"35[qy9gOnGF27$$\"3([$e9;ZK,*)F2$\"3'p#Q!f\\,gm#F27$$\"31
,++]Qk\\*)F2$\"3m@Xc7_$yN#F27$$\"3%\\L3-.B]+*F2$\"3#4IAo]7]*=F27$$\"3#
)omT5ASg!*F2$\"3eQY$f(3$oM\"F27$$\"3q-]i!R\"y:\"*F2$\"3\\1S>pIU@vFJ7$$
\"3![LL3dg6<*F2$\"3f(z')[P=!G:FJ7$$\"3<n;zpBp?#*F2$!3hzqypRuINFJ7$$\"3
K,+voTAq#*F2$!3#ybMS-ji,)FJ7$$\"3[N$3x'fv>$*F2$!3!Rk3Q\\7!p6F27$$\"3%y
mmmw(Gp$*F2$!3Xb?j48lO9F27$$\"3Q%e*)4Q$p&R*F2$!3je,`Ak2M:F27$$\"3%4]7`
**)4A%*F2$!3]b+>$*f_)f\"F27$$\"3LeRZ-=IN%*F2$!3:<_VGKA=;F27$$\"3]<aj4Y
][%*F2$!32Z,7FwbH;F27$$\"3mwoz;uqh%*F2$!3to#o\\&ycK;F27$$\"3/M$eRA5\\Z
*F2$!3ks.9x*Hti\"F27$$\"39nTg_9sF&*F2$!3W'RlVT-n_\"F27$$\"3C++D\"oK0e*
F2$!3oM(y$>N_48F27$$\"3d**\\il(z5j*F2$!3F<m#\\5!)>,\"F27$$\"3m+++]oi\"
o*F2$!3=O_>7Il9lFJ7$$\"3y,]PMR<K(*F2$!31RMC$Rt_c#FJ7$$\"35,+v=5s#y*F2$
\"3!f#o-#[#H?9FJ7$$\"3Q+D1k2/P)*F2$\"3anF7c+5*Q&FJ7$$\"3W,]P40O\"*)*F2
$\"3U+nw\"yr$z')FJ7$$\"3s+voa-oX**F2$\"3?_NnuB)z4\"F27$$\"#5F)$\"3(\\Y
jvy@`?\"F2-%'COLOURG6&%$RGBG$F^io!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q!F[j
o-%%VIEWG6$;F(F]io%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "fourier(ft,t,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
$*&#\"\"\"\"\"#F&*&%#PiGF&,.*&\"\"%F&-%&DiracG6#,&%\"wGF&\"\"&F&F&F&*&
F,F&-F.6#,&F1F&F2!\"\"F&F&*&-F.6#,&F1F&*$F2F%F&F&^#F&F&F&*&^#F7F&-F.6#
,&F1F&F<F7F&F&*&^#F'F&-F.6#,&F1F&F&F&F&F&*&^#!\"#F&-F.6#,&F1F&F&F7F&F&
F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Now, we can't graph th
e result, but in real life we never want to integrate infinite wave tr
ains. So what happens for a finite signal?" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "fw:=int(ft*exp(I*w*t),t=-20..20):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "plot([Re(fw),Im(fw)],w=-6..6,color=[red,b
lue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 904 240 240 {PLOTDATA 2 "6&-%'CUR
VESG6$7icl7$$!\"'\"\"!$\")>'>%=!\"(7$$!+Q&3Y$f!\"*$!+8:Gr]!#57$$!+vq@p
eF1$!+/$)=%R#F17$$!+8c#Q!eF1$!+\\(>0y(F47$$!+]TVQdF1$\"++-LbBF17$$!+E1
kCcF1$!+J-%*fSF47$$!+-r%3^&F1$!+DHKOFF17$$!+BR&)yaF1$!+jA,I[F47$$!+V2'
oW&F1$\"+u*Q\")G#F17$$!+kv'[T&F1$\"+tN96XF17$$!+$QuGQ&F1$\"+JQL#=&F17$
$!+/7)3N&F1$\"+BYxdPF17$$!+C!)))=`F1$\"+!>1oK%F47$$!+W[*oG&F1$!+R5**4Q
F17$$!+l;!\\D&F1$!+w--AuF17$$!+Q!*zQ_F1$!+k%QTY)F17$$!+8kpA_F1$!+;e,J(
)F17$$!+(y$f1_F1$!+00*33)F17$$!+i6\\!>&F1$!+\"p$)*HkF17$$!+P&)Qu^F1$!+
f.RiPF17$$!+6fGe^F1$!+.DM_8F47$$!+*fM-:&F1$\"+G2N+?F17$$!+'G$=U^F1$\"+
,%[=K%F17$$!+t>8M^F1$\"+[To-oF17$$!+g13E^F1$\"+!*4*GT*F17$$!+Z$H!=^F1$
\"+;@&>@\"!\")7$$!+M!y*4^F1$\"+n`r)[\"Fgs7$$!+@n#>5&F1$\"+t)Qyw\"Fgs7$
$!+4a(Q4&F1$\"+xA[X?Fgs7$$!+'4Ce3&F1$\"+Rkt<BFgs7$$!+$ysx2&F1$\"+!G&p!
e#Fgs7$$!+r9sp]F1$\"+Qm`IGFgs7$$!+e,nh]F1$\"+okgjIFgs7$$!+LvcX]F1$\"+A
o%eY$Fgs7$$!+2\\YH]F1$\"+EVRjPFgs7$$!+#GiL,&F1$\"+q(*HQRFgs7$$!+b'fs*
\\F1$\"+&*R**zRFgs7$$!+jNB\")\\F1$\"+e$))o)QFgs7$$!+qu?l\\F1$\"+vp?lOF
gs7$$!+x8=\\\\F1$\"+7jYGLFgs7$$!+&GbJ$\\F1$\"+4k\"p*GFgs7$$!+QA9D\\F1$
\"+juS`EFgs7$$!+#>Hr\"\\F1$\"+[@3'R#Fgs7$$!+Yh64\\F1$\"+Z,mG@Fgs7$$!+*
4.6!\\F1$\"+()G%\\&=Fgs7$$!+a+4$*[F1$\"+77xy:Fgs7$$!+2q2&)[F1$\"+)4ARI
\"Fgs7$$!+hR1x[F1$\"+_33M5Fgs7$$!+940p[F1$\"+oBaFxF17$$!+ny.h[F1$\"+w/
IK_F17$$!+@[-`[F1$\"+e85&)GF17$$!+u<,X[F1$\"+Dk:FrF47$$!+H()*p$[F1$!+p
-uh7F17$$!+OE(4#[F1$!+$)pmWXF17$$!+Vl%\\![F1$!+27SioF17$$!+^/#*)y%F1$!
+&pZZ=)F17$$!+eV*Gx%F1$!+r\"o0b)F17$$!+l#oov%F1$!+$*QPh!)F17$$!+s@%3u%
F1$!+mB**poF17$$!+$pD6r%F1$!+LqdiMF17$$!+9#49o%F1$\"+wFkqOF47$$!+NFp^Y
F1$\"+6[o>MF17$$!+di(>i%F1$\"+K?'\\\"\\F17$$!+*HVDc%F1$\"+ih/QHF17$$!+
U.6.XF1$!+%Rg]w\"F17$$!+s:.!Q%F1$!+aP$e-(F47$$!+-G&pD%F1$\"+&*H$H1#F17
$$!+#o3L>%F1$!+oB-u!)F47$$!+iXmHTF1$!+D9u(>#F17$$!+U/-mSF1$!+Kq0R`F47$
$!+AjP-SF1$\"+p'*>0;F17$$!+>j^')RF1$\"+2Yl9=F17$$!+:jlqRF1$\"+fWET=F17
$$!+7jzaRF1$\"+?2N)o\"F17$$!+4j$*QRF1$\"+HwRw8F17$$!+-j@2RF1$\"+!zVqE%
F47$$!+(H'\\vQF1$!+>m\"oD'F47$$!+$HO'fQF1$!+r(*[i5F17$$!+!HwP%QF1$!++W
R$Q\"F17$$!+(G;z#QF1$!+\\dsg:F17$$!+%Gc?\"QF1$!+q1Q\"e\"F17$$!+yiL!y$F
1$!+G6Bx6F17$$!+sih[PF1$!+'\\SZa$F47$$!+Z$))fr$F1$\"+ZoC:eF47$$!+A/O$o
$F1$\"+(Q_$Q7F17$$!+fk/nOF1$\"+w-nx8F17$$!+(\\K2l$F1$\"+:wwr8F17$$!+M&
=Wj$F1$\"+gO)\\A\"F17$$!+rX5=OF1$\"+j\"=&f&*F47$$!+@([Gb$F1$!+!*3TfiF4
7$$!+qGf([$F1$!+9Nb\">\"F17$$!+\"[:,V$F1$!+?Eu#G#F47$$!+$4QEP$F1$\"+v
\"*>\"=*F47$$!+)R**QM$F1$\"+]>946F17$$!+/2;:LF1$\"+>eQ#[*F47$$!+5?U'G$
F1$\"+m'HG)\\F47$$!+:LodKF1$!+sj@e!*!#67$$!+`.LDKF1$!+:fFgoF47$$!+!RxH
>$F1$!+cyt.**F47$$!+44!o<$F1$!+]M/O**F47$$!+FWigJF1$!++e=i*)F47$$!+YzW
WJF1$!+Q9(**4(F47$$!+j9FGJF1$!+5$*[aXF47$$!++&=f4$F1$\"+1I%*p9F47$$!+P
bcjIF1$\"+j,h+nF47$$!+a!*QZIF1$\"+A#[*\\$)F47$$!+tD@JIF1$\"+p%[]7*F47$
$!+\"4O],$F1$\"+,;*H'*)F47$$!+5'f))*HF1$\"+Y#Hr*yF47$$!+&*z)Q$HF1$!+GY
Nj>F47$$!+!Q;*oGF1$!+k=XI%)F47$$!+lZ%R!GF1$!+r&pWr#F47$$!+]J(*QFF1$\"+
iK$RV'F47$$!+bqm2FF1$\"+Xg;(z(F47$$!+g4OwEF1$\"+\"[BqD'F47$$!+l[0XEF1$
\"+3^PkCF47$$!+q([Ph#F1$!+_xvC@F47$$!+Bd4)f#F1$!+r(>k<%F47$$!+vEW#e#F1
$!+tTb.eF47$$!+G'*ymDF1$!+?UDeoF47$$!+!eO6b#F1$!+E!e([sF47$$!+KN[NDF1$
!++=xZpF47$$!+&[I)>DF1$!+(eVF*fF47$$!+Pu</DF1$!+K'oO[%F47$$!+!RC&)[#F1
$!+5yQqDF47$$!+da4gCF1$\"+x56=8F47$$!+BlmJCF1$\"+1mI6ZF47$$!+!fPKS#F1$
\"+ohw(e'F47$$!+c'3[P#F1$\"+Y7r6kF47$$!+*y]zJ#F1$\"+*eYm,*F[jl7$$!+AH4
hAF1$!+JRMl`F47$$!+r\")GFAF1$!+&R'e>jF47$$!+>M[$>#F1$!+_8njXF47$$!+o'y
'f@F1$!+WL@5\"*F[jl7$$!+;R(e7#F1$\"+)y&>PIF47$$!+S:(*3@F1$\"+qX$zd%F47
$$!+k\"p?4#F1$\"+;^)ff&F47$$!+)ym^2#F1$\"+/\"ei)fF47$$!+8WEe?F1$\"+V28
8dF47$$!+h'fW-#F1$\"+Agr)R$F47$$!+5\\l!*>F1$!+U<jeIF[jl7$$!+^y,i>F1$!+
;V&)RLF47$$!+#z!QL>F1$!+!>HIG&F47$$!+MPu/>F1$!+,*)[dbF47$$!+vm5w=F1$!+
\"o/16%F47$$!+eD$)==F1$\"+\\N#fl\"F47$$!+S%e:w\"F1$\"+./uX`F47$$!+E]$
\\p\"F1$\"+tr[,?F47$$!+6;JG;F1$!+=C*R9%F47$$!+/***\\f\"F1$!+9fzJ^F47$$
!+'>)oh:F1$!+sM-#)RF47$$!+*[w$G:F1$!+0_R47F47$$!+#yk]\\\"F1$\"+T\\(>,#
F47$$!+I:K![\"F1$\"+OI)\\A$F47$$!+x#ybY\"F1$\"+DXieTF47$$!+C]$3X\"F1$
\"+9B6PZF47$$!+s<4O9F1$\"+CE#[\"\\F47$$!+>&[8U\"F1$\"+)G$G!o%F47$$!+m_
g19F1$\"+5='e0%F47$$!+9?'=R\"F1$\"+DCC(4$F47$$!+h(=rP\"F1$\"+/tj')=F47
$$!+cAjZ8F1$!+Cw'=m)F[jl7$$!+]d9=8F1$!+)[alH$F47$$!+X#f')G\"F1$!+B^N<Y
F47$$!+SF<f7F1$!+7E%QT%F47$$!+++Z%>\"F1$\"+5/KKFF[jl7$$!+hswH6F1$\"+EF
EYWF47$$!+#*eT(4\"F1$\"+B76%=%F47$$!+AX1l5F1$\"+)zuMH#F47$$!+_JrK5F1$!
+sx%)pYF[jl7$$!+#yh.+\"F1$!+M!o.2$F47$$!+];Pr()F4$\"+o$R$[6F[jl7$$!*[D
\"RvF1$\"+PkDBFF47$$!+wbp<sF4$\"+'G&RMCF[jl7$$!+scE'*oF4$!+&fq!3BF47$$
!+od$[d'F4$!+h--\\RF47$$!+leS`iF4$!+f%ee0%F47$$!+dga5cF4$!+#p:G`\"F[jl
7$$!*D'on\\F1$\"+R9s2RF47$$!+fK^sYF4$\"+NQ\"=-%F47$$!+n-MxVF4$\"+)>(\\
@GF47$$!+ws;#3%F4$\"+PPBbqF[jl7$$!+&G%*py$F4$!+C,'4j\"F47$$!+$H@=\\$F4
$!+y2MCMF47$$!+-$[m>$F4$!+v9f\"4%F47$$!+6`Z,HF4$!+#*oI>MF47$$!*K-jg#F1
$!+JQxJ;F47$$!+u$=zG#F4$\"+t,&*f')F[jl7$$!+GW`p>F4$\"+!*3yGIF47$$!+#[]
6l\"F4$\"+6bgRSF47$$!+NlwK8F4$\"+(oO(=NF47$$!+Dk)*fpF[jl$!+)4b5:)F[jl7
$$!(vI#fF1$!+mN[BSF47$$\"+D8#>-'F[jl$!+=R4H:F47$$\"+:]hj7F4$\"+M4v6KF4
7$$\"+ckK%f\"F4$\"+yJ!41%F47$$\"+)*y.D>F4$\"+['*o`KF47$$\"+R$\\dD#F4$
\"+w@c:6F47$$\"*ygke#F1$!+fvj%[\"F47$$\"+ggMuGF4$!+aIy%H$F47$$\"+S8BiJ
F4$!+\"f&z\"3%F47$$\"+?m6]MF4$!+LhZ'f$F47$$\"++>+QPF4$!+-^R&)>F47$$\"+
!=()e-%F4$\"+]#>0`#F[jl7$$\"+gCx8VF4$\"+q)>EU#F47$$\"+Sxl,YF4$\"+#*y5X
QF47$$\"*-V&*)[F1$\"+f8JsSF47$$\"+-5R6bF4$\"+!RO\"zlF[jl7$$\"+&)*QK8'F
4$!+#QRan$F47$$\"+wH;WkF4$!+([7Q>%F47$$\"+np3bnF4$!+8Tm*=$F47$$\"+f4,m
qF4$!+L--;5F47$$\"*&\\$pP(F1$\"+Au=Y:F47$$\"+.CaPvF4$\"+hq%Hr#F47$$\"+
d)\\\")p(F4$\"+))et:OF47$$\"+5tveyF4$\"+4#pR;%F47$$\"+jZO>!)F4$\"+HmA,
VF47$$\"+<A(*z\")F4$\"+6347SF47$$\"+q'z0M)F4$\"+Rx4BLF47$$\"+Br=,&)F4$
\"+tpI+BF47$$\"+vXzh')F4$\"+\"e^O/\"F47$$\"+#[4I)*)F4$!+$30cm\"F47$$\"
+)QCUI*F4$!+MUGdPF47$$\"+%HRai*F4$!+&z-9T%F47$$\"*?am%**F1$!+D.zbLF47$
$\"+WY^d5F1$\"+$pZ&*o\"F47$$\"+oQO?6F1$\"+'4!*)eXF47$$\"+#48K=\"F1$\"+
z/$>G\"F47$$\"+:B1Y7F1$!+7W@QQF47$$\"+UsYw7F1$!+a`rCZF47$$\"+q@(oI\"F1
$!+%pPa'RF47$$\"+)4xsL\"F1$!++@<-=F47$$\"+E?on8F1$\"+?XmE5F47$$\"+ap3)
R\"F1$\"+vc1QNF47$$\"+#)=\\G9F1$\"+s8tW[F47$$\"+5o*)e9F1$\"+JPLoWF47$$
\"+Q<I*[\"F1$\"+/PZ8DF47$$\"+E71B:F1$!+waD)*oF[jl7$$\"+:2#ob\"F1$!+h()
fbOF47$$\"+/-e!f\"F1$!+))*y)*4&F47$$\"+#pRVi\"F1$!++-')pVF47$$\"+o'e=p
\"F1$\"+8%)Q*p\"F47$$\"+XwPf<F1$\"+A]&zO&F47$$\"+[k/?=F1$\"+5$[/`\"F47
$$\"+__r!)=F1$!+=&*\\XWF47$$\"+a'\\5\">F1$!+G3y^cF47$$\"+cSQT>F1$!+I@$
>*[F47$$\"+e%=<(>F1$!+[,a#R#F47$$\"+gG0-?F1$\"+W!)R.5F47$$\"+48WM?F1$
\"+%e*))*G%F47$$\"+e(Ho1#F1$\"+;CzLfF47$$\"+$)R-$3#F1$\"+>T>()eF47$$\"
+2#=#*4#F1$\"+y#fsB&F47$$\"+KCT:@F1$\"+_x]WSF47$$\"+bmgJ@F1$\"+V/&QU#F
47$$\"+/^*R;#F1$!+Q\"o8V\"F47$$\"+`NQ'>#F1$!+J_#p![F47$$\"+,?xGAF1$!+-
`8MjF47$$\"+]/;hAF1$!+$>$Gf`F47$$\"+Zz&)>BF1$\"+Z6[Z6F47$$\"+WabyBF1$
\"+@yDblF47$$\"+=)HKR#F1$\"+g@BonF47$$\"+#>/zS#F1$\"+ao#QT'F47$$\"+m&y
DU#F1$\"+enx8bF47$$\"+THDPCF1$\"+6&4x8%F47$$\"+*o,mY#F1$\"+!f#3*G%F[jl
7$$\"+Q/&f\\#F1$!+&)yW<NF47$$\"+uTHCEF1$!+$=Og(fF[jl7$$\"+5zj_FF1$\"+)
=\"H**\\F47$$\"+P'=I\"GF1$!+*H[y1%F47$$\"+k$*RtGF1$!+!\\#fu$)F47$$\"+
\"4!yLHF1$!+bgV\")>F47$$\"+=3;%*HF1$\"+CilLuF47$$\"+Z%=d-$F1$\"+clKv\"
*F47$$\"+vgFdIF1$\"+p(\\yV(F47$$\"+/P$))3$F1$\"+wDqlFF47$$\"+L8R?JF1$!
+X.CYJF47$$\"+Y,<OJF1$!+K%Hq'eF47$$\"+h*[>:$F1$!+JQki!)F47$$\"+vxsnJF1
$!+nrR3&*F47$$\"+!f1N=$F1$!+3R![+\"F17$$\"+0aG*>$F1$!+V$G:h*F47$$\"+>U
1:KF1$!+]<*GA)F47$$\"+MI%3B$F1$!+lMp**fF47$$\"+[=iYKF1$!+&ovO9$F47$$\"
+)>vuF$F1$\"+S``7KF47$$\"+[&G$3LF1$\"+UK$)H')F47$$\"+B_vBLF1$\"+:h_J5F
17$$\"+)*==RLF1$\"+8>^16F17$$\"+t&3YN$F1$\"+ML**y5F17$$\"+\\_.qLF1$\"+
f]b%\\*F47$$\"+\\>uJMF1$!+2%fsl#F47$$\"+]'[M\\$F1$!+%os:A\"F17$$\"+Z)
\\!eNF1$!+$Gf!*3&F47$$\"+W5lAOF1$\"+9$R;/\"F17$$\"+V8!)QOF1$\"+!zRuF\"
F17$$\"+U;&\\l$F1$\"+T3C(Q\"F17$$\"+T>5rOF1$\"+yjWc8F17$$\"+TAD(o$F1$
\"+WX`%=\"F17$$\"+RGb>PF1$\"+zBln[F47$$\"+QM&=v$F1$!+:&Rn[%F47$$\"+`F'
Hy$F1$!+#*pPI7F17$$\"+p?29QF1$!+8i^(e\"F17$$\"+EniHQF1$!+%*R.\\:F17$$
\"+%Q\"=XQF1$!+E$f/O\"F17$$\"+UgtgQF1$!+gn`N5F17$$\"+*p!HwQF1$!++gJ9gF
47$$\"+:+S2RF1$\"+(pR!HVF47$$\"+I$4&QRF1$\"+(R'4m8F17$$\"+()R1aRF1$\"+
o#Gsn\"F17$$\"+X'='pRF1$\"+d*3m$=F17$$\"+-L<&)RF1$\"+(QqS#=F17$$\"*'zs
+SFgs$\"+?v>N;F17$$\"+&\\bV1%F1$!+1JquYF47$$\"+II)z7%F1$!++74*=#F17$$
\"+l0h\">%F1$!+\")QM1))F47$$\"*5Q_D%Fgs$\"+q9N.?F17$$\"+oK,rUF1$\"+%4>
\\Y#F17$$\"+N%)y'G%F1$\"+BbO,FF17$$\"+?gn%H%F1$\"+/]pBFF17$$\"+.Oc-VF1
$\"+0:#)yEF17$$\"+(=^/J%F1$\"+p)oic#F17$$\"+q(Q$=VF1$\"+'Q%>(Q#F17$$\"
+0\"*))\\VF1$\"+$Gfa3\"F17$$\"+S%R9Q%F1$!+?>2&*yF47$$\"+v(*)HT%F1$!+;Y
*\\b#F17$$\"+5,aWWF1$!+U,'e\\$F17$$\"+X/4wWF1$!+3iFPJF17$$\"*ySw]%Fgs$
!+Xtqb9F17$$\"+SPilXF1$\"+`t'>;$F17$$\"++ngBYF1$\"+\\*z%z[F17$$\"+!=)f
_YF1$\"+oI1ZLF17$$\"+g'*e\"o%F1$\"+d53^MF47$$\"+S6e5ZF1$!+HL8#R$F17$$
\"*is&RZFgs$!+Q=0^nF17$$\"+uj=cZF1$!+3lvB!)F17$$\"+G,!Gx%F1$!+-03]&)F1
7$$\"+#)QT*y%F1$!+\")>ne\")F17$$\"+Nw-1[F1$!+n2TPnF17$$\"+!RTE#[F1$!+&
RwmC%F17$$\"+V^DR[F1$!+RqmpsF47$$\"+??cZ[F1$\"+![aOQ\"F17$$\"+(*)oe&[F
1$\"+co\\*p$F17$$\"+td<k[F1$\"+uAR$>'F17$$\"+]E[s[F1$\"+Bu8M))F17$$\"+
F&*y!)[F1$\"+,')oe6Fgs7$$\"+.k4*)[F1$\"+i@RT9Fgs7$$\"+!G.u*[F1$\"+P=]F
<Fgs7$$\"+d,r0\\F1$\"+YG'G,#Fgs7$$\"+Lq,9\\F1$\"+-\"4KH#Fgs7$$\"+5RKA
\\F1$\"+&o'GkDFgs7$$\"+(yI1$\\F1$\"+;&4>#GFgs7$$\"+lw$*Q\\F1$\"+$fU?1$
Fgs7$$\"+=9bb\\F1$\"+)eL\\Z$Fgs7$$\"+s^;s\\F1$\"+M!Rlx$Fgs7$$\"+E*y())
\\F1$\"+YwKZRFgs7$$\"*o#R0]Fgs$\"+Hf\"f(RFgs7$$\"+?AD?]F1$\"+A%>$zQFgs
7$$\"+g<6N]F1$\"+$GNAn$Fgs7$$\"++8(*\\]F1$\"+y?LlLFgs7$$\"+S3$[1&F1$\"
+CMMuHFgs7$$\"+51Es]F1$\"+T!)Q`FFgs7$$\"+!Q!pz]F1$\"+H>6>DFgs7$$\"+],7
(3&F1$\"+h\\VuAFgs7$$\"+?*\\X4&F1$\"+MqPA?Fgs7$$\"+!pz>5&F1$\"+2H+m<Fg
s7$$\"+g%4%4^F1$\"+9CR3:Fgs7$$\"+I#Ro6&F1$\"+\\vc_7Fgs7$$\"++!pU7&F1$
\"+1hZ,5Fgs7$$\"+S&G\"R^F1$\"+Dy(eC&F17$$\"+!3))R:&F1$\"+(e$o,)*F47$$
\"+?w%)o^F1$!+6L[<EF17$$\"+grq$=&F1$!+:yWGaF17$$\"++nc)>&F1$!+j))p&Q(F
17$$\"+SiU8_F1$!+@.ev%)F17$$\"+5g&3A&F1$!+/TF0()F17$$\"+!y&GG_F1$!+k$y
ht)F17$$\"+]brN_F1$!+;6K!e)F17$$\"*KXJC&Fgs$!+#))[@D)F17$$\"+e?$[F&F1$
!+(QG)Q`F17$$\"+&z=lI&F1$!+pz0v6F17$$\"+Lb?Q`F1$\"+%))H%HEF17$$\"+qA*)
p`F1$\"+S3Mk[F17$$\"+bR\"yP&F1$\"+b%\\65&F17$$\"+Rct&Q&F1$\"+B[E/_F17$
$\"+Ctl$R&F1$\"+`few^F17$$\"+2!z:S&F1$\"+F\"pQ-&F17$$\"+wBU<aF1$\"+(p7
$zVF17$$\"+XdELaF1$\"+Up/kLF17$$\"+#[_\\Y&F1$\"+s)ol;(F47$$\"*ARm\\&Fg
s$!+*3%oe=F17$$\"+?A&zh&F1$!+*o%zg%)F47$$\"*Al#RdFgs$\"+fpFFBF17$$\"+:
*[W!eF1$!+_o!*z!)F47$$\"+5EjpeF1$!+:nC#R#F17$$\"+0j\"[$fF1$!+R&p+)\\F4
7$$\"\"'F*$\"+_M'>%=F1-%'COLOURG6&%$RGBG$\"*++++\"Fgs$F*F*F^hr-F$6$7]d
l7$F($\"+ti+lJF47$F@$!(39J\"F-7$$!+pd)*4dF1$\"('y`8F-7$$!+)QP:o&F1$\"'
-BOF)7$$!+2!*3`cF1$\"'#\\x%F)7$FE$\"'QKWF)7$$!+kQunbF1$\"&0\"pF-7$FJ$!
'S$[%F)7$FO$!'3A\\F)7$FT$!'-8MF)7$FY$!'lt_F-7$Fhn$\"'?4EF)7$$!+%zxoO&F
1$\"'EuQF)7$F]o$\"'[bZF)7$$!+9Y)[L&F1$\"'%3;&F)7$Fbo$\"'gX]F)7$Fgo$\"'
\"pL$F)7$F\\p$\"';yFF-7$Ffp$!'TcHF)7$F`q$!'E^]F)7$Feq$!']u`F)7$Fjq$!'>
Z^F)7$Fdr$!'g*Q%F)7$F^s$!')\\<$F)7$Fct$\"'?]5F-7$Fgu$\"'G&R$F)7$F\\v$
\"')ph%F)7$Fav$\"'zs`F)7$Ffv$\"'w%e&F)7$F[w$\"'ND_F)7$F_x$!'=7[F-7$Fgz
$!'iycF)7$Fa[l$!'z!z&F)7$F[\\l$!'s9`F)7$F`\\l$!'2%H%F)7$Fe\\l$!(At#GF-
7$F_]l$\"'>)H)F-7$Fi]l$\"'S?UF)7$F]_l$!&L7&F-7$Fg_l$!'cVWF)7$$!+d4dTWF
1$\"';IHF)7$F\\`l$\"'2GmF)7$$!+(=#\\=VF1$\")ep9:Fgs7$Fa`l$!'G_eF)7$$!+
U28DUF1$!'J/qF)7$Ff`l$!'uGaF)7$$!+Am[hTF1$!)ze&o\"Fgs7$F[al$\"'d-GF)7$
$!+KNv8TF1$\"'SuZF)7$$!+-D%y4%F1$\"'1!H'F)7$$!+s9$>3%F1$\"'w#>(F)7$F`a
l$\"'@&Q(F)7$$!+#Q)>MSF1$\"'c.cF)7$Feal$\"(V\"y:F-7$F_bl$!'(*[JF)7$Fib
l$!'qgnF)7$$!+0j2BRF1$!'cmwF)7$F^cl$!'!4#yF)7$$!+*Hc8*QF1$!'E*>(F)7$Fc
cl$!'h`eF)7$Fgdl$\"'hpMF)7$Fael$\"'TG$)F)7$F[fl$\"(Q'H7F-7$F_gl$!'*H9)
F)7$$!+41z,OF1$!'%=$*)F)7$$!+YmZ&e$F1$!'=%z)F)7$$!+$oi\"pNF1$!'MIxF)7$
Fdgl$!'qPeF)7$$!+&z?-_$F1$!&@y$F)7$Figl$\"'J;aF)7$$!+BNAtMF1$\"'HWuF)7
$$!+vT&)eMF1$\"'@&*))F)7$$!+G[[WMF1$\"'wU'*F)7$F^hl$\"'a:'*F)7$$!+'yw8
S$F1$\"'lisF)7$Fchl$\"(st\\#F-7$Fhhl$!'DEKF)7$F]il$!'_,\")F)7$$!+d8z+L
F1$!'i*p*F)7$Fbil$!(MN0\"F)7$$!+jE0sKF1$!(/G0\"F)7$Fgil$!'lj'*F)7$Fbjl
$\"'*G$>F)7$Ff[m$\"(xs;\"F)7$F`\\m$\")NCe[Fgs7$Fd]m$!'yh**F)7$$!+2nh#)
HF1$!(BXA\"F)7$$!+.QPmHF1$!(GCL\"F)7$$!++48]HF1$!(3bI\"F)7$Fi]m$!(@F9
\"F)7$$!+)=-9!HF1$!(ult%F-7$F^^m$\"')GK%F)7$$!+s0VOGF1$\"(@EA\"F)7$Fc^
m$\"(/*o:F)7$$!+d*e9x#F1$\"(+()H\"F)7$Fh^m$\"'%ot%F)7$F]_m$!'S*y&F)7$F
b_m$!'E2:!\"&7$$!+7zqgEF1$!'O3=Fh^t7$Fg_m$!'O_>Fh^t7$$!+S$Gsj#F1$!'Fe>
Fh^t7$$!+<=SHEF1$!(6\"=>F)7$$!+$Hv:i#F1$!([:$=F)7$F\\`m$!(G%*p\"F)7$F`
am$\"&3)eFh^t7$Fdbm$\"'`RFFh^t7$$!+B*4VZ#F1$\"'*4$GFh^t7$Fibm$\"'G6FFh
^t7$$!+!*4)eW#F1$\"(&\\pBF)7$F^cm$\"(yr!=F)7$Fccm$\"%<*)Fh^t7$Fhcm$!'7
'>#Fh^t7$$!+A(zjM#F1$!'=sYFh^t7$F]dm$!'><pFh^t7$$!+b=_*G#F1$!'$zb)Fh^t
7$Fbdm$!'pY$*Fh^t7$Fgdm$!&C3*!\"%7$F\\em$!&&*p(Fbct7$Faem$!'bGcFh^t7$F
fem$!(+pQ$F)7$Fjfm$\"%a5Fh^t7$Fdgm$\"'8i6Fh^t7$Fhhm$\"&He$Fh^t7$Fbim$!
(Ny+\"F)7$$!+'e-\\u\"F1$!(+60\"F)7$$!+LnCG<F1$!(,I,\"F)7$$!+!)3f6<F1$!
'qo))F)7$Fgim$!'DMnF)7$$!+=Lih;F1$!&:6$F)7$F\\jm$\"'/btF)7$$!+edl6;F1$
\"(r.3\"F)7$Fajm$\"(D%\\8F)7$$!+]SMy:F1$\"(s%3:F)7$Ffjm$\"(x3`\"F)7$F[
[n$\"()e=6F)7$F`[n$\"(I^m\"F-7$Fj[n$!'Fj*)F)7$Fd\\n$!(=$R=F)7$Fi\\n$!(
9$f@F)7$F^]n$!(7%GBF)7$$!+SOB*R\"F1$!([uM#F)7$Fc]n$!(#R>BF)7$$!+)Q!\\%
Q\"F1$!(/FC#F)7$Fh]n$!(8n6#F)7$Fb^n$\"'$oL%F)7$F\\_n$\"($QFOF)7$$!+)*[
3^7F1$\"(C&zQF)7$$!+bq*HC\"F1$\"(W70%F)7$$!+7#4\\B\"F1$\"(IG8%F)7$$!+q
8#oA\"F1$\"(*y:TF)7$$!+&oX1@\"F1$\"(e*ePF)7$Fa_n$\"(DU%HF)7$$!+:VHy6F1
$\"*%Q#[m\"Fgs7$$!+I'=@;\"F1$!&'y`F)7$$!+YH%f9\"F1$!(\\S:#F)7$Ff_n$!(%
y[XF)7$$!+w:f86F1$!(xf7(F)7$F[`n$!(>dv*F)7$$!+2-C\"3\"F1$!)5&)H7F)7$F`
`n$!)iXh9F)7$Fe`n$!*@Zi!=F-7$Fj`n$!)S*>$>F)7$$!+yib&p*F4$!)d3@=F)7$$!+
NZ\\(Q*F4$!)7\"*3:F)7$$!+kRYL#*F4$!)%>jH\"F)7$$!+$>L%z!*F4$!)/3g5F)7$$
!+@CSD*)F4$!(!H?\")F)7$F_an$!)\"f<k&F-7$$!+z3M<')F4$!*5-+G$Fgs7$$!+2,J
j%)F4$!)#)yP6F-7$$!+O$z#4$)F4$\"'d**pF)7$$!+l&[_:)F4$\"(t.<#F)7$$!+$z<
7+)F4$\"((\\NKF)7$$!+Aq=ZyF4$\"(tL)QF)7$$!+P;<qxF4$\"(_S0%F)7$$!+^i:$p
(F4$\"(ps7%F)7$$!+m39;wF4$\"(V#3TF)7$Fdan$\"(QM+%F)7$F^bn$\"(4qi*F-7$F
hbn$!(nS1#F)7$$!+)Q[I<'F4$!(K7B#F)7$$!+74p#4'F4$!(`+M#F)7$$!+OML7gF4$!
(y1R#F)7$$!+gf(>$fF4$!($[%Q#F)7$$!+35ErdF4$!),y7AF-7$F]cn$!)b3d=F-7$$!
+`h6*G&F4$!(W!>yF-7$Fbcn$\"(lf7%F-7$Fgcn$\"(*Hk7F)7$F\\dn$\")TYj;F-7$$
!+sPvHUF4$\")=8o;F-7$Fadn$\")Ho[:F-7$$!+!y!eMRF4$\"*^N.K\"Fgs7$Ffdn$\"
)X:05F-7$F`en$!(.*GbF-7$Fjen$!)NLC8F-7$Fdfn$!(5)fcF-7$F^gn$\"(%G!p(F-7
$$!+iXdt6F4$\"(FF!**F-7$$!+*e#Q95F4$\"*J`b7\"Fgs7$$!+dh!>b)F[jl$\"*:la
;\"Fgs7$Fcgn$\"*he#36Fgs7$$!+ip9wPF[jl$\")P&*GtFgs7$Fhgn$\")*H*R7Fgs7$
$\"+7p![r#F[jl$!)!y>Z&Fgs7$F]hn$!*LO6.\"Fgs7$$\"+I&yan(F[jl$!+#*3yX6F1
7$$\"+Pd.H$*F[jl$!*;$>e6Fgs7$$\"+%Hf#)4\"F4$!)Bul5F-7$Fbhn$!(2zu)F-7$F
\\in$\"(M;r%F-7$Ffin$\")&)GC8F-7$F`jn$\"(#*3L'F-7$Fjjn$!(%))e))F-7$F_[
o$!)HQt9F-7$Fd[o$!)#G8o\"F-7$Fi[o$!)mH19F-7$F^\\o$!'DFnF)7$Fc\\o$\"*'G
0l:Fgs7$Fh\\o$\"(bCH#F)7$Fb]o$!((4/7F-7$F\\^o$!(\")o`$F)7$Fa^o$!(S/+%F
)7$Ff^o$!(z`7%F)7$F[_o$!(^!\\QF)7$F`_o$!()pJJF)7$Fe_o$!(!)4'>F)7$Fj_o$
!'_XNF)7$F_`o$\")=ER;F-7$Fd`o$\"*=5L%RFgs7$$\"+H?SA))F4$\")mSbkF-7$Fi`
o$\"(lT0*F)7$$\"+NphV\"*F4$\")jkg6F)7$F^ao$\")tg(R\"F)7$Fcao$\")#yfw\"
F)7$Fhao$\")wIG>F)7$$\"+K+4E5F1$\"*Ln:&=F-7$F]bo$\")41e:F)7$$\"+]pAt5F
1$\")`E[8F)7$$\"+c#R*)3\"F1$\")M!36\"F)7$$\"+i:l/6F1$\"(2<e)F)7$Fbbo$
\"(!zKgF)7$$\"+uh2O6F1$\"(2ke$F)7$$\"+![)y^6F1$\"(JfN\"F)7$$\"+'y+v;\"
F1$!(R!RcF-7$Fgbo$!)'e@5#F-7$$\"+)RD*)>\"F1$!(#H:KF)7$$\"+/xj97F1$!(I'
))QF)7$$\"+dQ\\A7F1$!(*fjSF)7$$\"+5+NI7F1$!(Qe8%F)7$$\"+ih?Q7F1$!(J56%
F)7$F\\co$!(zi*RF)7$Ffco$!()4:6F)7$F`do$\"(vR)=F)7$$\"+!\\%)GQ\"F1$\"(
l%>AF)7$Fedo$\"(?iM#F)7$$\"+'=)o09F1$\"(wKL#F)7$$\"+=%*G89F1$\"(GCF#F)
7$$\"+]1*3U\"F1$\"(-q;#F)7$Fjdo$\"(d5-#F)7$F_eo$\"(NH8\"F)7$Fdeo$\"'Vn
MF-7$Fieo$!'!*z**F)7$F^fo$!(x!4:F)7$$\"+g/qt:F1$!(I$H:F)7$Fcfo$!(@OS\"
F)7$$\"+[*fug\"F1$!(bn:\"F)7$Fhfo$!'HE#)F)7$$\"+!=*4e;F1$!&x$[F)7$F]go
$\"'N]iF)7$$\"+7%Q(3<F1$\"'Wj&)F)7$$\"+c\"=cs\"F1$\"'>!***F)7$$\"++z\\
U<F1$\"(b30\"F)7$Fbgo$\"(?x,\"F)7$F\\ho$!&P<%Fh^t7$F`io$!'<,6Fh^t7$Fji
o$\"&:%HFh^t7$F^[p$\"';ePFh^t7$Fc[p$\"';8fFh^t7$Fh[p$\"&C&yFbct7$$\"+w
xd7AF1$\"&pf)Fbct7$F]\\p$\"&,7*Fbct7$$\"+7\"poB#F1$\"%v#*!\"$7$$\"+Di'
\\C#F1$\"&&y$*Fbct7$$\"+PL1`AF1$\"&)*R*Fbct7$Fb\\p$\"')fM*Fh^t7$$\"+*>
40H#F1$\"',9&)Fh^t7$Fg\\p$\"'&>y'Fh^t7$$\"+'p1#\\BF1$\"'ZHWFh^t7$F\\]p
$\"'0v=Fh^t7$Ff]p$!&k=%Fh^t7$F`^p$!(#z`?F)7$$\"+9t#>X#F1$!(0Aa#F)7$Fe^
p$!'H$z#Fh^t7$$\"+w)QRZ#F1$!'kIGFh^t7$$\"+jgF\"[#F1$!'U6GFh^t7$$\"+^Kh
)[#F1$!'MQFFh^t7$Fj^p$!')[h#Fh^t7$$\"+1B7gDF1$!&N!=Fh^t7$F__p$\"(uo'=F
)7$$\"+UgY)o#F1$\"'%y=\"Fh^t7$Fd_p$!(REu)F-7$$\"+\"4Lxw#F1$!(5'G7F)7$$
\"+t#GGy#F1$!(%zk9F)7$$\"+aM#zz#F1$!(6pc\"F)7$Fi_p$!()zI:F)7$$\"++!4K%
GF1$!(0]3\"F)7$F^`p$!'0eIF)7$$\"+G(*e.HF1$\"(jpH&F-7$Fc`p$\"(379\"F)7$
$\"+u_()[HF1$\"(2xH\"F)7$$\"+b/(R'HF1$\"(OrL\"F)7$$\"+Pc1zHF1$\"($4f7F
)7$Fh`p$\"(#Gt5F)7$Fbap$!(.4V$F-7$F\\bp$!(*y!=\"F)7$F`cp$!'.\"*RF)7$Fd
dp$\"'\"oW)F)7$$\"+B&[?E$F1$\"(\"\\,5F)7$Fidp$\"(BK1\"F)7$$\"+t=!HH$F1
$\"(cf-\"F)7$F^ep$\"'x[*)F)7$Fhep$\"'%G8%F)7$Fbfp$!(JA*>F-7$$\"+)f))3S
$F1$!'h)>(F)7$Fgfp$!'Ne'*F)7$$\"+C'orW$F1$!'tg&*F)7$$\"+*H&fiMF1$!')4e
)F)7$$\"+v>-yMF1$!'dBoF)7$F\\gp$!'biWF)7$$\"+\\#\\d_$F1$\"(pVR\"F-7$Fa
gp$\"'\\@lF)7$$\"+Z,?uNF1$\"'q`\")F)7$$\"+Y/N!f$F1$\"'?M*)F)7$$\"+X2]1
OF1$\"'x'z)F)7$Ffgp$\"'xoxF)7$Fjhp$!(7U)=F-7$Fdip$!'ap$)F)7$F^jp$!'?nJ
F)7$Fb[q$\"'iOfF)7$$\"+e`%=*QF1$\"'pHsF)7$Fg[q$\"'kByF)7$$\"+tY&H#RF1$
\"'rqwF)7$F\\\\q$\"'>%z'F)7$Ff\\q$\"'^%H$F)7$F`]q$!(lVL\"F-7$$\"+G<aKS
F1$!'5RaF)7$Fe]q$!']itF)7$$\"+!Qi-3%F1$!'OZsF)7$$\"+j#ph4%F1$!'6;kF)7$
$\"+Zh27TF1$!'&)e\\F)7$Fj]q$!'\"o-$F)7$$\"+(z'zfTF1$\"(y2X\"F-7$F_^q$
\"'8u_F)7$$\"+[u^2UF1$\"'ndkF)7$$\"+KVUBUF1$\"'i))pF)7$$\"+;7LRUF1$\"'
^@oF)7$Fd^q$\"')*zfF)7$Fb`q$!)t[%\\\"Fgs7$F\\aq$!'([k'F)7$Ffaq$!'4wDF)
7$F`bq$\"'6Q[F)7$Fjbq$!'O0:F-7$F^dq$!'!*GVF)7$Fhdq$!'k4%)F-7$Fbeq$\"(I
u$HF-7$Fgeq$\"'@BWF)7$F\\fq$\"'i;aF)7$Fffq$\"'99eF)7$F`gq$\"']ybF)7$Fh
iq$!''zw\"F-7$F\\[r$!')QZ&F)7$Fa[r$!')[c&F)7$Ff[r$!'Nn^F)7$F[\\r$!'xBV
F)7$F`\\r$!'B6JF)7$Fd]r$!&K7$F-7$Fh^r$\"(*o9IF-7$F]_r$\"'L\">%F)7$Fb_r
$\"'\"\\*\\F)7$Fg_r$\"'[e`F)7$F\\`r$\"'L`_F)7$Ff`r$\"'REPF)7$Fjar$\"'D
n&*F-7$F_br$!(=`G#F-7$Fdbr$!'m-YF)7$Fibr$!'#)=^F)7$F^cr$!'&=m$F)7$Fbdr
$!'=*=)F-7$F\\er$\"(t\")H#F-7$$\"+8\"4\"\\aF1$\"'kuNF)7$Faer$\"'$))[%F
)7$$\"+^ez![&F1$\"'*G&\\F)7$Ffer$\"'@C\\F)7$$\"+?dHdbF1$\"'(Q_*F-7$F[f
r$!'pMTF)7$$\"+q/G[cF1$!'nD[F)7$$\"+?(3'ycF1$!'(H!QF)7$$\"+qp$*3dF1$!'
`\\9F)7$F`fr$\"($e'Q\"F-7$Fdgr$!+3P+lJF4-Figr6&F[hrF^hrF^hrF\\hr-%+AXE
SLABELSG6$Q\"w6\"Q!Fjgx-%%VIEWG6$;F(Fdgr%(DEFAULTG" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 67 "Clearly the information about the frequen
cy contributions is there." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT 258 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 136 "Ext
end the time over which the signal is integrated, i.e., consider a lon
ger wavetrain, and graph the results. What is your observation?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 11 "Exercise
 3:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Choose different s
ignals and perform the Fourier analysis using the infinite domain, and
 then different finite domains." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 377 "Note how
 the amplitudes of the signal contributions at different frequencies i
n the time and frequency domains are preserved. This is a consequence \+
of the linearity of the FT. The figure above displays also how we can \+
generate a sequence of oscillatory functions which approximate the Dir
ac delta (in the limit of an infinite time domain the Dirac delta func
tion is recovered)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 26 "Discrete Fouri
er Transform" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 620 "In the discretized version of the FT we approximate the integr
al by a Riemann sum. This approximation is not unreasonable, because t
he integrand is assumed to be periodic, which means that the function \+
values and derivatives at the left and right boundaries of the integra
tion interval are equal. For this case the Riemann sum is more accurat
e than some of the usual quadrature rules on an equidistant grid, such
 as the trapezoid or Simpson rule. The reduction of the infinite domai
n to a finite one means that we are using a Fourier series representat
ion instead of a true Fourier transform [investigate the worksheets " 
}{TEXT 19 17 "FourierSeries.mws" }{TEXT -1 2 ", " }{TEXT 19 11 "HeatEq
n.mws" }{TEXT -1 2 ", " }{TEXT 19 11 "WaveEqn.mws" }{TEXT -1 28 " to f
ind out more about it]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 722 "In the discrete FT we not only have a finite domain
, but in fact we will discretize time and frequency (position and mome
ntum). Let us use the position-momentum language familiar from quantum
 mechanics, although the same arguments apply in the time-frequency pa
ir of variables. If we have a spatial domain of finite extent, then th
is implies that there is a lowest wavenumber (momentum) that can be re
presented (a sine wave spanning the entire domain with one positive, a
nd one negative lobe). Likewise, when we discretize position, there wi
ll be a maximum wavenumber that can be resolved: a sequence of alterna
ting +1, -1, can be thought of as the highest momentum wave representa
ble for a given choice of discretization " }{TEXT 19 2 "dx" }{TEXT -1 
103 ". Let us pick a spacing, and a number of points (which for techni
cal reasons has to be a power of two)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "dx:=0.4; n:=8; N:=2^n;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#dxG$\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\")
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$c#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "dp:=evalhf(2*Pi/(N*dx));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#dpG$\"3okDa^J#f8'!#>" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 107 "The spacing in the momentum domain follows from the argu
ments given above: The size of the position domain " }{TEXT 19 4 "N*dx
" }{TEXT -1 167 " controls not only the smallest momentum that can be \+
represented, but also the resolution in the momentum domain. The large
st momentum that can be observed is given by" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "p_max:=N*dp/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%&p_maxG$\"+N;)R&y!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 411 "The
 momentum and coordinate meshes are arranged in a somewhat unusual way
: we begin with the positive domains to be stored in the first halves \+
of the arrays, followed by the negative values attached in a periodic \+
way (following the largest positive position/momentum come the largest
 negative positions/momenta). We use vectors from the LinearAlgebra pa
ckage for storage, but lists or arrays would work as well." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "xv:=Vector(N): pv:=Vector(N):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "for i from 1 to N do:\nxv[i
]:=evalhf((i-N/2-1)*dx):\nif i < N/2+1 then\npv[i]:=evalhf((i-1)*dp): \+
pv[N-i+1]:=evalhf(-i*dp): end if:\nod:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "pv[20];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3K(3$z)R
De;\"!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "pointplot(\{seq
([i,xv[i]],i=1..N)\});" }}{PARA 13 "" 1 "" {GLPLOT2D 608 236 236 
{PLOTDATA 2 "6#-%'POINTSG6\\[l7$$\"$B\"\"\"!$!+++++C!\"*7$$\"$C\"F)$!
\"#F)7$$\"$D\"F)$!+++++;F,7$$\"$E\"F)$!+++++7F,7$$\"$F\"F)$!+++++!)!#5
7$$\"$[#F)$\"++++gZ!\")7$$\"$\\#F)$\"#[F)7$$\"$]#F)$\"++++S[FG7$$\"$^#
F)$\"++++!)[FG7$$\"$Z#F)$\"++++?ZFG7$$\"$+\"F)$!++++g6FG7$$\"$,\"F)$!+
+++?6FG7$$\"$-\"F)$!++++!3\"FG7$$\"$.\"F)$!++++S5FG7$$\"#**F)$!#7F)7$$
\"$6#F)$\"++++!G$FG7$$\"$7#F)$\"++++?LFG7$$\"$8#F)$\"++++gLFG7$$\"$9#F
)$\"#MF)7$$\"$5#F)$\"++++SKFG7$$\"$X\"F)$\"+++++kF,7$$\"$Y\"F)$\"+++++
oF,7$$\"$Z\"F)$\"+++++sF,7$$\"$[\"F)$\"+++++wF,7$$\"$\\\"F)$\"\")F)7$$
\"$]\"F)$\"+++++%)F,7$$\"$^\"F)$\"+++++))F,7$$\"$_\"F)$\"+++++#*F,7$$
\"$`\"F)$\"+++++'*F,7$$\"$a\"F)$\"#5F)7$$\"$b\"F)$\"++++S5FG7$$\"$c\"F
)$\"++++!3\"FG7$$\"$d\"F)$\"++++?6FG7$$\"$e\"F)$\"++++g6FG7$$\"$f\"F)$
\"#7F)7$$\"$g\"F)$\"++++S7FG7$$\"$h\"F)$\"++++!G\"FG7$$\"$i\"F)$\"++++
?8FG7$$\"$j\"F)$\"++++g8FG7$$\"$k\"F)$\"#9F)7$$\"$l\"F)$\"++++S9FG7$$
\"$m\"F)$\"++++![\"FG7$$\"$n\"F)$\"++++?:FG7$$\"$o\"F)$\"++++g:FG7$$\"
$p\"F)$\"#;F)7$$\"$q\"F)$\"++++S;FG7$$\"$r\"F)$\"++++!o\"FG7$$\"$s\"F)
$\"++++?<FG7$$\"$t\"F)$\"++++g<FG7$$\"$u\"F)$\"#=F)7$$\"$v\"F)$\"++++S
=FG7$$\"$w\"F)$\"++++!)=FG7$$\"$x\"F)$\"++++?>FG7$$\"$y\"F)$\"++++g>FG
7$$\"$z\"F)$\"#?F)7$$\"$!=F)$\"++++S?FG7$$\"$\"=F)$\"++++!3#FG7$$\"$#=
F)$\"++++?@FG7$$\"$$=F)$\"++++g@FG7$$\"$%=F)$\"#AF)7$$\"$&=F)$\"++++SA
FG7$$\"$'=F)$\"++++!G#FG7$$\"$(=F)$\"++++?BFG7$$\"$)=F)$\"++++gBFG7$$
\"$*=F)$\"#CF)7$$\"$!>F)$\"++++SCFG7$$\"$\">F)$\"++++![#FG7$$\"$#>F)$
\"++++?DFG7$$\"$$>F)$\"++++gDFG7$$\"$%>F)$\"#EF)7$$\"$&>F)$\"++++SEFG7
$$\"$'>F)$\"++++!o#FG7$$\"$(>F)$\"++++?FFG7$$\"$)>F)$\"++++gFFG7$$\"$*
>F)$\"#GF)7$$\"$+#F)$\"++++SGFG7$$\"$,#F)$\"++++!)GFG7$$\"$-#F)$\"++++
?HFG7$$\"$.#F)$\"++++gHFG7$$\"$/#F)$\"#IF)7$$\"$0#F)$\"++++SIFG7$$\"$1
#F)$\"++++!3$FG7$$\"$2#F)$\"++++?JFG7$$\"$3#F)$\"++++gJFG7$$\"$4#F)$\"
#KF)7$$\"$G\"F)$!+++++SFA7$$\"$H\"F)$F)F)7$$\"$I\"F)$\"+++++SFA7$$\"$J
\"F)$\"+++++!)FA7$$\"$K\"F)$\"+++++7F,7$$\"$L\"F)$\"+++++;F,7$$\"$M\"F
)$\"\"#F)7$$\"$N\"F)$\"+++++CF,7$$\"$O\"F)$\"+++++GF,7$$\"$P\"F)$\"+++
++KF,7$$\"$Q\"F)$\"+++++OF,7$$\"$R\"F)$\"\"%F)7$$\"$S\"F)$\"+++++WF,7$
$\"$T\"F)$\"+++++[F,7$$\"$U\"F)$\"+++++_F,7$$\"$V\"F)$\"+++++cF,7$$\"$
W\"F)$\"\"'F)7$$\"$/\"F)$FAF)7$$\"$0\"F)$!+++++'*F,7$$\"$1\"F)$!+++++#
*F,7$$\"$2\"F)$!+++++))F,7$$\"$3\"F)$!+++++%)F,7$$\"$4\"F)$FGF)7$$\"$5
\"F)$!+++++wF,7$$\"$6\"F)$!+++++sF,7$$\"$7\"F)$!+++++oF,7$$\"$8\"F)$!+
++++kF,7$$\"$9\"F)$!\"'F)7$$\"$:\"F)$!+++++cF,7$$\"$;\"F)$!+++++_F,7$$
\"$<\"F)$!+++++[F,7$$\"$=\"F)$!+++++WF,7$$\"$>\"F)$!\"%F)7$$\"$?\"F)$!
+++++OF,7$$\"$@\"F)$!+++++KF,7$$\"$A\"F)$!+++++GF,7$Faq$!#QF)7$$\"#NF)
$!++++gPFG7$$\"#OF)$!++++?PFG7$$\"#PF)$!++++!o$FG7$$\"#QF)$!++++SOFG7$
$\"#RF)$!#OF)7$$\"#SF)$!++++gNFG7$$\"#TF)$!++++?NFG7$$\"#UF)$!++++![$F
G7$$\"#VF)$!++++SMFG7$$\"#WF)$!#MF)7$$\"#XF)$!++++gLFG7$$\"#YF)$!++++?
LFG7$$\"#ZF)$!++++!G$FG7$FK$!++++SKFG7$$\"#\\F)$!#KF)7$$\"#]F)$!++++gJ
FG7$$\"#^F)$!++++?JFG7$$\"#_F)$!++++!3$FG7$$\"#`F)$!++++SIFG7$$\"#aF)$
!#IF)7$$\"#bF)$!++++gHFG7$$\"#cF)$!++++?HFG7$$\"#dF)$!++++!)GFG7$$\"#e
F)$!++++SGFG7$$\"#fF)$!#GF)7$$\"#gF)$!++++gFFG7$$\"#hF)$!++++?FFG7$$\"
#iF)$!++++!o#FG7$$\"#jF)$!++++SEFG7$$\"#kF)$!#EF)7$$\"#lF)$!++++gDFG7$
$\"#mF)$!++++?DFG7$$\"#nF)$!++++![#FG7$$\"#oF)$!++++SCFG7$$\"#pF)$!#CF
)7$$\"#qF)$!++++gBFG7$$\"#rF)$!++++?BFG7$$\"#sF)$!++++!G#FG7$$\"#tF)$!
++++SAFG7$$\"#uF)$!#AF)7$$\"#vF)$!++++g@FG7$$\"#wF)$!++++?@FG7$$\"#xF)
$!++++!3#FG7$$\"#yF)$!++++S?FG7$$\"#zF)$!#?F)7$$\"#!)F)$!++++g>FG7$$\"
#\")F)$!++++?>FG7$$\"##)F)$!++++!)=FG7$$\"#$)F)$!++++S=FG7$$\"#%)F)$!#
=F)7$$\"#&)F)$!++++g<FG7$$\"#')F)$!++++?<FG7$$\"#()F)$!++++!o\"FG7$$\"
#))F)$!++++S;FG7$$\"#*)F)$!#;F)7$$\"#!*F)$!++++g:FG7$$\"#\"*F)$!++++?:
FG7$$\"##*F)$!++++![\"FG7$$\"#$*F)$!++++S9FG7$$\"#%*F)$!#9F)7$$\"#&*F)
$!++++g8FG7$$\"#'*F)$!++++?8FG7$$\"#(*F)$!++++!G\"FG7$$\"#)*F)$!++++S7
FG7$$\"\"\"F)$!++++?^FG7$F]hl$!++++!3&FG7$$\"\"$F)$!++++S]FG7$Ffil$!#]
F)7$$\"\"&F)$!++++g\\FG7$F_[m$!++++?\\FG7$$\"\"(F)$!++++!)[FG7$F_s$!++
++S[FG7$$\"\"*F)$!#[F)7$Fht$!++++gZFG7$$\"#6F)$!++++?ZFG7$Fav$!++++!o%
FG7$$\"#8F)$!++++SYFG7$Fjw$!#YF)7$$\"#:F)$!++++gXFG7$Fcy$!++++?XFG7$$
\"#<F)$!++++![%FG7$F\\[l$!++++SWFG7$$\"#>F)$!#WF)7$Fe\\l$!++++gVFG7$$
\"#@F)$!++++?VFG7$F^^l$!++++!G%FG7$$\"#BF)$!++++SUFG7$Fg_l$!#UF)7$$\"#
DF)$!++++gTFG7$F`al$!++++?TFG7$$\"#FF)$!++++!3%FG7$Fibl$!++++SSFG7$$\"
#HF)$!#SF)7$Fbdl$!++++gRFG7$$\"#JF)$!++++?RFG7$F[fl$!++++!)QFG7$$\"#LF
)$!++++SQFG7$$\"$`#F)$\"++++g\\FG7$$\"$a#F)F[fm7$$\"$b#F)$\"++++S]FG7$
$\"$c#F)$\"++++!3&FG7$$\"$_#F)$\"++++?\\FG7$$\"$:#F)$\"++++SMFG7$$\"$;
#F)$\"++++![$FG7$$\"$<#F)$\"++++?NFG7$$\"$=#F)$\"++++gNFG7$$\"$>#F)Fga
m7$$\"$?#F)$\"++++SOFG7$$\"$@#F)$\"++++!o$FG7$$\"$A#F)$\"++++?PFG7$$\"
$B#F)$\"++++gPFG7$$\"$C#F)Fabm7$$\"$D#F)$\"++++SQFG7$$\"$E#F)$\"++++!)
QFG7$$\"$F#F)$\"++++?RFG7$$\"$G#F)$\"++++gRFG7$$\"$H#F)F[cm7$$\"$I#F)$
\"++++SSFG7$$\"$J#F)$\"++++!3%FG7$$\"$K#F)$\"++++?TFG7$$\"$L#F)$\"++++
gTFG7$$\"$M#F)Fecm7$$\"$N#F)$\"++++SUFG7$$\"$O#F)$\"++++!G%FG7$$\"$P#F
)$\"++++?VFG7$$\"$Q#F)$\"++++gVFG7$$\"$R#F)F_dm7$$\"$S#F)$\"++++SWFG7$
$\"$T#F)$\"++++![%FG7$$\"$U#F)$\"++++?XFG7$$\"$V#F)$\"++++gXFG7$$\"$W#
F)Fidm7$$\"$X#F)$\"++++SYFG7$$\"$Y#F)$\"++++!o%FG" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "pointplot(\{seq([i,pv[i]],i=1..N)\});" }}
{PARA 13 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 640 222 
222 {PLOTDATA 2 "6#-%'POINTSG6\\[l7$$\"$/#\"\"!$!+q#R?D$!\"*7$$\"$0#F)
$!+R+o!>$F,7$$\"$1#F)$!+23KHJF,7$$\"$2#F)$!+w:'z1$F,7$$\"$.#F)$!+-&)R8
LF,7$$\"$`#F)$!+h#pVX#!#57$$\"$a#F)$!+XpxS=FF7$$\"$b#F)$!+IY=F7FF7$$\"
$c#F)$!+_J#f8'!#67$$\"#FF)$\"+>+M&f\"F,7$$\"#GF)$\"+^#*pc;F,7$$\"#IF)$
\"+9xTz<F,7$$\"#JF)$\"+XpxS=F,7$$\"#KF)$\"+xh8->F,7$$\"#LF)$\"+3a\\j>F
,7$$\"#HF)$\"+#[e!=<F,7$$\"#]F)$\"+WBg1IF,7$$\"#^F)$\"+w:'z1$F,7$$\"#_
F)$\"+23KHJF,7$$\"#`F)$\"+R+o!>$F,7$$\"#\\F)$\"+8JCXHF,7$$\"#XF)$\"+(=
1)*p#F,7$$\"#YF)$\"+=a;hFF,7$$\"#ZF)$\"+]Y_AGF,7$$\"#[F)$\"+\")Q)Q)GF,
7$$\"#WF)$\"+bpWQEF,7$$\"$1\"F)$\"+4$>FW'F,7$$\"$2\"F)$\"+T&yS]'F,7$$
\"$3\"F)$\"+sxVllF,7$$\"$4\"F)$\"+/qzEmF,7$$\"$5\"F)$\"+Ni:)o'F,7$$\"$
4#F)$!+8JCXHF,7$$\"$5#F)$!+\")Q)Q)GF,7$$\"$6#F)$!+]Y_AGF,7$$\"$7#F)$!+
=a;hFF,7$$\"$3#F)$!+WBg1IF,7$$\"#DF)$\"+c:is9F,7$$\"#EF)$\"+)y!)R`\"F,
7$$\"#BF)$\"+$4.*\\8F,7$$\"#CF)$\"+DBE69F,7$$\"#@F)$\"+IY=F7F,7$$\"#AF
)$\"+iQa)G\"F,7$$\"\"\"F)$F)F)7$$\"\"#F)$\"+_J#f8'FV7$$\"\"$F)$F`xFF7$
$\"\"%F)$FdoFF7$$\"\"&F)$\"+h#pVX#FF7$$\"\"'F)$F]qFF7$$\"\"(F)$\"+\"*Q
b\"o$FF7$$\"\")F)$\"+1i9&H%FF7$$\"\"*F)$\"+@&Q(3\\FF7$$\"#5F)$\"+O3LAb
FF7$$\"#6F)$F^yFF7$$\"#7F)$\"+na^\\nFF7$$\"#8F)$\"+#y2JO(FF7$$\"#9F)$
\"+(4+n(zFF7$$\"#:F)$\"+7CH!f)FF7$$\"#;F)$\"+FZ)Q?*FF7$$\"#<F)$\"+UqZ<
)*FF7$$\"#=F)$\"+Op5V5F,7$$\"#>F)$\"+nhY/6F,7$$\"#?F)$\"+*RDe;\"F,7$$
\"$9#F)$!+bpWQEF,7$$\"$:#F)$!+Cx3xDF,7$$\"$;#F)$!+#\\Gd^#F,7$$\"$<#F)$
FEF,7$$\"$=#F)$!+H+,$R#F,7$$\"$>#F)$!+)z];L#F,7$$\"$?#F)$!+m:HqAF,7$$
\"$@#F)$!+NB$*3AF,7$$\"$A#F)$!+.JdZ@F,7$$\"$B#F)$!+sQ@'3#F,7$$\"$C#F)$
!+SY&[-#F,7$$\"$D#F)$!+3a\\j>F,7$$\"$E#F)$!+xh8->F,7$$\"$F#F)$FKF,7$$
\"$G#F)$!+9xTz<F,7$$\"$H#F)$!+#[e!=<F,7$$\"$I#F)$!+^#*pc;F,7$$\"$J#F)$
!+>+M&f\"F,7$$\"$K#F)$!+)y!)R`\"F,7$$\"$L#F)$!+c:is9F,7$$\"$M#F)$!+DBE
69F,7$$\"$N#F)$!+$4.*\\8F,7$$\"$O#F)$!+iQa)G\"F,7$$\"$P#F)$FPF,7$$\"$Q
#F)$!+*RDe;\"F,7$$\"$R#F)$!+nhY/6F,7$$\"$S#F)$!+Op5V5F,7$$\"$T#F)$!+Uq
Z<)*FF7$$\"$U#F)$!+FZ)Q?*FF7$$\"$V#F)$!+7CH!f)FF7$$\"$W#F)$!+(4+n(zFF7
$$\"$X#F)$!+#y2JO(FF7$$\"$Y#F)$!+na^\\nFF7$$\"$Z#F)$FUFF7$$\"$[#F)$!+O
3LAbFF7$$\"$\\#F)$!+@&Q(3\\FF7$$\"$]#F)$!+1i9&H%FF7$$\"$^#F)$!+\"*Qb\"
o$FF7$$\"$_#F)$F;FF7$$\"$8#F)$!+(=1)*p#F,7$$\"$Q\"F)$!+]&[<I(F,7$$\"$R
\"F)$!+>$*QSsF,7$$\"$S\"F)$!+(3I!zrF,7$$\"$T\"F)$!+c3n<rF,7$$\"$U\"F)$
!+C;JcqF,7$$\"$V\"F)$!+$R_\\*pF,7$$\"$W\"F)$!+hJfLpF,7$$\"$X\"F)$!+IRB
soF,7$$\"$Y\"F)$!+)pu3\"oF,7$$\"$Z\"F)$FfhlF,7$$\"$[\"F)$!+Ni:)o'F,7$$
\"$\\\"F)$!+/qzEmF,7$$\"$]\"F)$!+sxVllF,7$$\"$^\"F)$!+T&yS]'F,7$$\"$_
\"F)$!+4$>FW'F,7$$\"$`\"F)$!+y+O\"Q'F,7$$\"$a\"F)$!+Y3+?jF,7$$\"$b\"F)
$!+:;keiF,7$$\"$c\"F)$!+$Q#G(>'F,7$$\"$d\"F)$FUF,7$$\"$e\"F)$!+?RcugF,
7$$\"$f\"F)$!+*o/K,'F,7$$\"$g\"F)$!+da%=&fF,7$$\"$h\"F)$!+Di[!*eF,7$$
\"$i\"F)$!+%*p7HeF,7$$\"$j\"F)$!+ixwndF,7$$\"$k\"F)$!+J&3kq&F,7$$\"$l
\"F)$!+*H\\]k&F,7$$\"$m\"F)$!+o+p$e&F,7$$\"$n\"F)$F_ilF,7$$\"$o\"F)$!+
0;(4Y&F,7$$\"$p\"F)$!+tBh*R&F,7$$\"$q\"F)$!+UJDQ`F,7$$\"$r\"F)$!+5R*oF
&F,7$$\"$s\"F)$!+zY`:_F,7$$\"$t\"F)$!+Za<a^F,7$$\"$u\"F)$!+;i\"G4&F,7$
$\"$v\"F)$!+%)pXJ]F,7$$\"$w\"F)$!+`x4q\\F,7$$\"$x\"F)$FdilF,7$$\"$y\"F
)$!+!Hzt%[F,7$$\"$z\"F)$!+e+-'y%F,7$$\"$!=F)$!+F3mCZF,7$$\"$\"=F)$!+&f
,Lm%F,7$$\"$#=F)$!+kB%>g%F,7$$\"$$=F)$!+KJeSXF,7$$\"$%=F)$!+,RAzWF,7$$
\"$&=F)$!+pY'yT%F,7$$\"$'=F)$!+Qa]cVF,7$$\"$(=F)$FiilF,7$$\"$)=F)$!+vp
yLUF,7$$\"$*=F)$!+VxUsTF,7$$\"$!>F)$!+7&o56%F,7$$\"$\">F)$!+!G4(\\SF,7
$$\"$#>F)$!+\\+N))RF,7$$\"$$>F)$!+<3*p#RF,7$$\"$%>F)$!+&eJc'QF,7$$\"$&
>F)$!+aBF/QF,7$$\"$'>F)$!+AJ\"Hu$F,7$$\"$(>F)$F^jlF,7$$\"$)>F)$!+fY>?O
F,7$$\"$*>F)$!+Ga$)eNF,7$$\"$+#F)$!+'>wu\\$F,7$$\"$,#F)$!+lp6OMF,7$$\"
$-#F)$!+LxvuLF,7$$\"$6\"F)$F\\\\lF,7$$\"$7\"F)$\"+)pu3\"oF,7$$\"$8\"F)
$\"+IRBsoF,7$$\"$9\"F)$\"+hJfLpF,7$$\"$:\"F)$\"+$R_\\*pF,7$$\"$;\"F)$
\"+C;JcqF,7$$\"$<\"F)$\"+c3n<rF,7$$\"$=\"F)$\"+(3I!zrF,7$$\"$>\"F)$\"+
>$*QSsF,7$$\"$?\"F)$\"+]&[<I(F,7$$\"$@\"F)$Fa\\lF,7$$\"$A\"F)$\"+8qYCu
F,7$$\"$B\"F)$\"+Xi#e[(F,7$$\"$C\"F)$\"+wa=ZvF,7$$\"$D\"F)$\"+3Za3wF,7
$$\"$E\"F)$\"+RR!*pwF,7$$\"$F\"F)$\"+rJEJxF,7$$\"$G\"F)$\"+-Ci#z(F,7$$
\"$H\"F)$!+M;)R&yF,7$$\"$I\"F)$!+-Ci#z(F,7$$\"$J\"F)$!+rJEJxF,7$$\"$K
\"F)$!+RR!*pwF,7$$\"$L\"F)$!+3Za3wF,7$$\"$M\"F)$!+wa=ZvF,7$$\"$N\"F)$!
+Xi#e[(F,7$$\"$O\"F)$!+8qYCuF,7$$\"$P\"F)$FahlF,7$$\"#gF)$\"+fY>?OF,7$
$\"#hF)$FdzF,7$$\"#iF)$\"+AJ\"Hu$F,7$$\"#jF)$\"+aBF/QF,7$$\"#kF)$\"+&e
Jc'QF,7$$\"#lF)$\"+<3*p#RF,7$$\"#mF)$\"+\\+N))RF,7$$\"#nF)$\"+!G4(\\SF
,7$$\"#oF)$\"+7&o56%F,7$$\"#pF)$\"+VxUsTF,7$$\"#qF)$\"+vpyLUF,7$$\"#rF
)$FizF,7$$\"#sF)$\"+Qa]cVF,7$$\"#tF)$\"+pY'yT%F,7$$\"#uF)$\"+,RAzWF,7$
$\"#vF)$\"+KJeSXF,7$$\"#wF)$\"+kB%>g%F,7$$\"#xF)$\"+&f,Lm%F,7$$\"#yF)$
\"+F3mCZF,7$$\"#zF)$\"+e+-'y%F,7$$\"#!)F)$\"+!Hzt%[F,7$$\"#\")F)$F^[lF
,7$$\"##)F)$\"+`x4q\\F,7$$\"#$)F)$\"+%)pXJ]F,7$$\"#%)F)$\"+;i\"G4&F,7$
$\"#&)F)$\"+Za<a^F,7$$\"#')F)$\"+zY`:_F,7$$\"#()F)$\"+5R*oF&F,7$$\"#))
F)$\"+UJDQ`F,7$$\"#*)F)$\"+tBh*R&F,7$$\"#!*F)$\"+0;(4Y&F,7$$\"#\"*F)$F
c[lF,7$$\"##*F)$\"+o+p$e&F,7$$\"#$*F)$\"+*H\\]k&F,7$$\"#%*F)$\"+J&3kq&
F,7$$\"#&*F)$\"+ixwndF,7$$\"#'*F)$\"+%*p7HeF,7$$\"#(*F)$\"+Di[!*eF,7$$
\"#)*F)$\"+da%=&fF,7$$\"#**F)$\"+*o/K,'F,7$$\"$+\"F)$\"+?RcugF,7$$\"$,
\"F)$F^yF,7$$\"$-\"F)$\"+$Q#G(>'F,7$$\"$.\"F)$\"+:;keiF,7$$\"$/\"F)$\"
+Y3+?jF,7$$\"$0\"F)$\"+y+O\"Q'F,7$$\"#aF)$\"+q#R?D$F,7$$\"#bF)$\"+-&)R
8LF,7$$\"#cF)$\"+LxvuLF,7$$\"#dF)$\"+lp6OMF,7$$\"#eF)$\"+'>wu\\$F,7$$
\"#fF)$\"+Ga$)eNF,7$$\"#NF)$\"+sQ@'3#F,7$$\"#OF)$\"+.JdZ@F,7$$\"#PF)$
\"+NB$*3AF,7$$\"#QF)$\"+m:HqAF,7$$\"#MF)$\"+SY&[-#F,7$$\"#SF)$\"+H+,$R
#F,7$$\"#TF)$F[zF,7$$\"#UF)$\"+#\\Gd^#F,7$$\"#VF)$\"+Cx3xDF,7$$\"#RF)$
\"+)z];L#F," 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "
Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 226 "The meshes include z
ero as their first point. These are not the only possible choices. In \+
particular, one can choose meshes that avoid the origin, i.e., are spa
ced a half-step around it, if one has reasons to avoid the origin." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 192 "Let us c
hoose a time sequence and generate the transform. We pick a real funct
ion with sine and cosine contributions. The function is real-valued, b
ut we prepare for a real and imaginary part " }{TEXT 19 2 "yR" }{TEXT 
-1 5 " and " }{TEXT 19 2 "yI" }{TEXT -1 14 " respectively." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fx:=sin(x)+2*cos(2*x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#fxG,&-%$sinG6#%\"xG\"\"\"*&\"\"#F*-%$cosG6#,
$*&F,F*F)F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "yR:=Ve
ctor(N): yI:=Vector(N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "for i fr
om 1 to N do: yR[i]:=evalf(subs(x=xv[i],Re(fx))); yI[i]:=evalf(subs(x=
xv[i],Im(fx))); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot
([seq([xv[i],yR[i]],i=1..N)],style=point);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 824 208 208 {PLOTDATA 2 "6&-%'CURVESG6$7\\[l7$$!3H++++++?^!#
;$!3))*****R$o)>R\"!#<7$$!3W++++++!3&F*$\"3%)*****H@\\^_%!#=7$$!3d++++
++S]F*$\"3!******RTVRz\"F-7$$!#]\"\"!$\"31+++*f7q)>F-7$$!39++++++g\\F*
$\"35+++\\k^#4\"F-7$$!3H++++++?\\F*$!3'*******fZNw=F37$$!3W++++++!)[F*
$!3)********3,Mh*F37$$!3d++++++S[F*$!35+++&zho0(F37$$!#[F<$\"3'******p
iPR2%F37$$!39++++++gZF*$\"30+++ml2=;F-7$$!3H++++++?ZF*$\"31+++&R5H0#F-
7$$!3W++++++!o%F*$\"3%******>m%fw7F-7$$!3d++++++SYF*$!3;+++uo!*oTF37$$
!#YF<$!3;+++Wsna@F-7$$!39++++++gXF*$!3))*****fl0,*HF-7$$!3H++++++?XF*$
!3A+++J6**fCF-7$$!3W++++++![%F*$!3!******R+cqe)F37$$!3d++++++SWF*$\"3U
+++9*=4O*F37$$!#WF<$\"33+++UY/\")>F-7$$!39++++++gVF*$\"31+++\"Hzj\"=F-
7$$!3H++++++?VF*$\"3'******R5YJ<(F37$$!3W++++++!G%F*$!3R+++L/k0]F37$$!
3d++++++SUF*$!3'******p\"3P&***F37$$!#UF<$!3F+++JWDNWF37$$!39++++++gTF
*$\"3p*****f\\Qx#zF37$$!3H++++++?TF*$\"33+++7H3d=F-7$$!3W++++++!3%F*$
\"3!******z(Ho_>F-7$$!3d++++++SSF*$\"3w*****>QBbX)F37$$!#SF<$!3_+++\"[
w)e'*F37$$!39++++++gRF*$!34+++4=xDDF-7$$!3H++++++?RF*$!3;+++N1/yHF-7$$
!3W++++++!)QF*$!3)******f`#[r?F-7$$!3d++++++SQF*$!3=+++v\"RY3$F37$$!#Q
F<$\"33+++$3%H_8F-7$$!39++++++gPF*$\"31+++>byf?F-7$$!3H++++++?PF*$\"3'
******pDQAc\"F-7$$!3W++++++!o$F*$\"34+++uvI=LF37$$!3')************ROF*
$!3/+++b>B'[(F37$$!#OF<$!3n*****fBBsU*F37$$!39++++++gNF*$!3-+++(p!f*>
\"F37$$!3H++++++?NF*$\"31+++E-5j6F-7$$!3W++++++![$F*$\"3;+++!G(\\4?F-7
$$!3')************RMF*$\"3\"******4Q+Zu\"F-7$$!#MF<$\"3&)******)eL?^$F
37$$!39++++++gLF*$!3)******H%Gt#\\\"F-7$$!3H++++++?LF*$!3')*****z0O\")
z#F-7$$!3W++++++!G$F*$!3<+++\\%pZ%GF-7$$!3')************RKF*$!3,+++FHr
1;F-7$$!#KF<$\"3%******fzxGK#F37$$!39++++++gJF*$\"31+++ff&Ho\"F-7$$!3H
++++++?JF*$\"3#******f$fjI?F-7$$!33++++++!3$F*$\"3,+++ajpU7F-7$$!3A+++
+++SIF*$!3C+++!*p/#)R!#>7$$!#IF<$!31+++pL%z;*F37$$!39++++++gHF*$!35+++
V![J%zF37$$!3H++++++?HF*$\"35+++`=!GX#F37$$!33++++++!)GF*$\"3-+++\"H>Y
\\\"F-7$$!3A++++++SGF*$\"3)*******[&*\\i?F-7$$!#GF<$\"3*)*****pUMbV\"F
-7$$!39++++++gFF*$!3\"******RkVw$=F37$$!3H++++++?FF*$!35+++Ivvr>F-7$$!
33++++++!o#F*$!3-+++$RY#eHF-7$$!3A++++++SEF*$!3!*******f_#pf#F-7$$!#EF
<$!3%******>,S&)3\"F-7$$!39++++++gDF*$\"3c*****p*[sztF37$$!3H++++++?DF
*$\"3++++FUs9>F-7$$!33++++++![#F*$\"33+++/J()**=F-7$$!3A++++++SCF*$\"3
C+++#H%[\"z)F37$$!#CF<$!3;+++qJ5ZPF37$$!39++++++gBF*$!35+++>/M\\**F37$
$!3H++++++?BF*$!3\"******zE/Ej&F37$$!33++++++!G#F*$\"3')******y5;(H'F3
7$$!3A++++++SAF*$\"3#******zf``w\"F-7$$!#AF<$\"3\")*****f#z`3?F-7$$!39
++++++g@F*$\"3&******zvTt.\"F-7$$!3H++++++?@F*$!3C+++GOiTtF37$$!33++++
++!3#F*$!3!******4eP%zBF-7$$!3A++++++S?F*$!33+++ti8)*HF-7$$!#?F<$!3/++
+u8#oC#F-7$$!39++++++g>F*$!3j*****\\O6EU&F37$$!3H++++++?>F*$\"31+++5XA
&=\"F-7$$!33++++++!)=F*$\"34+++A0sR?F-7$$!3A++++++S=F*$\"3++++-l<z;F-7
$$F3F<$\"3#)*****fn)f]\\F37$$!39++++++g<F*$!3`*****4F!zBlF37$$!3$*****
********><F*$!3%)******4!*Q$y*F37$$!33++++++!o\"F*$!34+++C#R`j#F37$$!3
A++++++S;F*$\"3********HhQ45F-7$$F*F<$\"31+++Q+Nc>F-7$$!39++++++g:F*$
\"3%*******\\H!e%=F-7$$!36++++++?:F*$\"3[*****4tlEn&F37$$!33++++++![\"
F*$!3)******RN<OF\"F-7$$!3/++++++S9F*$!3/+++Uvj&p#F-7$$!#9F<$!3?+++*3>
e\"HF-7$$!39++++++g8F*$!3)******4\"zz6=F-7$$!36++++++?8F*$\"3F++++\")H
AdF^dl7$$!33++++++!G\"F*$\"31+++zH5a:F-7$$!3/++++++S7F*$\"3-+++._!f0#F
-7$$!#7F<$\"3#******H$4$\\Q\"F-7$$!39++++++g6F*$\"31+++#35k9\"F37$$!36
++++++?6F*$!3E+++=Pyf&)F37$$!33++++++!3\"F*$!3=+++*>t+o)F37$$!3/++++++
S5F*$\"3Q******\\<)Rm)F^\\l7$$!#5F<$\"3-+++M_=g8F-7$$!3U,++++++'*F-$\"
3#******RZnF0#F-7$$!31,++++++#*F-$\"35+++J/Ry:F-7$$!3q+++++++))F-$\"3)
)*******oJql%F^\\l7$$!3M+++++++%)F-$!33+++$3cfx\"F-7$$!\")F<$!3%)*****
z?xY!HF-7$$!3_+++++++wF-$!36+++QOR:FF-7$$!3;+++++++sF-$!3*)*****fd-LJ
\"F-7$$!3q+++++++oF-$\"3w******Qj%HH&F37$$!3M+++++++kF-$\"31+++Ef6H=F-
7$$!\"'F<$\"35+++:M7n>F-7$$!3_+++++++cF-$\"3&******fOws.\"F-7$$!3;++++
+++_F-$!3#******HfQ^Q#F37$$!3q+++++++[F-$!3R+++K56K(*F37$$!3M+++++++WF
-$!3Y+++T&Req'F37$$!\"%F<$\"3A+++xU-eYF37$$!33+++++++OF-$\"33+++sIAf;F
-7$$!3;+++++++KF-$\"3))*****4)RuW?F-7$$!3E+++++++GF-$\"3,+++2O9;7F-7$$
!3M+++++++CF-$!3n*****p8_Y+&F37$$!\"#F<$!3&*******oYe;AF-7$$!33+++++++
;F-$!33+++bJ;'*HF-7$$!3;+++++++7F-$!3$)*****p^EoS#F-7$$!3U+++++++!)F3$
!3q*****\\N^vv(F37$$!3A+++++++SF3$\"3!******p2&*R+\"F-7$$F<F<$\"\"#F<7
$$\"3A+++++++SF3$\"3'******4wJGy\"F-7$$\"3U+++++++!)F3$\"3g*****HYq&*e
'F37$$\"3;+++++++7F-$!3@+++]M[FaF37$$\"33+++++++;F-$!37+++!\\f,(**F37$
Ffcm$!3')*****>:)*)zRF37$$\"3M+++++++CF-$\"39+++v9h/&)F37$$\"3E+++++++
GF-$\"35+++2*>h)=F-7$$\"3;+++++++KF-$\"3!******\\p&*z#>F-7$$\"33++++++
+OF-$\"3p*****p&=#=u(F37$$\"\"%F<$!33+++jD!y/\"F-7$$\"3M+++++++WF-$!3'
******>5)ytDF-7$$\"3q+++++++[F-$!3?+++@.alHF-7$$\"3;+++++++_F-$!3*)***
**4<Ba+#F-7$$\"3_+++++++cF-$!3-+++.\"pDD#F37$$\"\"'F<$\"3\"*******=CH3
9F-7$$\"3M+++++++kF-$\"3#)*****fL9A1#F-7$$\"3q+++++++oF-$\"3%******fL@
v^\"F-7$$\"3;+++++++sF-$\"32+++9:LSFF37$$\"3_+++++++wF-$!3!)******RHa&
z(F37$$\"\")F<$!3_+++Wrgf#*F37$$\"3M+++++++%)F-$!3#*******pm#en'F^\\l7
$$\"3q+++++++))F-$\"3.+++.ZS;7F-7$$\"31,++++++#*F-$\"3#*******e-<C?F-7
$$\"3U,++++++'*F-$\"3&******>\"R6/<F-7$$\"#5F<$\"3?+++F,V@FF37$$\"3/++
++++S5F*$!3-+++@J,p:F-7$$\"33++++++!3\"F*$!3=+++!yz)HGF-7$$\"36++++++?
6F*$!3#)******HQL9GF-7$$\"39++++++g6F*$!3*)*****>=;5`\"F-7$$\"#7F<$\"3
y*****f'4&y6$F37$$\"3/++++++S7F*$\"3(******H&opC<F-7$$\"33++++++!G\"F*
$\"3<+++HE7<?F-7$$\"36++++++?8F*$\"3))*****zKp)*=\"F-7$$\"39++++++g8F*
$!3g+++]\"GuM*F^\\l7$$\"#9F<$!35+++tP/Y$*F37$$\"3/++++++S9F*$!3l*****
\\*)>Kk(F37$$\"33++++++![\"F*$\"3;+++1y')GIF37$$\"36++++++?:F*$\"3/+++
^R1S:F-7$$\"39++++++g:F*$\"3!******RD581#F-7$$\"#;F<$\"3)******RSV0Q\"
F-7$$\"3A++++++S;F*$!35+++XBFoEF37$$\"33++++++!o\"F*$!3*********)zmQ?F
-7$$\"3$*************><F*$!3%******H.R@(HF-7$$\"39++++++g<F*$!37+++B!o
+b#F-7$$\"#=F<$!3#******fi9p+\"F-7$$\"3A++++++S=F*$\"3P+++zDX+\")F37$$
\"33++++++!)=F*$\"3%*******R#\\1%>F-7$$\"3H++++++?>F*$\"3%******zOa=(=
F-7$$\"39++++++g>F*$\"3R+++P5m;#)F37$$\"#?F<$!3\")*****Hs3$4UF37$$\"3A
++++++S?F*$!3')******GZ]&)**F37$$\"33++++++!3#F*$!3l******G6Z=_F37$$\"
3H++++++?@F*$\"3y******H3g\")oF37$$\"39++++++g@F*$\"3/+++UF%)*z\"F-7$$
\"#AF<$\"33+++3`$3*>F-7$$\"3A++++++SAF*$\"3H+++>BU-(*F37$$\"33++++++!G
#F*$!3W+++TSts\")F37$$\"3H++++++?BF*$!3z*****pD!oLCF-7$$\"39++++++gBF*
$!3!)*****f-'[$*HF-7$$\"#CF<$!3%)*****4/ne=#F-7$$\"3A++++++SCF*$!3;+++
;@r'e%F37$$\"33++++++![#F*$\"35+++_GgY7F-7$$\"3H++++++?DF*$\"3=+++t.9
\\?F-7$$\"39++++++gDF*$\"3/+++yO&)Q;F-7$$\"#EF<$\"39+++*))odO%F37$$\"3
A++++++SEF*$!3q*****\\r]N)oF37$$\"33++++++!o#F*$!35+++J=Wv'*F37$$\"3H+
+++++?FF*$!3*)*****H19@8#F37$$\"39++++++gFF*$\"35+++1**)\\1\"F-7$$\"#G
F<$\"33+++.gMx>F-7$$\"3A++++++SGF*$\"3#******pHG=\"=F-7$$\"33++++++!)G
F*$\"3(******f)oS5\\F37$$\"3H++++++?HF*$!3%******>xiFN\"F-7$$\"39+++++
+gHF*$!34+++si_MFF-7$$\"#IF<$!3y*****\\edG*GF-7$$\"3A++++++SIF*$!3,+++
i&e(R<F-7$$\"33++++++!3$F*$\"3$)******>liE()F^\\l7$$\"3H++++++?JF*$\"3
0+++c38-;F-7$$\"39++++++gJF*$\"3-+++<u-\\?F-7$$\"#KF<$\"3*)*****>9T^L
\"F-7$$\"3')************RKF*$\"3/+++!R/1)eF^\\l7$$\"3W++++++!G$F*$!3q*
****f%pL&z)F37$$\"3H++++++?LF*$!3b*****fL5XV)F37$$\"39++++++gLF*$\"3*)
*****Hm'*zU\"F37$$\"#MF<$\"3/+++J(o$49F-7$$\"3')************RMF*$\"3!)
*****4dP%e?F-7$$\"3W++++++![$F*$\"34+++%3u#H:F-7$$\"3H++++++?NF*$!3t**
****RsmENF^\\l7$$\"39++++++gNF*$!31+++()QFZ=F-7$$\"#OF<$!3.+++I+GEHF-7
$$\"3')************ROF*$!3;+++WOEvEF-7$$\"3W++++++!o$F*$!3,+++q#=PB\"F
-7$$\"3H++++++?PF*$\"3Y+++%>k%\\gF37$$\"39++++++gPF*$\"33+++.i)='=F-7$
$\"#QF<$\"3\"******4CJ]%>F-7$$\"3d++++++SQF*$\"3l*****H[&H8)*F37$$\"3W
++++++!)QF*$!3!*********yi#)GF37$$\"3H++++++?RF*$!3a+++rzDH)*F37$$\"39
++++++gRF*$!3w*****RVmuL'F37$$\"#SF<$\"3C+++HnQV_F37$$\"3d++++++SSF*$
\"3'******f%eq)p\"F-7$$\"3W++++++!3%F*$\"3)******>\\pS.#F-7$$\"3H+++++
+?TF*$\"3)******>H`Q:\"F-7$$\"39++++++gTF*$!37+++'4u/%eF37$$\"#UF<$!3!
*******Q&olF#F-7$$\"3d++++++SUF*$!3-+++9[S**HF-7$$\"3W++++++!G%F*$!3=+
++r7_^BF-7$$\"3H++++++?VF*$!3s*****Rkm]#pF37$$\"39++++++gVF*$\"31+++\\
PFq5F-7$$\"#WF<$\"3()*****>\\[k,#F-7$$\"3d++++++SWF*$\"3-+++XcTZ<F-7$$
\"3W++++++![%F*$\"3E+++W(yW+'F37$$\"3H++++++?XF*$!3n*****p%=wLeF37$$\"
39++++++gXF*$!3Y+++VZ0B**F37$$\"#YF<$!3!******R[05^$F37$$\"3d++++++SYF
*$\"3#******f=ur2*F37$$\"3W++++++!o%F*$\"34+++%o2J\">F-7$$\"3H++++++?Z
F*$\"3'******H#p$3!>F-7$$\"39++++++gZF*$\"3%******\\ccU,(F37$$\"#[F<$!
3-+++gb6H6F-7$$\"3d++++++S[F*$!3%)*****z(GZ>EF-7$$\"3W++++++!)[F*$!33+
++X$f-&HF-7$$\"3H++++++?\\F*$!3!******Rj!oP>F-7$$\"39++++++g\\F*$!34++
+3)4OU\"F37$$\"#]F<$\"3'******4*GEi9F-7$$\"3d++++++S]F*$\"37+++qy;i?F-
7$$\"3W++++++!3&F*$\"35+++'zm8Z\"F--%'COLOURG6&%$RGBG$F`[n!\"\"FecmFec
m-%+AXESLABELSG6$Q!6\"Fj[p-%&STYLEG6#%&POINTG-%%VIEWG6$%(DEFAULTGFc\\p
" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "It is hard to see the signal! Ch
oose " }{TEXT 19 10 "style=line" }{TEXT -1 17 " to get the idea." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "N_x:=dx*add(yR[i]^2+yI[i]^2,
i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N_xG$\"+s_*[b#!\"(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalhf(FFT(n, var(yR), var(y
I)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$c#\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "N_p:=dp*add(yR[i]^2+yI[i]^2,i=1..N);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N_pG$\"+G]I.5!\"&" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 68 "This is not normalized correctly, so let us def
ine a normalized DFT:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "y
Rn:=Vector(N): yIn:=Vector(N): sq:=evalf(1/sqrt(2*Pi)): for i from 1 t
o N do: yRn[i]:=dx*sq*yR[i]: yIn[i]:=dx*sq*yI[i]: od:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "N_p:=dp*add(yRn[i]^2+yIn[i]^2,i=1..
N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N_pG$\"+s_*[b#!\"(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Now we understand that the FFT fun
ction carried out the sum, but did not multiply by " }{TEXT 19 2 "dx" 
}{TEXT -1 79 " (to turn it into the Riemann sum), and that it does not
 include the customary " }{TEXT 19 12 "1/sqrt(2*Pi)" }{TEXT -1 8 " fac
tor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 224 "
Observe the efficiency: in one GO the FFT function has calculated N in
tegrals, one for each of the frequency (momentum) values represented o
n the p-mesh. The reason why it is called fast is that the effort does
 not scale as " }{TEXT 19 3 "N^2" }{TEXT -1 59 " as would be normally \+
expected from the DFT, but rather as " }{TEXT 19 8 "N*log(N)" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
28 "Now let us graph the result:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "plot([seq([pv[i],yRn[i]],i=1..N)],style=point);" }}
{PARA 13 "" 1 "" {GLPLOT2D 824 208 208 {PLOTDATA 2 "6&-%'CURVESG6$7\\[
l7$$\"\"!F)$\"3R+++!>n>$f!#=7$$\"3okDa^J#f8'!#>$\"3$*******\\&R\"RfF,7
$$\"3%H^3.j%=F7F,$\"3'******z-Q2'fF,7$$\"3EpFYXpxS=F,$\"3a******e$4q*f
F,7$$\"3)e-<1EpVX#F,$\"3e*****f`s$[gF,7$$\"3[#Grddhz1$F,$\"3M+++^6V:hF
,7$$\"3_Qb#4*Qb\"o$F,$\"3Q+++w8**)>'F,7$$\"37&zzg?Y^H%F,$\"3b*******)o
3+jF,7$$\"3u^SB@&Q(3\\F,$\"3M+++'=7+U'F,7$$\"3N3$)QO3LAbF,$\"3_+++xgOg
lF,7$$\"3'\\cU::Bf8'F,$\"3F+++5)3Js'F,7$$\"3c@opma^\\nF,$\"3`******3ij
5pF,7$$\"31x5&=y2JO(F,$\"3()*****f'p(e7(F,7$$\"3mL`+(4+n(zF,$\"3y*****
\\$3UstF,7$$\"3E!ff@T#H!f)F,$\"3#)*****\\A*oawF,7$$\"3)o%QJFZ)Q?*F,$\"
3%******4%\\;yzF,7$$\"3\\.\"oC/xu\")*F,$\"3u*****\\$eq\\$)F,7$$\"3,OAw
Np5V5!#<$\"3a+++I*zzx)F,7$$\"3nhwFnhY/6F[q$\"3Y*****p?!3u#*F,7$$\"3K(3
$z)RDe;\"F[q$\"3-+++68T_)*F,7$$\"3*H^3.j%=F7F[q$\"3'******z#**>`5F[q7$
$\"3mQR#='Qa)G\"F[q$\"3/+++uz%Q8\"F[q7$$\"3Kk$RL4.*\\8F[q$\"3/+++c7sI7
F[q7$$\"3(**ya[Ki7T\"F[q$\"3-+++eW$)[8F[q7$$\"3U:-Pc:is9F[q$\"3/+++K2a
&\\\"F[q7$$\"32Tc)yy!)R`\"F[q$\"35+++H22#o\"F[q7$$\"3sm5S>+M&f\"F[q$\"
3\"******>*=ZE>F[q7$$\"3S#\\;4D*pc;F[q$\"3#*******e)*pfAF[q7$$\"30=>V#
[e!=<F[q$\"3()*****\\2k'RFF[q7$$\"3sVt%Rr<%z<F[q$\"3!******Rt:()[$F[q7
$$\"3PpFYXpxS=F[q$\"3O+++ha\"y\"[F[q7$$\"3/&>yp<O@!>F[q$\"3F+++_$4y\"y
F[q7$$\"3q?O\\3a\\j>F[q$\"3%)*****H3$Q\"4#!#;7$$\"39Y!4+ka[-#F[q$!31++
+QHjmIFiu7$$\"3-sW_rQ@'3#F[q$!3H+++We_I))F[q7$$\"3Y(*)RI5tv9#F[q$!31++
+C]Cb^F[q7$$\"3MB`bMB$*3AF[q$!3*******\\_w-k$F[q7$$\"3y[22m:HqAF[q$!3y
*****HSDV\"GF[q7$$\"3muhe(z];L#F[q$!3-+++2a#\\H#F[q7$$\"35+;5H+,$R#F[q
$!3)******HWk%Q>F[q7$$\"3)f-<1EpVX#F[q$!3/+++sp!*y;F[q7$$\"3U^C8#\\Gd^
#F[q$!3#******>CX;[\"F[q7$$\"3IxykBx3xDF[q$!34+++48zE8F[q7$$\"3u-L;bpW
QEF[q$!3(*******)f.@?\"F[q7$$\"3iG(ym=1)*p#F[q$!3++++rYj*4\"F[q7$$\"31
aT>=a;hFF[q$!3/+++w.+95F[q7$$\"3&*z&4(\\Y_AGF[q$!3a*****HLPUT*F,7$$\"3
Q0]A\")Q)Q)GF[q$!3s*****\\\\q<z)F,7$$\"3#3VSF6V_%HF[q$!37+++3KT_#)F,7$
$\"3rceDWBg1IF[q$!3O+++r@!4y(F,7$$\"39#Grddhz1$F[q$!3))*****4I&[ltF,7$
$\"3.3nG23KHJF[q$!3_+++.w'p*pF,7$$\"3YL@!)Q+o!>$F[q$!3Y*****pJ_!omF,7$
$\"3OfvJq#R?D$F[q$!3a*****fNvGP'F,7$$\"3z%)H$=])R8LF[q$!37+++)zqm5'F,7
$$\"3m5%[LtdZP$F[q$!3G+++!fDb'eF,7$$\"36OQ'['p6OMF[q$!3o*****R?(>YcF,7
$$\"3+i#zj>wu\\$F[q$!3?+++pi(fW&F,7$$\"3W(o%*yUN)eNF[q$!37+++/:ei_F,7$
$\"3J8,TfY>?OF[q$!3%)*****HjyS4&F,7$$\"3uQb#4*Qb\"o$F[q$!3,+++\\p\")Q
\\F,7$$\"3kk4WAJ\"Hu$F[q$!3E+++q$z`z%F,7$$\"32!RcRNsU!QF[q$!39+++KFaiY
F,7$$\"3'f\"=Z&eJc'QF[q$!3y*****4uX#RXF,7$$\"3RTs)p\"3*p#RF[q$!3?+++TK
cCWF,7$$\"3%om-&[+N))RF[q$!3F+++/^o<VF,7$$\"3G#4=+G4(\\SF[q$!3<+++5\"*
*y@%F,7$$\"3:=N`6&o56%F[q$!3:+++%\\wX7%F,7$$\"3/W*[IuFC<%F[q$!38+++G.;
PSF,7$$\"3#*pVcupyLUF[q$!3!******H^b^&RF,7$$\"3#\\zzg?Y^H%F[q$!3'*****
*pZ?\"yQF,7$$\"3!3A&fPa]cVF[q$!3)*******R.m0QF,7$$\"3oY16pY'yT%F[q$!3&
******>w?ut$F,7$$\"3csgi+RAzWF[q$!3%******f2$3tOF,7$$\"3c(\\T@8$eSXF[q
$!3;+++\"))fBh$F,7$$\"3WBpljB%>g%F[q$!3w*****\\S\"*\\b$F,7$$\"3K\\B<&f
,Lm%F[q$!32+++AAu+NF,7$$\"3@vxoE3mCZF[q$!3++++o&)R\\MF,7$$\"3?+K?e+-'y
%F[q$!3')*****R'fw+MF,7$$\"33E'=(*Gzt%[F[q$!3A+++*3nYN$F,7$$\"3(>0M7_Q
(3\\F[q$!3C+++c*R4J$F,7$$\"3'pZ\\Fv(4q\\F[q$!3')*****pMO%pKF,7$$\"3%G!
\\E%)pXJ]F[q$!35+++O/-IKF,7$$\"3sG.y:i\"G4&F[q$!3/+++$[nD>$F,7$$\"3iad
HZa<a^F[q$!3!)*****>(G'p:$F,7$$\"3gz6\")yY`:_F[q$!3F+++**45BJF,7$$\"3[
0mK5R*oF&F[q$!3#******4b%)34$F,7$$\"3PJ?%=9`#Q`F[q$!3')*****HnB-1$F,7$
$\"3CduNtBh*R&F[q$!3++++>`.JIF,7$$\"3C#)G([gr4Y&F[q$!3!)******HEC.IF,7
$$\"383$)QO3LAbF[q$!3')******HWxwHF,7$$\"3-MP!z1!p$e&F[q$!3A+++HZc^HF,
7$$\"3!*f\">%*H\\]k&F[q$!3A+++XAbFHF,7$$\"3*[eM4`3kq&F[q$!3<+++H,o/HF,
7$$\"3y5+XixwndF[q$!37+++Fa*G)GF,7$$\"3lOa'R*p7HeF[q$!3A+++j*[@'GF,7$$
\"3kh3[Di[!*eF[q$!3=+++J\\RUGF,7$$\"3a(G'*pXX=&fF[q$!3;+++$o!fBGF,7$$
\"3T8<^)o/K,'F[q$!30+++Lkp0GF,7$$\"3IRr-?RcugF[q$!33+++q^n)y#F,7$$\"3H
kDa^J#f8'F[q$!3++++7B\\sFF,7$$\"3;!*z0$Q#G(>'F[q$!3'******ph:rv#F,7$$
\"31;Md9;keiF[q$!3%)*****R0:Du#F,7$$\"3%>%))3Y3+?jF[q$!3u******fDmGFF,
7$$\"3%pE/w2g8Q'F[q$!3/+++\")>`:FF,7$$\"3\"Gp>\"4$>FW'F[q$!3:+++H*)4.F
F,7$$\"3q=^jS&yS]'F[q$!3C+++'pS8p#F,7$$\"3eW0:sxVllF[q$!3%)*****\\0O-o
#F,7$$\"3epfm.qzEmF[q$!3G+++=`wpEF,7$$\"3Y&R\"=Ni:)o'F[q$!3')*****>95*
fEF,7$$\"3M@opma^\\nF[q$!3-+++.Nl]EF,7$$\"3BZA@)pu3\"oF[q$!3>+++\"ez>k
#F,7$$\"3AswsHRBsoF[q$!3))*****4ttQj#F,7$$\"35)4V7;$fLpF[q$!3:+++SCKEE
F,7$$\"3)R_eFR_\\*pF[q$!3;+++EKJ>EF,7$$\"3)*[RFC;JcqF[q$!3#******HgMGh
#F,7$$\"3'[P*yb3n<rF[q$!3\"******R7wog#F,7$$\"3u+[I(3I!zrF[q$!3$)*****
z8G9g#F,7$$\"3jE-#)=$*QSsF[q$!3)******p+#['f#F,7$$\"3i^cL]&[<I(F[q$!3)
******p()H?f#F,7$$\"3]x5&=y2JO(F[q$!3y*****Huk!)e#F,7$$\"3R.lO8qYCuF[q
$!3#)*****zT!e%e#F,7$$\"3GH>)[CEe[(F[q$!37+++h9d\"e#F,7$$\"3EatRwa=ZvF
[q$!3!*******>K.zDF,7$$\"3:!y7zqW&3wF[q$!3y*****4xhpd#F,7$$\"3-1#G%RR!
*pwF[q$!3u******>RNvDF,7$$\"3#>jV4<j7t(F[q$!3!*******>s?uDF,7$$\"3!p0f
CSAEz(F[q$!3)******f*)>Nd#F,7$$!3y#[uRj\")R&yF[q$!3/+++'*3HtDF,7$$!3!p
0fCSAEz(F[qFccm7$$!3#>jV4<j7t(F[qF^cm7$$!3-1#G%RR!*pwF[qFibm7$$!3:!y7z
qW&3wF[qFdbm7$$!3EatRwa=ZvF[qF_bm7$$!3GH>)[CEe[(F[qFjam7$$!3R.lO8qYCuF
[qFeam7$$!3]x5&=y2JO(F[qF`am7$$!3i^cL]&[<I(F[qF[am7$$!3jE-#)=$*QSsF[qF
f`m7$$!3u+[I(3I!zrF[qFa`m7$$!3'[P*yb3n<rF[qF\\`m7$$!3)*[RFC;JcqF[qFg_m
7$$!3)R_eFR_\\*pF[qFb_m7$$!35)4V7;$fLpF[qF]_m7$$!3AswsHRBsoF[qFh^m7$$!
3BZA@)pu3\"oF[qFc^m7$$!3M@opma^\\nF[qF^^m7$$!3Y&R\"=Ni:)o'F[qFi]m7$$!3
epfm.qzEmF[qFd]m7$$!3eW0:sxVllF[qF_]m7$$!3q=^jS&yS]'F[qFj\\m7$$!3\"Gp>
\"4$>FW'F[qFe\\m7$$!3%pE/w2g8Q'F[qF`\\m7$$!3%>%))3Y3+?jF[qF[\\m7$$!31;
Md9;keiF[qFf[m7$$!3;!*z0$Q#G(>'F[qFa[m7$$!3HkDa^J#f8'F[qF\\[m7$$!3IRr-
?RcugF[qFgjl7$$!3T8<^)o/K,'F[qFbjl7$$!3a(G'*pXX=&fF[qF]jl7$$!3kh3[Di[!
*eF[qFhil7$$!3lOa'R*p7HeF[qFcil7$$!3y5+XixwndF[qF^il7$$!3*[eM4`3kq&F[q
Fihl7$$!3!*f\">%*H\\]k&F[qFdhl7$$!3-MP!z1!p$e&F[qF_hl7$$!383$)QO3LAbF[
qFjgl7$$!3C#)G([gr4Y&F[qFegl7$$!3CduNtBh*R&F[qF`gl7$$!3PJ?%=9`#Q`F[qF[
gl7$$!3[0mK5R*oF&F[qFffl7$$!3gz6\")yY`:_F[qFafl7$$!3iadHZa<a^F[qF\\fl7
$$!3sG.y:i\"G4&F[qFgel7$$!3%G!\\E%)pXJ]F[qFbel7$$!3'pZ\\Fv(4q\\F[qF]el
7$$!3(>0M7_Q(3\\F[qFhdl7$$!33E'=(*Gzt%[F[qFcdl7$$!3?+K?e+-'y%F[qF^dl7$
$!3@vxoE3mCZF[qFicl7$$!3K\\B<&f,Lm%F[qFdcl7$$!3WBpljB%>g%F[qF_cl7$$!3c
(\\T@8$eSXF[qFjbl7$$!3csgi+RAzWF[qFebl7$$!3oY16pY'yT%F[qF`bl7$$!3!3A&f
Pa]cVF[qF[bl7$$!3#\\zzg?Y^H%F[qFfal7$$!3#*pVcupyLUF[qFaal7$$!3/W*[IuFC
<%F[qF\\al7$$!3:=N`6&o56%F[qFg`l7$$!3G#4=+G4(\\SF[qFb`l7$$!3%om-&[+N))
RF[qF]`l7$$!3RTs)p\"3*p#RF[qFh_l7$$!3'f\"=Z&eJc'QF[qFc_l7$$!32!RcRNsU!
QF[qF^_l7$$!3kk4WAJ\"Hu$F[qFi^l7$$!3uQb#4*Qb\"o$F[qFd^l7$$!3J8,TfY>?OF
[qF_^l7$$!3W(o%*yUN)eNF[qFj]l7$$!3+i#zj>wu\\$F[qFe]l7$$!36OQ'['p6OMF[q
F`]l7$$!3m5%[LtdZP$F[qF[]l7$$!3z%)H$=])R8LF[qFf\\l7$$!3OfvJq#R?D$F[qFa
\\l7$$!3YL@!)Q+o!>$F[qF\\\\l7$$!3.3nG23KHJF[qFg[l7$$!39#Grddhz1$F[qFb[
l7$$!3rceDWBg1IF[qF][l7$$!3#3VSF6V_%HF[qFhz7$$!3Q0]A\")Q)Q)GF[qFcz7$$!
3&*z&4(\\Y_AGF[qF^z7$$!31aT>=a;hFF[qFiy7$$!3iG(ym=1)*p#F[qFdy7$$!3u-L;
bpWQEF[qF_y7$$!3IxykBx3xDF[qFjx7$$!3U^C8#\\Gd^#F[qFex7$$!3)f-<1EpVX#F[
qF`x7$$!35+;5H+,$R#F[qF[x7$$!3muhe(z];L#F[qFfw7$$!3y[22m:HqAF[qFaw7$$!
3MB`bMB$*3AF[qF\\w7$$!3Y(*)RI5tv9#F[qFgv7$$!3-sW_rQ@'3#F[qFbv7$$!39Y!4
+ka[-#F[qF]v7$$!3q?O\\3a\\j>F[qFgu7$$!3/&>yp<O@!>F[qFbu7$$!3PpFYXpxS=F
[qF]u7$$!3sVt%Rr<%z<F[qFht7$$!30=>V#[e!=<F[qFct7$$!3S#\\;4D*pc;F[qF^t7
$$!3sm5S>+M&f\"F[qFis7$$!32Tc)yy!)R`\"F[qFds7$$!3U:-Pc:is9F[qF_s7$$!3(
**ya[Ki7T\"F[qFjr7$$!3Kk$RL4.*\\8F[qFer7$$!3mQR#='Qa)G\"F[qF`r7$$!3*H^
3.j%=F7F[qF[r7$$!3K(3$z)RDe;\"F[qFfq7$$!3nhwFnhY/6F[qFaq7$$!3,OAwNp5V5
F[qF\\q7$$!3\\.\"oC/xu\")*F,Ffp7$$!3)o%QJFZ)Q?*F,Fap7$$!3E!ff@T#H!f)F,
F\\p7$$!3mL`+(4+n(zF,Fgo7$$!31x5&=y2JO(F,Fbo7$$!3c@opma^\\nF,F]o7$$!3'
\\cU::Bf8'F,Fhn7$$!3N3$)QO3LAbF,FY7$$!3u^SB@&Q(3\\F,FT7$$!37&zzg?Y^H%F
,FO7$$!3_Qb#4*Qb\"o$F,FJ7$$!3[#Grddhz1$F,FE7$$!3)e-<1EpVX#F,F@7$$!3EpF
YXpxS=F,F;7$$!3%H^3.j%=F7F,F67$$!3okDa^J#f8'F0F1-%'COLOURG6&%$RGBG$\"#
5!\"\"F(F(-%+AXESLABELSG6$Q!6\"Fa\\o-%&STYLEG6#%&POINTG-%%VIEWG6$%(DEF
AULTGFj\\o" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot([seq([pv[i]
,yIn[i]],i=1..N)],style=point,color=blue);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 824 208 208 {PLOTDATA 2 "6'-%'CURVESG6#7\\[l7$$\"\"!F)F(7$$
\"3okDa^J#f8'!#>$!3=+++Sr_0SF-7$$\"3%H^3.j%=F7!#=$!3Y*****H-6<5)F-7$$
\"3EpFYXpxS=F3$!3%******4QF'Q7F37$$\"3)e-<1EpVX#F3$!3%******4xDsp\"F37
$$\"3[#Grddhz1$F3$!33+++!G,**>#F37$$\"3_Qb#4*Qb\"o$F3$!3-+++=X$[w#F37$
$\"37&zzg?Y^H%F3$!3.+++6B,<MF37$$\"3u^SB@&Q(3\\F3$!35+++!GjD>%F37$$\"3
N3$)QO3LAbF3$!3W+++>\\fY^F37$$\"3'\\cU::Bf8'F3$!39+++jPWojF37$$\"3c@op
ma^\\nF3$!3o*****p(4:9!)F37$$\"31x5&=y2JO(F3$!3!******4_B%Q5!#<7$$\"3m
L`+(4+n(zF3$!3!******Rb$H99Fbo7$$\"3E!ff@T#H!f)F3$!3')*****Ha(*46#Fbo7
$$\"3)o%QJFZ)Q?*F3$!3++++r\"p8(QFbo7$$\"3\\.\"oC/xu\")*F3$!3%******pb^
Eu\"!#;7$$\"3,OAwNp5V5Fbo$\"3i*******H3<f(Fbo7$$\"3nhwFnhY/6Fbo$\"3$**
****z\"eZ9KFbo7$$\"3K(3$z)RDe;\"Fbo$\"3)******HXsI2#Fbo7$$\"3*H^3.j%=F
7Fbo$\"31+++Gn$ea\"Fbo7$$\"3mQR#='Qa)G\"Fbo$\"30+++J-4T7Fbo7$$\"3Kk$RL
4.*\\8Fbo$\"33+++8I)=/\"Fbo7$$\"3(**ya[Ki7T\"Fbo$\"3o******33^5!*F37$$
\"3U:-Pc:is9Fbo$\"3[*****\\xm\"fzF37$$\"32Tc)yy!)R`\"Fbo$\"38+++=^?UrF
37$$\"3sm5S>+M&f\"Fbo$\"3c*****4cLv['F37$$\"3S#\\;4D*pc;Fbo$\"3o*****4
yr*\\fF37$$\"30=>V#[e!=<Fbo$\"3A+++kjv*\\&F37$$\"3sVt%Rr<%z<Fbo$\"3V++
+h\\Z;^F37$$\"3PpFYXpxS=Fbo$\"3y*****\\Jccy%F37$$\"3/&>yp<O@!>Fbo$\"3%
)*****>[^n\\%F37$$\"3q?O\\3a\\j>Fbo$\"3/+++tj)=C%F37$$\"39Y!4+ka[-#Fbo
$\"3!******>7i],%F37$$\"3-sW_rQ@'3#Fbo$\"3%*******fki6QF37$$\"3Y(*)RI5
tv9#Fbo$\"3A+++@X\"zi$F37$$\"3MB`bMB$*3AFbo$\"3!******H!*)*4Y$F37$$\"3
y[22m:HqAFbo$\"36+++\"=7&3LF37$$\"3muhe(z];L#Fbo$\"31+++h$=&oJF37$$\"3
5+;5H+,$R#Fbo$\"3$******HI=%RIF37$$\"3)f-<1EpVX#Fbo$\"3F+++n*y)>HF37$$
\"3U^C8#\\Gd^#Fbo$\"3&)******3*z(3GF37$$\"3IxykBx3xDFbo$\"3x******p@<0
FF37$$\"3u-L;bpWQEFbo$\"3#******fQY#3EF37$$\"3iG(ym=1)*p#Fbo$\"30+++#y
2t^#F37$$\"31aT>=a;hFFbo$\"31+++6kvJCF37$$\"3&*z&4(\\Y_AGFbo$\"37+++_8
2^BF37$$\"3Q0]A\")Q)Q)GFbo$\"3/+++#)yzuAF37$$\"3#3VSF6V_%HFbo$\"32+++$
GPD?#F37$$\"3rceDWBg1IFbo$\"37+++Z#QR8#F37$$\"39#Grddhz1$Fbo$\"3#*****
*H/!po?F37$$\"3.3nG23KHJFbo$\"33+++(o;l+#F37$$\"3YL@!)Q+o!>$Fbo$\"3!**
*****o@<Z>F37$$\"3OfvJq#R?D$Fbo$\"33+++.kV!*=F37$$\"3z%)H$=])R8LFbo$\"
3++++z=6O=F37$$\"3m5%[LtdZP$Fbo$\"3++++\"z?Sy\"F37$$\"36OQ'['p6OMFbo$
\"38+++oD+M<F37$$\"3+i#zj>wu\\$Fbo$\"3))*****4\"=\"fo\"F37$$\"3W(o%*yU
N)eNFbo$\"3!******R`;'R;F37$$\"3J8,TfY>?OFbo$\"3)******zf'*\\f\"F37$$
\"3uQb#4*Qb\"o$Fbo$\"3!******RSU>b\"F37$$\"3kk4WAJ\"Hu$Fbo$\"3/+++qON5
:F37$$\"32!RcRNsU!QFbo$\"33+++o&Q,Z\"F37$$\"3'f\"=Z&eJc'QFbo$\"3&*****
***p77V\"F37$$\"3RTs)p\"3*p#RFbo$\"3))*****\\U)\\$R\"F37$$\"3%om-&[+N)
)RFbo$\"33+++LT#pN\"F37$$\"3G#4=+G4(\\SFbo$\"3&******\\pB9K\"F37$$\"3:
=N`6&o56%Fbo$\"33+++:f$pG\"F37$$\"3/W*[IuFC<%Fbo$\"3#******p./MD\"F37$
$\"3#*pVcupyLUFbo$\"3&******HSv2A\"F37$$\"3#\\zzg?Y^H%Fbo$\"31+++$*4+*
=\"F37$$\"3!3A&fPa]cVFbo$\"3/+++)>N!e6F37$$\"3oY16pY'yT%Fbo$\"3(******
zVNy7\"F37$$\"3csgi+RAzWFbo$\"3)*******H>O)4\"F37$$\"3c(\\T@8$eSXFbo$
\"33+++#\\x&p5F37$$\"3WBpljB%>g%Fbo$\"3/+++jsWT5F37$$\"3K\\B<&f,Lm%Fbo
$\"30+++b&QR,\"F37$$\"3@vxoE3mCZFbo$\"3%)******4o?q)*F-7$$\"3?+K?e+-'y
%Fbo$\"3R*****4YZmg*F-7$$\"33E'=(*Gzt%[Fbo$\"3o*****pbN%[$*F-7$$\"3(>0
M7_Q(3\\Fbo$\"3O*****zC:`4*F-7$$\"3'pZ\\Fv(4q\\Fbo$\"3C+++dT/Z))F-7$$
\"3%G!\\E%)pXJ]Fbo$\"3@+++nUR.')F-7$$\"3sG.y:i\"G4&Fbo$\"3a******y*[TO
)F-7$$\"3iadHZa<a^Fbo$\"3l+++7P5H\")F-7$$\"3gz6\")yY`:_Fbo$\"3_*****zS
k!)*yF-7$$\"3[0mK5R*oF&Fbo$\"3[*****>*o%3n(F-7$$\"3PJ?%=9`#Q`Fbo$\"3m*
****43wsW(F-7$$\"3CduNtBh*R&Fbo$\"3f+++zb=FsF-7$$\"3C#)G([gr4Y&Fbo$\"3
s*****Hv;/,(F-7$$\"383$)QO3LAbFbo$\"3w*****\\w=oz'F-7$$\"3-MP!z1!p$e&F
bo$\"3/+++XtC'e'F-7$$\"3!*f\">%*H\\]k&Fbo$\"3o+++4]cyjF-7$$\"3*[eM4`3k
q&Fbo$\"3I+++***RO<'F-7$$\"3y5+XixwndFbo$\"3I+++HmMrfF-7$$\"3lOa'R*p7H
eFbo$\"3I+++kScrdF-7$$\"3kh3[Di[!*eFbo$\"3/+++-l<ubF-7$$\"3a(G'*pXX=&f
Fbo$\"33+++;J2z`F-7$$\"3T8<^)o/K,'Fbo$\"3))*****44Zh=&F-7$$\"3IRr-?Rcu
gFbo$\"3s*****Rm&H&*\\F-7$$\"3HkDa^J#f8'Fbo$\"3o******p+U1[F-7$$\"3;!*
z0$Q#G(>'Fbo$\"33+++^[U>YF-7$$\"31;Md9;keiFbo$\"3=+++M\"=UV%F-7$$\"3%>
%))3Y3+?jFbo$\"3!******R<62D%F-7$$\"3%pE/w2g8Q'Fbo$\"3A+++Xw\")oSF-7$$
\"3\"Gp>\"4$>FW'Fbo$\"3C+++zXX))QF-7$$\"3q=^jS&yS]'Fbo$\"3I+++'GT&4PF-
7$$\"3eW0:sxVllFbo$\"3q*****HY**>`$F-7$$\"3epfm.qzEmFbo$\"3A++++IvbLF-
7$$\"3Y&R\"=Ni:)o'Fbo$\"3/+++%3G2=$F-7$$\"3M@opma^\\nFbo$\"3$******HY_
o+$F-7$$\"3BZA@)pu3\"oFbo$\"3.+++@h0MGF-7$$\"3AswsHRBsoFbo$\"35+++M/Fi
EF-7$$\"35)4V7;$fLpFbo$\"3!******\\FG9\\#F-7$$\"3)R_eFR_\\*pFbo$\"3***
****4Pk9K#F-7$$\"3)*[RFC;JcqFbo$\"31+++SUJ_@F-7$$\"3'[P*yb3n<rFbo$\"3/
+++R^\"R)>F-7$$\"3u+[I(3I!zrFbo$\"3)******p\"\\?;=F-7$$\"3jE-#)=$*QSsF
bo$\"3=+++[G7\\;F-7$$\"3i^cL]&[<I(Fbo$\"3'******\\**3E[\"F-7$$\"3]x5&=
y2JO(Fbo$\"3*******>,/mJ\"F-7$$\"3R.lO8qYCuFbo$\"35+++e(\\5:\"F-7$$\"3
GH>)[CEe[(Fbo$\"3;+++Ng))e)*!#?7$$\"3EatRwa=ZvFbo$\"3$******4\"*H1@)Fj
am7$$\"3:!y7zqW&3wFbo$\"3!******\\nl^c'Fjam7$$\"3-1#G%RR!*pwFbo$\"3=++
+K#H>#\\Fjam7$$\"3#>jV4<j7t(Fbo$\"3/+++E0O!G$Fjam7$$\"3!p0fCSAEz(Fbo$
\"30+++$\\.*R;Fjam7$$!3y#[uRj\")R&yFboF(7$$!3!p0fCSAEz(Fbo$!30+++$\\.*
R;Fjam7$$!3#>jV4<j7t(Fbo$!3/+++E0O!G$Fjam7$$!3-1#G%RR!*pwFbo$!3=+++K#H
>#\\Fjam7$$!3:!y7zqW&3wFbo$!3!******\\nl^c'Fjam7$$!3EatRwa=ZvFbo$!3$**
****4\"*H1@)Fjam7$$!3GH>)[CEe[(Fbo$!3;+++Ng))e)*Fjam7$$!3R.lO8qYCuFbo$
!35+++e(\\5:\"F-7$$!3]x5&=y2JO(Fbo$!3*******>,/mJ\"F-7$$!3i^cL]&[<I(Fb
o$!3'******\\**3E[\"F-7$$!3jE-#)=$*QSsFbo$!3=+++[G7\\;F-7$$!3u+[I(3I!z
rFbo$!3)******p\"\\?;=F-7$$!3'[P*yb3n<rFbo$!3/+++R^\"R)>F-7$$!3)*[RFC;
JcqFbo$!31+++SUJ_@F-7$$!3)R_eFR_\\*pFbo$!3*******4Pk9K#F-7$$!35)4V7;$f
LpFbo$!3!******\\FG9\\#F-7$$!3AswsHRBsoFbo$!35+++M/FiEF-7$$!3BZA@)pu3
\"oFbo$!3.+++@h0MGF-7$$!3M@opma^\\nFbo$!3$******HY_o+$F-7$$!3Y&R\"=Ni:
)o'Fbo$!3/+++%3G2=$F-7$$!3epfm.qzEmFbo$!3A++++IvbLF-7$$!3eW0:sxVllFbo$
!3q*****HY**>`$F-7$$!3q=^jS&yS]'Fbo$!3I+++'GT&4PF-7$$!3\"Gp>\"4$>FW'Fb
o$!3C+++zXX))QF-7$$!3%pE/w2g8Q'Fbo$!3A+++Xw\")oSF-7$$!3%>%))3Y3+?jFbo$
!3!******R<62D%F-7$$!31;Md9;keiFbo$!3=+++M\"=UV%F-7$$!3;!*z0$Q#G(>'Fbo
$!33+++^[U>YF-7$$!3HkDa^J#f8'Fbo$!3o******p+U1[F-7$$!3IRr-?RcugFbo$!3s
*****Rm&H&*\\F-7$$!3T8<^)o/K,'Fbo$!3))*****44Zh=&F-7$$!3a(G'*pXX=&fFbo
$!33+++;J2z`F-7$$!3kh3[Di[!*eFbo$!3/+++-l<ubF-7$$!3lOa'R*p7HeFbo$!3I++
+kScrdF-7$$!3y5+XixwndFbo$!3I+++HmMrfF-7$$!3*[eM4`3kq&Fbo$!3I+++***RO<
'F-7$$!3!*f\">%*H\\]k&Fbo$!3o+++4]cyjF-7$$!3-MP!z1!p$e&Fbo$!3/+++XtC'e
'F-7$$!383$)QO3LAbFbo$!3w*****\\w=oz'F-7$$!3C#)G([gr4Y&Fbo$!3s*****Hv;
/,(F-7$$!3CduNtBh*R&Fbo$!3f+++zb=FsF-7$$!3PJ?%=9`#Q`Fbo$!3m*****43wsW(
F-7$$!3[0mK5R*oF&Fbo$!3[*****>*o%3n(F-7$$!3gz6\")yY`:_Fbo$!3_*****zSk!
)*yF-7$$!3iadHZa<a^Fbo$!3l+++7P5H\")F-7$$!3sG.y:i\"G4&Fbo$!3a******y*[
TO)F-7$$!3%G!\\E%)pXJ]Fbo$!3@+++nUR.')F-7$$!3'pZ\\Fv(4q\\Fbo$!3C+++dT/
Z))F-7$$!3(>0M7_Q(3\\Fbo$!3O*****zC:`4*F-7$$!33E'=(*Gzt%[Fbo$!3o*****p
bN%[$*F-7$$!3?+K?e+-'y%Fbo$!3R*****4YZmg*F-7$$!3@vxoE3mCZFbo$!3%)*****
*4o?q)*F-7$$!3K\\B<&f,Lm%Fbo$!30+++b&QR,\"F37$$!3WBpljB%>g%Fbo$!3/+++j
sWT5F37$$!3c(\\T@8$eSXFbo$!33+++#\\x&p5F37$$!3csgi+RAzWFbo$!3)*******H
>O)4\"F37$$!3oY16pY'yT%Fbo$!3(******zVNy7\"F37$$!3!3A&fPa]cVFbo$!3/+++
)>N!e6F37$$!3#\\zzg?Y^H%Fbo$!31+++$*4+*=\"F37$$!3#*pVcupyLUFbo$!3&****
**HSv2A\"F37$$!3/W*[IuFC<%Fbo$!3#******p./MD\"F37$$!3:=N`6&o56%Fbo$!33
+++:f$pG\"F37$$!3G#4=+G4(\\SFbo$!3&******\\pB9K\"F37$$!3%om-&[+N))RFbo
$!33+++LT#pN\"F37$$!3RTs)p\"3*p#RFbo$!3))*****\\U)\\$R\"F37$$!3'f\"=Z&
eJc'QFbo$!3&********p77V\"F37$$!32!RcRNsU!QFbo$!33+++o&Q,Z\"F37$$!3kk4
WAJ\"Hu$Fbo$!3/+++qON5:F37$$!3uQb#4*Qb\"o$Fbo$!3!******RSU>b\"F37$$!3J
8,TfY>?OFbo$!3)******zf'*\\f\"F37$$!3W(o%*yUN)eNFbo$!3!******R`;'R;F37
$$!3+i#zj>wu\\$Fbo$!3))*****4\"=\"fo\"F37$$!36OQ'['p6OMFbo$!38+++oD+M<
F37$$!3m5%[LtdZP$Fbo$!3++++\"z?Sy\"F37$$!3z%)H$=])R8LFbo$!3++++z=6O=F3
7$$!3OfvJq#R?D$Fbo$!33+++.kV!*=F37$$!3YL@!)Q+o!>$Fbo$!3!*******o@<Z>F3
7$$!3.3nG23KHJFbo$!33+++(o;l+#F37$$!39#Grddhz1$Fbo$!3#******H/!po?F37$
$!3rceDWBg1IFbo$!37+++Z#QR8#F37$$!3#3VSF6V_%HFbo$!32+++$GPD?#F37$$!3Q0
]A\")Q)Q)GFbo$!3/+++#)yzuAF37$$!3&*z&4(\\Y_AGFbo$!37+++_82^BF37$$!31aT
>=a;hFFbo$!31+++6kvJCF37$$!3iG(ym=1)*p#Fbo$!30+++#y2t^#F37$$!3u-L;bpWQ
EFbo$!3#******fQY#3EF37$$!3IxykBx3xDFbo$!3x******p@<0FF37$$!3U^C8#\\Gd
^#Fbo$!3&)******3*z(3GF37$$!3)f-<1EpVX#Fbo$!3F+++n*y)>HF37$$!35+;5H+,$
R#Fbo$!3$******HI=%RIF37$$!3muhe(z];L#Fbo$!31+++h$=&oJF37$$!3y[22m:HqA
Fbo$!36+++\"=7&3LF37$$!3MB`bMB$*3AFbo$!3!******H!*)*4Y$F37$$!3Y(*)RI5t
v9#Fbo$!3A+++@X\"zi$F37$$!3-sW_rQ@'3#Fbo$!3%*******fki6QF37$$!39Y!4+ka
[-#Fbo$!3!******>7i],%F37$$!3q?O\\3a\\j>Fbo$!3/+++tj)=C%F37$$!3/&>yp<O
@!>Fbo$!3%)*****>[^n\\%F37$$!3PpFYXpxS=Fbo$!3y*****\\Jccy%F37$$!3sVt%R
r<%z<Fbo$!3V+++h\\Z;^F37$$!30=>V#[e!=<Fbo$!3A+++kjv*\\&F37$$!3S#\\;4D*
pc;Fbo$!3o*****4yr*\\fF37$$!3sm5S>+M&f\"Fbo$!3c*****4cLv['F37$$!32Tc)y
y!)R`\"Fbo$!38+++=^?UrF37$$!3U:-Pc:is9Fbo$!3[*****\\xm\"fzF37$$!3(**ya
[Ki7T\"Fbo$!3o******33^5!*F37$$!3Kk$RL4.*\\8Fbo$!33+++8I)=/\"Fbo7$$!3m
QR#='Qa)G\"Fbo$!30+++J-4T7Fbo7$$!3*H^3.j%=F7Fbo$!31+++Gn$ea\"Fbo7$$!3K
(3$z)RDe;\"Fbo$!3)******HXsI2#Fbo7$$!3nhwFnhY/6Fbo$!3$******z\"eZ9KFbo
7$$!3,OAwNp5V5Fbo$!3i*******H3<f(Fbo7$$!3\\.\"oC/xu\")*F3$\"3%******pb
^Eu\"Fgp7$$!3)o%QJFZ)Q?*F3$\"3++++r\"p8(QFbo7$$!3E!ff@T#H!f)F3$\"3')**
***Ha(*46#Fbo7$$!3mL`+(4+n(zF3$\"3!******Rb$H99Fbo7$$!31x5&=y2JO(F3$\"
3!******4_B%Q5Fbo7$$!3c@opma^\\nF3$\"3o*****p(4:9!)F37$$!3'\\cU::Bf8'F
3$\"39+++jPWojF37$$!3N3$)QO3LAbF3$\"3W+++>\\fY^F37$$!3u^SB@&Q(3\\F3$\"
35+++!GjD>%F37$$!37&zzg?Y^H%F3$\"3.+++6B,<MF37$$!3_Qb#4*Qb\"o$F3$\"3-+
++=X$[w#F37$$!3[#Grddhz1$F3$\"33+++!G,**>#F37$$!3)e-<1EpVX#F3$\"3%****
**4xDsp\"F37$$!3EpFYXpxS=F3$\"3%******4QF'Q7F37$$!3%H^3.j%=F7F3$\"3Y**
***H-6<5)F-7$$!3okDa^J#f8'F-$\"3=+++Sr_0SF--%+AXESLABELSG6$Q!6\"Fe[p-%
'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%&STYLEG6#%&POINTG-%%VIEWG6$%(DEFAU
LTGFe\\p" 1 5 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot([seq([pv[i],y
Rn[i]^2+yIn[i]^2],i=1..N)],style=point,color=black);" }}{PARA 13 "" 1 
"" {GLPLOT2D 824 208 208 {PLOTDATA 2 "6'-%'CURVESG6#7\\[l7$$\"\"!F)$\"
3<+++uM#)=N!#=7$$\"3okDa^J#f8'!#>$\"3*)*****p5#QVNF,7$$\"3%H^3.j%=F7F,
$\"3#)*****f]x'=OF,7$$\"3EpFYXpxS=F,$\"3++++/>$)\\PF,7$$\"3)e-<1EpVX#F
,$\"3!******\\cQj%RF,7$$\"3[#Grddhz1$F,$\"3v******zj!QA%F,7$$\"3_Qb#4*
Qb\"o$F,$\"3()******)R!=2YF,7$$\"37&zzg?Y^H%F,$\"30+++&z1n8&F,7$$\"3u^
SB@&Q(3\\F,$\"3Q+++LVTzeF,7$$\"3N3$)QO3LAbF,$\"3w*****>B%e_pF,7$$\"3'
\\cU::Bf8'F,$\"3++++.osv&)F,7$$\"3c@opma^\\nF,$\"3,+++(3N)>6!#<7$$\"31
x5&=y2JO(F,$\"3-+++MY5'e\"F_o7$$\"3mL`+(4+n(zF,$\"3=+++Z@vVDF_o7$$\"3E
!ff@T#H!f)F,$\"3L+++)H`A/&F_o7$$\"3)o%QJFZ)Q?*F,$\"3%******>/,Cc\"!#;7
$$\"3\\.\"oC/xu\")*F,$\"3u*****z?1Q/$!#:7$$\"3,OAwNp5V5F_o$\"3$)*****H
Uc/%eFdp7$$\"3nhwFnhY/6F_o$\"3%******>/%H>6Fdp7$$\"3K(3$z)RDe;\"F_o$\"
3K+++\"[H$o_F_o7$$\"3*H^3.j%=F7F_o$\"3++++y7%))\\$F_o7$$\"3mQR#='Qa)G
\"F_o$\"3(*******)=;f#GF_o7$$\"3Kk$RL4.*\\8F_o$\"3%******H-&>+EF_o7$$
\"3(**ya[Ki7T\"F_o$\"33+++XuCJEF_o7$$\"3U:-Pc:is9F_o$\"33+++Qa7qGF_o7$
$\"32Tc)yy!)R`\"F_o$\"3')*****f(GZRLF_o7$$\"3sm5S>+M&f\"F_o$\"3*)*****
z'[<KTF_o7$$\"3S#\\;4D*pc;F_o$\"3c*****\\4l-Y&F_o7$$\"30=>V#[e!=<F_o$
\"3!)*****\\WK#3yF_o7$$\"3sVt%Rr<%z<F_o$\"3++++j?HV7Fdp7$$\"3PpFYXpxS=
F_o$\"3(******z3PSM#Fdp7$$\"3/&>yp<O@!>F_o$\"3++++!3N?8'Fdp7$$\"3q?O\\
3a\\j>F_o$\"3?+++gDovVFjp7$$\"39Y!4+ka[-#F_o$\"3m*****zk\\eS*Fjp7$$\"3
-sW_rQ@'3#F_o$\"3`+++krM7yFdp7$$\"3Y(*)RI5tv9#F_o$\"3/+++!*o\"3n#Fdp7$
$\"3MB`bMB$*3AF_o$\"3++++J)RrL\"Fdp7$$\"3y[22m:HqAF_o$\"39+++-+*)H!)F_
o7$$\"3muhe(z];L#F_o$\"3A+++5x2n`F_o7$$\"35+;5H+,$R#F_o$\"35+++L]-]QF_
o7$$\"3)f-<1EpVX#F_o$\"30+++`b)R!HF_o7$$\"3U^C8#\\Gd^#F_o$\"3%*******o
];uAF_o7$$\"3IxykBx3xDF_o$\"35+++VZbL=F_o7$$\"3u-L;bpWQEF_o$\"3++++bD3
8:F_o7$$\"3iG(ym=1)*p#F_o$\"33+++&zkDF\"F_o7$$\"31aT>=a;hFF_o$\"3++++b
6L(3\"F_o7$$\"3&*z&4(\\Y_AGF_o$\"3\"******f5SbT*F,7$$\"3Q0]A\")Q)Q)GF_
o$\"3#)*****RQ$*pC)F,7$$\"3#3VSF6V_%HF_o$\"3_+++C%\\`H(F,7$$\"3rceDWBg
1IF_o$\"3S+++.Jh4lF,7$$\"39#Grddhz1$F_o$\"3'******4A&)H&eF,7$$\"3.3nG2
3KHJF_o$\"39+++'[m$)H&F,7$$\"3YL@!)Q+o!>$F_o$\"3*)*****f6Sa#[F,7$$\"3O
fvJq#R?D$F_o$\"3w******3!H(=WF,7$$\"3z%)H$=])R8LF_o$\"3-+++2NFmSF,7$$
\"3m5%[LtdZP$F_o$\"3#)*****H17(ePF,7$$\"36OQ'['p6OMF_o$\"3w*****zxI')[
$F,7$$\"3+i#zj>wu\\$F_o$\"3))*****\\h&4]KF,7$$\"3W(o%*yUN)eNF_o$\"3')*
****pk5$QIF,7$$\"3J8,TfY>?OF_o$\"3y*****p7l$\\GF,7$$\"3uQb#4*Qb\"o$F_o
$\"3'******z!Q/!o#F,7$$\"3kk4WAJ\"Hu$F_o$\"3u*****4:$oFDF,7$$\"32!RcRN
sU!QF_o$\"3%******R@h+R#F,7$$\"3'f\"=Z&eJc'QF_o$\"3)*******o@JlAF,7$$
\"3RTs)p\"3*p#RF_o$\"3)******Hxf=:#F,7$$\"3%om-&[+N))RF_o$\"37+++wZO[?
F,7$$\"3G#4=+G4(\\SF_o$\"3))*****zM$o`>F,7$$\"3:=N`6&o56%F_o$\"3'*****
**HN$o'=F,7$$\"3/W*[IuFC<%F_o$\"33+++>&opy\"F,7$$\"3#*pVcupyLUF_o$\"3.
+++sZN8<F,7$$\"3#\\zzg?Y^H%F_o$\"33+++zTNX;F,7$$\"3!3A&fPa]cVF_o$\"3#*
*****Rh4Ce\"F,7$$\"3oY16pY'yT%F_o$\"3)******>nKS_\"F,7$$\"3csgi+RAzWF_
o$\"3++++@Qzp9F,7$$\"3c(\\T@8$eSXF_o$\"3-+++#*RJ>9F,7$$\"3WBpljB%>g%F_
o$\"3#*******GwDs8F,7$$\"3K\\B<&f,Lm%F_o$\"3.+++\\nKG8F,7$$\"3@vxoE3mC
ZF_o$\"3%*******HgD(G\"F,7$$\"3?+K?e+-'y%F_o$\"3-+++*e3)[7F,7$$\"33E'=
(*Gzt%[F_o$\"3'******\\PsF@\"F,7$$\"3(>0M7_Q(3\\F_o$\"3)******z4d*y6F,
7$$\"3'pZ\\Fv(4q\\F_o$\"3)******Hf\">Z6F,7$$\"3%G!\\E%)pXJ]F_o$\"3(***
***\\f@t6\"F,7$$\"3sG.y:i\"G4&F_o$\"3%******4q2#*3\"F,7$$\"3iadHZa<a^F
_o$\"3++++&yBF1\"F,7$$\"3gz6\")yY`:_F_o$\"3%******fSbx.\"F,7$$\"3[0mK5
R*oF&F_o$\"3(******Hi)>95F,7$$\"3PJ?%=9`#Q`F_o$\"3U+++R\")e>**F07$$\"3
CduNtBh*R&F_o$\"3N+++>s\\4(*F07$$\"3C#)G([gr4Y&F_o$\"3k*****fsD4^*F07$
$\"383$)QO3LAbF_o$\"3*)******fM:B$*F07$$\"3-MP!z1!p$e&F_o$\"39+++!*3_X
\"*F07$$\"3!*f\">%*H\\]k&F_o$\"3K*****H1Bu(*)F07$$\"3*[eM4`3kq&F_o$\"3
O******f\\I=))F07$$\"3y5+XixwndF_o$\"3)*******[eln')F07$$\"3lOa'R*p7He
F_o$\"3S+++.i+D&)F07$$\"3kh3[Di[!*eF_o$\"39+++\"QB**Q)F07$$\"3a(G'*pXX
=&fF_o$\"3!******4i3?E)F07$$\"3T8<^)o/K,'F_o$\"3j+++!p$*39)F07$$\"3IRr
-?RcugF_o$\"3j*****>**Qi-)F07$$\"3HkDa^J#f8'F_o$\"3y******e.t<zF07$$\"
3;!*z0$Q#G(>'F_o$\"3Y*****fQx]\"yF07$$\"31;Md9;keiF_o$\"3-+++E<,=xF07$
$\"3%>%))3Y3+?jF_o$\"3x*****4#[GEwF07$$\"3%pE/w2g8Q'F_o$\"3]******4nmR
vF07$$\"3\"Gp>\"4$>FW'F_o$\"3?+++5Y%zX(F07$$\"3q=^jS&yS]'F_o$\"3I+++3<
#4Q(F07$$\"3eW0:sxVllF_o$\"3[*****H^:%3tF07$$\"3epfm.qzEmF_o$\"3v*****
\\jd-C(F07$$\"3Y&R\"=Ni:)o'F_o$\"3'******\\F#HwrF07$$\"3M@opma^\\nF_o$
\"3I+++5cP;rF07$$\"3BZA@)pu3\"oF_o$\"3_+++Z[PgqF07$$\"3AswsHRBsoF_o$\"
3!)******pw;3qF07$$\"35)4V7;$fLpF_o$\"3'******pqT'fpF07$$\"3)R_eFR_\\*
pF_o$\"3.+++WPp9pF07$$\"3)*[RFC;JcqF_o$\"3m******G#HK(oF07$$\"3'[P*yb3
n<rF_o$\"3@+++>B;NoF07$$\"3u+[I(3I!zrF_o$\"3r******fVT+oF07$$\"3jE-#)=
$*QSsF_o$\"3s*****\\([\"*onF07$$\"3i^cL]&[<I(F_o$\"39+++x,gSnF07$$\"3]
x5&=y2JO(F_o$\"3k*****\\d8ar'F07$$\"3R.lO8qYCuF_o$\"33+++#40Lp'F07$$\"
3GH>)[CEe[(F_o$\"3I+++r4BumF07$$\"3EatRwa=ZvF_o$\"3Y+++%z`\"emF07$$\"3
:!y7zqW&3wF_o$\"3K+++4@/XmF07$$\"3-1#G%RR!*pwF_o$\"3k*****>Nq[j'F07$$
\"3#>jV4<j7t(F_o$\"3S+++*))=wi'F07$$\"3!p0fCSAEz(F_o$\"3-+++$etKi'F07$
$!3y#[uRj\")R&yF_o$\"3A+++Ng#=i'F07$$!3!p0fCSAEz(F_oFdcm7$$!3#>jV4<j7t
(F_oF_cm7$$!3-1#G%RR!*pwF_oFjbm7$$!3:!y7zqW&3wF_oFebm7$$!3EatRwa=ZvF_o
F`bm7$$!3GH>)[CEe[(F_oF[bm7$$!3R.lO8qYCuF_oFfam7$$!3]x5&=y2JO(F_oFaam7
$$!3i^cL]&[<I(F_oF\\am7$$!3jE-#)=$*QSsF_oFg`m7$$!3u+[I(3I!zrF_oFb`m7$$
!3'[P*yb3n<rF_oF]`m7$$!3)*[RFC;JcqF_oFh_m7$$!3)R_eFR_\\*pF_oFc_m7$$!35
)4V7;$fLpF_oF^_m7$$!3AswsHRBsoF_oFi^m7$$!3BZA@)pu3\"oF_oFd^m7$$!3M@opm
a^\\nF_oF_^m7$$!3Y&R\"=Ni:)o'F_oFj]m7$$!3epfm.qzEmF_oFe]m7$$!3eW0:sxVl
lF_oF`]m7$$!3q=^jS&yS]'F_oF[]m7$$!3\"Gp>\"4$>FW'F_oFf\\m7$$!3%pE/w2g8Q
'F_oFa\\m7$$!3%>%))3Y3+?jF_oF\\\\m7$$!31;Md9;keiF_oFg[m7$$!3;!*z0$Q#G(
>'F_oFb[m7$$!3HkDa^J#f8'F_oF][m7$$!3IRr-?RcugF_oFhjl7$$!3T8<^)o/K,'F_o
Fcjl7$$!3a(G'*pXX=&fF_oF^jl7$$!3kh3[Di[!*eF_oFiil7$$!3lOa'R*p7HeF_oFdi
l7$$!3y5+XixwndF_oF_il7$$!3*[eM4`3kq&F_oFjhl7$$!3!*f\">%*H\\]k&F_oFehl
7$$!3-MP!z1!p$e&F_oF`hl7$$!383$)QO3LAbF_oF[hl7$$!3C#)G([gr4Y&F_oFfgl7$
$!3CduNtBh*R&F_oFagl7$$!3PJ?%=9`#Q`F_oF\\gl7$$!3[0mK5R*oF&F_oFgfl7$$!3
gz6\")yY`:_F_oFbfl7$$!3iadHZa<a^F_oF]fl7$$!3sG.y:i\"G4&F_oFhel7$$!3%G!
\\E%)pXJ]F_oFcel7$$!3'pZ\\Fv(4q\\F_oF^el7$$!3(>0M7_Q(3\\F_oFidl7$$!33E
'=(*Gzt%[F_oFddl7$$!3?+K?e+-'y%F_oF_dl7$$!3@vxoE3mCZF_oFjcl7$$!3K\\B<&
f,Lm%F_oFecl7$$!3WBpljB%>g%F_oF`cl7$$!3c(\\T@8$eSXF_oF[cl7$$!3csgi+RAz
WF_oFfbl7$$!3oY16pY'yT%F_oFabl7$$!3!3A&fPa]cVF_oF\\bl7$$!3#\\zzg?Y^H%F
_oFgal7$$!3#*pVcupyLUF_oFbal7$$!3/W*[IuFC<%F_oF]al7$$!3:=N`6&o56%F_oFh
`l7$$!3G#4=+G4(\\SF_oFc`l7$$!3%om-&[+N))RF_oF^`l7$$!3RTs)p\"3*p#RF_oFi
_l7$$!3'f\"=Z&eJc'QF_oFd_l7$$!32!RcRNsU!QF_oF__l7$$!3kk4WAJ\"Hu$F_oFj^
l7$$!3uQb#4*Qb\"o$F_oFe^l7$$!3J8,TfY>?OF_oF`^l7$$!3W(o%*yUN)eNF_oF[^l7
$$!3+i#zj>wu\\$F_oFf]l7$$!36OQ'['p6OMF_oFa]l7$$!3m5%[LtdZP$F_oF\\]l7$$
!3z%)H$=])R8LF_oFg\\l7$$!3OfvJq#R?D$F_oFb\\l7$$!3YL@!)Q+o!>$F_oF]\\l7$
$!3.3nG23KHJF_oFh[l7$$!39#Grddhz1$F_oFc[l7$$!3rceDWBg1IF_oF^[l7$$!3#3V
SF6V_%HF_oFiz7$$!3Q0]A\")Q)Q)GF_oFdz7$$!3&*z&4(\\Y_AGF_oF_z7$$!31aT>=a
;hFF_oFjy7$$!3iG(ym=1)*p#F_oFey7$$!3u-L;bpWQEF_oF`y7$$!3IxykBx3xDF_oF[
y7$$!3U^C8#\\Gd^#F_oFfx7$$!3)f-<1EpVX#F_oFax7$$!35+;5H+,$R#F_oF\\x7$$!
3muhe(z];L#F_oFgw7$$!3y[22m:HqAF_oFbw7$$!3MB`bMB$*3AF_oF]w7$$!3Y(*)RI5
tv9#F_oFhv7$$!3-sW_rQ@'3#F_oFcv7$$!39Y!4+ka[-#F_oF^v7$$!3q?O\\3a\\j>F_
oFiu7$$!3/&>yp<O@!>F_oFdu7$$!3PpFYXpxS=F_oF_u7$$!3sVt%Rr<%z<F_oFjt7$$!
30=>V#[e!=<F_oFet7$$!3S#\\;4D*pc;F_oF`t7$$!3sm5S>+M&f\"F_oF[t7$$!32Tc)
yy!)R`\"F_oFfs7$$!3U:-Pc:is9F_oFas7$$!3(**ya[Ki7T\"F_oF\\s7$$!3Kk$RL4.
*\\8F_oFgr7$$!3mQR#='Qa)G\"F_oFbr7$$!3*H^3.j%=F7F_oF]r7$$!3K(3$z)RDe;
\"F_oFhq7$$!3nhwFnhY/6F_oFcq7$$!3,OAwNp5V5F_oF^q7$$!3\\.\"oC/xu\")*F,F
hp7$$!3)o%QJFZ)Q?*F,Fbp7$$!3E!ff@T#H!f)F,F]p7$$!3mL`+(4+n(zF,Fho7$$!31
x5&=y2JO(F,Fco7$$!3c@opma^\\nF,F]o7$$!3'\\cU::Bf8'F,Fhn7$$!3N3$)QO3LAb
F,FY7$$!3u^SB@&Q(3\\F,FT7$$!37&zzg?Y^H%F,FO7$$!3_Qb#4*Qb\"o$F,FJ7$$!3[
#Grddhz1$F,FE7$$!3)e-<1EpVX#F,F@7$$!3EpFYXpxS=F,F;7$$!3%H^3.j%=F7F,F67
$$!3okDa^J#f8'F0F1-%+AXESLABELSG6$Q!6\"F[\\o-%'COLOURG6&%$RGBGF)F)F)-%
&STYLEG6#%&POINTG-%%VIEWG6$%(DEFAULTGFh\\o" 1 5 0 1 10 0 2 6 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 462 "Evidently, unless the resolution is increased considerab
ly one can read off the accurate weight of the different frequency con
tributions only when a proper lineshape is fitted to the data near the
 peaks: the frequency (momentum) grid can easily miss the accurate loc
ation of the peak. Nevertheless, we can see in the given example that \+
the signal intensities for the two frequency contributions are approxi
mately 4:1, as is appropriate for the original function " }{TEXT 19 
20 "fx=sin(x)+2*cos(2*x)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 344 "Note that the frequency informati
on is relatively sharp, because the time interval sampled contains a g
ood number of cycles. The efficiency of the FFT to display information
 content in terms of population of frequencies is obvious, and this ex
plains the widespread use of FFTs in Electronics, Acoustics, and other
 branches of signal processing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 450 "There is some redundancy in the above re
sult: We have a degree of symmetry: the real part of the FT is symmetr
ic, while the imaginary part is antisymmetric in frequency with the en
d result of a symmetric intensity profile. The redundancy to some exte
nt is the result of using a real signal only. When adding an imaginary
 part to the function in the time domain (at a different frequency, e.
g., f=3), the real part of the transform loses its symmetry." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 11 "Exercise 4:" }
}{PARA 0 "" 0 "" {TEXT -1 65 "Play with the above code while keeping t
he same function: change " }{TEXT 19 2 "dx" }{TEXT -1 25 ", while keep
ing the same " }{TEXT 19 1 "N" }{TEXT -1 12 "; then keep " }{TEXT 19 
2 "dx" }{TEXT -1 17 ", while changing " }{TEXT 19 1 "N" }{TEXT -1 203 
". Observe the implications for the information content. If one wishes
 to work with a fixed total length interval, one has to use a window f
unction to cut off the signal outside the desired time interval." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 20 "
Filtering noisy data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 397 "A classical application of DFT's concerns the removal \+
of noise from data. The idea is that noise represents signal component
s of high frequencies which can be removed by a low-pass filter. In th
is section we will construct a signal with known smooth components and
 will add some randomness. Then we will analyze the DFT, remove high f
requencies and observe the result of an inverse transformation." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Maple has
 a built in random number generator for integer randoms. In the " }
{TEXT 19 5 "stats" }{TEXT -1 82 " package there are also more sophisti
cated generators, but this one will suffice. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rand()/10^1
2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"+r$z9=#\"+++D1R" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(rand()/10^12);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+u0*y:*!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 148 "Subsequent calls result in a sequence of pseudorandom numbers.
 One can changed the seed value for different deterministic sequences.
 By dividing by " }{TEXT 19 5 "10^12" }{TEXT -1 62 " we have uniformly
 distributed randoms on the interval [0, 1]." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 28 "fx:=(sin(x/4)+2*cos(1/2*x));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#fxG,&-%$sinG6#,$*&\"\"%!\"\"%\"xG\"\"\"F.F.*&\"\"#
F.-%$cosG6#,$*&F0F,F-F.F.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 29 "yR:=Vector(N): yI:=Vector(N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
118 "for i from 1 to N do: yR[i]:=evalf(subs(x=xv[i],Re(fx))+1*(rand()
/10^12-0.5)); yI[i]:=evalf(subs(x=xv[i],Im(fx))); od:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "P1:=plot([seq([xv[i],yR[i]],i=1..N)
],style=line): display(P1);" }}{PARA 13 "" 1 "" {GLPLOT2D 824 208 208 
{PLOTDATA 2 "6%-%'CURVESG6%7\\[l7$$!3H++++++?^!#;$\"34+++&p`j!=!#<7$$!
3W++++++!3&F*$\"3,+++J3^2AF-7$$!3d++++++S]F*$\"31+++\"Gm&*>#F-7$$!#]\"
\"!$\"3$******>@c_+#F-7$$!39++++++g\\F*$\"3>+++UQ1NAF-7$$!3H++++++?\\F
*$\"3:+++3\">4B#F-7$$!3W++++++!)[F*$\"3'********3Ho'=F-7$$!3d++++++S[F
*$\"3)*******H%>5b\"F-7$$!#[F;$\"3)******f,e'f:F-7$$!39++++++gZF*$\"3a
******pb*Ql*!#=7$$!3H++++++?ZF*$\"33+++`:a'>\"F-7$$!3W++++++!o%F*$!3e*
******4#o%>'!#?7$$!3d++++++SYF*$\"3%******zN3%pQFfn7$$!#YF;$\"3m******
4d<8Q!#>7$$!39++++++gXF*$!3=+++!y0X\"[F\\p7$$!3H++++++?XF*$!3!******\\
6+2,\"F-7$$!3W++++++![%F*$!3'******4J)[h5F-7$$!3d++++++SWF*$!3#)******
z%R&o#*Ffn7$$!#WF;$!3O+++L)Hu\"zFfn7$$!39++++++gVF*$!3#*******\\3\"HD
\"F-7$$!3H++++++?VF*$!3y******\\e+vZFfn7$$!3W++++++!G%F*$!3%)*****fZ:h
R*Ffn7$$!3d++++++SUF*$!3y*****4lo/e%Ffn7$$!#UF;$\"31+++`<jf=Ffn7$$!39+
+++++gTF*$\"3&)*****fC]Ei$Ffn7$$!3H++++++?TF*$\"39+++)f))zf%Ffn7$$!3W+
+++++!3%F*$\"3A+++%Gx[x(Ffn7$$!3d++++++SSF*$\"3!******f&)=@L\"F-7$$!#S
F;$\"35+++UZ!**e\"F-7$$!39++++++gRF*$\"33+++I'zkv\"F-7$$!3H++++++?RF*$
\"3@+++u:M/@F-7$$!3W++++++!)QF*$\"38+++1GNR?F-7$$!3d++++++SQF*$\"3-+++
2D!*4<F-7$$!#QF;$\"3))*****4\\J'y?F-7$$!39++++++gPF*$\"35+++^tl=:F-7$$
!3H++++++?PF*$\"37+++'o$po;F-7$$!3W++++++!o$F*$\"3!******>[5d*=F-7$$!3
')************ROF*$\"3-+++Wi)y[\"F-7$$!#OF;$\"33+++1V&QJ\"F-7$$!39++++
++gNF*$\"3-+++Es2#*HFfn7$$!3H++++++?NF*$\"3O+++IIh9XF\\p7$$!3W++++++![
$F*$!35+++u5^`XFfn7$$!3')************RMF*$!3_*****RCr%p%*Ffn7$$!#MF;$!
35+++m!=c3\"F-7$$!39++++++gLF*$!3%******RhB`x\"F-7$$!3H++++++?LF*$!3A+
++uu=_@F-7$$!3W++++++!G$F*$!3y*****pqAiq#F-7$$!3')************RKF*$!39
+++.['*)*GF-7$$!#KF;$!3&)******oWOFFF-7$$!39++++++gJF*$!3%******H)*[&*
\\#F-7$$!3H++++++?JF*$!3z*****4q'[EKF-7$$!33++++++!3$F*$!3/+++&3UFN$F-
7$$!3A++++++SIF*$!32+++?A(QA#F-7$$!#IF;$!3++++cTCw?F-7$$!39++++++gHF*$
!3*)******\\z[6@F-7$$!3H++++++?HF*$!3)******>&o4/:F-7$$!33++++++!)GF*$
!3'*******Q)R]G\"F-7$$!3A++++++SGF*$!3B+++YmuGWFfn7$$!#GF;$\"3O+++!)==
\")[F\\p7$$!39++++++gFF*$!3B+++G'[^@$Ffn7$$!3H++++++?FF*$\"3++++y1gJ6F
fn7$$!33++++++!o#F*$\"33+++c@6#=\"F-7$$!3A++++++SEF*$\"3]*****fW*y#\\*
Ffn7$$!#EF;$\"3&******p)zS<8F-7$$!39++++++gDF*$\"3)******\\p6Q(>F-7$$!
3H++++++?DF*$\"3A+++!pS3E#F-7$$!33++++++![#F*$\"39+++oD!R=#F-7$$!3A+++
+++SCF*$\"3=+++-W9@AF-7$$!#CF;$\"3$*******zu*o.#F-7$$!39++++++gBF*$\"3
/+++_1C'y\"F-7$$!3H++++++?BF*$\"3(******p**GUq\"F-7$$!33++++++!G#F*$\"
3#******R)3*GC\"F-7$$!3A++++++SAF*$\"3]******e=2\"*yFfn7$$!#AF;$\"3%)*
****HZ2mA'Ffn7$$!39++++++g@F*$\"3%******4j?fH#Ffn7$$!3H++++++?@F*$\"3'
******R@fd#[Ffn7$$!33++++++!3#F*$!3()*****R4CD]#Ffn7$$!3A++++++S?F*$!3
%*******zW!\\q#Ffn7$$FaoF;$!37+++i&ouv%Ffn7$$!39++++++g>F*$!3e*****RNn
dX&Ffn7$$!3H++++++?>F*$!3,+++l\\**4yFfn7$$!33++++++!)=F*$!3w*****Rev>E
)Ffn7$$!3A++++++S=F*$!3]+++v5NyvFfn7$$FfnF;$!3j*****Hb#e?nFfn7$$!39+++
+++g<F*$!3)******R4I:$HFfn7$$!3$*************><F*$!3++++Jtov\")Ffn7$$!
33++++++!o\"F*$\"39++++Q![l$Fao7$$!3A++++++S;F*$!3!******zp_$)f#Ffn7$$
F*F;$\"3!*******G$3ma\"Ffn7$$!39++++++g:F*$\"3=+++iZM,KFfn7$$!36++++++
?:F*$\"3-+++EX-![\"F-7$$!33++++++![\"F*$\"3)******4T(3\"Q\"F-7$$!3/+++
+++S9F*$\"3++++;)4\"G=F-7$$!#9F;$\"3/+++\\Xgl9F-7$$!39++++++g8F*$\"3#)
*****H4]lK#F-7$$!36++++++?8F*$\"3$******f8x.#>F-7$$!33++++++!G\"F*$\"3
-+++&*Q$)[?F-7$$!3/++++++S7F*$\"3/+++iX*)))>F-7$$!#7F;$\"3#******Rm\"3
*o\"F-7$$!39++++++g6F*$\"3/+++)=qfC\"F-7$$!36++++++?6F*$\"3))*****fuvF
S(Ffn7$$!33++++++!3\"F*$\"3y*****\\C7?a&Ffn7$$!3/++++++S5F*$\"3\"*****
*\\qN%=!)Ffn7$$!#5F;$\"3-+++h`'*>6Ffn7$$!3U,++++++'*F-$!3_+++i)[iu'Ffn
7$$!31,++++++#*F-$!33+++WFgY5F-7$$!3q+++++++))F-$!3)*******H_7y)*Ffn7$
$!3M+++++++%)F-$!3$******p@y)49F-7$$!\")F;$!3#******fRLa*>F-7$$!3_++++
+++wF-$!3?+++*fqB%HF-7$$!3;+++++++sF-$!3-+++,*zzj#F-7$$!3q+++++++oF-$!
3))******H%y:Y#F-7$$!3M+++++++kF-$!32+++V%z-!HF-7$$!\"'F;$!3;+++Ge-*\\
#F-7$$!3_+++++++cF-$!3%******pI.&*G$F-7$$!3;+++++++_F-$!3;+++4[FUCF-7$
$!3q+++++++[F-$!3%)*****4FNJq#F-7$$!3M+++++++WF-$!3!*******G#ooX#F-7$$
!\"%F;$!3++++i\\he<F-7$$!33+++++++OF-$!3/+++IP'y4\"F-7$$!3;+++++++KF-$
!3#)*****zY[]R$Ffn7$$!3E+++++++GF-$\"3++++!\\'*yX#F\\p7$$!3M+++++++CF-
$\"3Y*****\\O>4D'Ffn7$$!\"#F;$\"3y*****\\Fopb$Ffn7$$!33+++++++;F-$\"3(
*******4rS=9F-7$$!3;+++++++7F-$\"3#******p0CTf\"F-7$$!3U+++++++!)Ffn$
\"3)******H&Qs2@F-7$$!3A+++++++SFfn$\"3++++cKXJAF-7$$F;F;$\"3)*******>
)3O.#F-7$$\"3A+++++++SFfn$\"39+++onYj@F-7$$\"3U+++++++!)Ffn$\"3/+++)=E
\"*e\"F-7$$\"3;+++++++7F-$\"3-+++mz%e&=F-7$$\"33+++++++;F-$\"33+++*3V$
e:F-7$$\"\"#F;$\"31+++n$omu\"F-7$$\"3M+++++++CF-$\"3()*****pQ()4s)Ffn7
$$\"3E+++++++GF-$\"3++++.1r[8F-7$$\"3;+++++++KF-$\"32+++1V/F5F-7$$\"33
+++++++OF-$\"3i*****z!f^4jFfn7$$\"\"%F;$!3-+++9o%yI$Ffn7$$\"3M+++++++W
F-$!3%******H!=0%y\"Ffn7$$\"3q+++++++[F-$!33+++kuMg()Ffn7$$\"3;+++++++
_F-$!3G+++)RtvI%Ffn7$$\"3_+++++++cF-$!3i*****HX;_E(Ffn7$$\"\"'F;$!34++
+7UT<8F-7$$\"3M+++++++kF-$!3#******z(*y/2\"F-7$$\"3q+++++++oF-$!36+++/
f>?xFfn7$$\"3;+++++++sF-$!3;+++I<etVFfn7$$\"3_+++++++wF-$!3#*******Hu5
\\cFfn7$$\"\")F;$!31+++QQ\"y<#Ffn7$$\"3M+++++++%)F-$!3a+++q,R8vF\\p7$$
\"3q+++++++))F-$\"3D++++S/VTFao7$$\"31,++++++#*F-$\"37+++Led@%)Ffn7$$
\"3U,++++++'*F-$\"3?+++)y>O-%Ffn7$$\"#5F;$\"33+++sE%\\Z\"F-7$$\"3/++++
++S5F*$\"3/+++BBt$)=F-7$$\"33++++++!3\"F*$\"3?+++<4)o1#F-7$$\"36++++++
?6F*$\"3(******RU'p6=F-7$$\"39++++++g6F*$\"3?+++Duw#=#F-7$$\"#7F;$\"37
+++*3+BQ#F-7$$\"3/++++++S7F*$\"3/+++W)R9'>F-7$$\"33++++++!G\"F*$\"3))*
****R[xa4#F-7$$\"36++++++?8F*$\"3'******>;`s#>F-7$$\"39++++++g8F*$\"3$
******>H!zQ;F-7$$\"#9F;$\"36+++\"fWoB\"F-7$$\"3/++++++S9F*$\"3:++++x$3
B(Ffn7$$\"33++++++![\"F*$\"3.+++t^uB9Ffn7$$\"36++++++?:F*$!3)*******[4
Hg:Ffn7$$\"39++++++g:F*$!3++++m=f#o\"Ffn7$$\"#;F;$!35+++ra;\\6F-7$$\"3
A++++++S;F*$!3%******4P*4n6F-7$$\"33++++++!o\"F*$!3.+++1\\\"H/#F-7$$\"
3$*************><F*$!3')*****4\\H)\\AF-7$$\"39++++++g<F*$!3))*****pYj)
)H#F-7$$\"#=F;$!33+++9\"z!)H$F-7$$\"3A++++++S=F*$!32+++1%>BI$F-7$$\"33
++++++!)=F*$!3%*******>\\xXDF-7$$\"3H++++++?>F*$!3A+++?D@6LF-7$$\"39++
++++g>F*$!3)******RSv^#HF-7$$\"#?F;$!3!******Ho$prIF-7$$\"3A++++++S?F*
$!3>+++FPrEDF-7$$\"33++++++!3#F*$!3/+++#R!>b<F-7$$\"3H++++++?@F*$!3/++
+*RC]A\"F-7$$\"39++++++g@F*$!3a******3Rb\\')Ffn7$$\"#AF;$!3-+++ko$Q@#F
fn7$$\"3A++++++SAF*$!3&******pa15/#Ffn7$$\"33++++++!G#F*$\"3t******z^]
.KFfn7$$\"3H++++++?BF*$\"3;+++%)yO$=)Ffn7$$\"39++++++gBF*$\"3]+++g\\j]
$*Ffn7$$\"#CF;$\"3(******\\)4B1:F-7$$\"3A++++++SCF*$\"3-+++Yw(*Q8F-7$$
\"33++++++![#F*$\"3%******>[o'3>F-7$$\"3H++++++?DF*$\"3\")******QhjSAF
-7$$\"39++++++gDF*$\"3-+++3t$f*>F-7$$\"#EF;$\"3)******Rx&GCDF-7$$\"3A+
+++++SEF*$\"3)******H[P@#>F-7$$\"33++++++!o#F*$\"3/+++\\`5a<F-7$$\"3H+
+++++?FF*$\"3++++u;f$e\"F-7$$\"39++++++gFF*$\"3#******>CkNY\"F-7$$\"#G
F;$\"3++++Ev?$H\"F-7$$\"3A++++++SGF*$\"3Q+++#RR$e&*Ffn7$$\"33++++++!)G
F*$\"31+++Y@+!G$Ffn7$$\"3H++++++?HF*$!3'******f3z3N&Ffn7$$\"39++++++gH
F*$\"3L+++]]tS]F\\p7$$\"#IF;$!35+++0pJ!3\"F-7$$\"3A++++++SIF*$!3<+++'4
9^!QFfn7$$\"33++++++!3$F*$!3\"******>TKR%[Ffn7$$\"3H++++++?JF*$!3u****
*\\[G=%yFfn7$$\"39++++++gJF*$!3)******p)>.-9F-7$$\"#KF;$!3]+++8d8z_Ffn
7$$\"3')************RKF*$!3,+++ka)Q8%Ffn7$$\"3W++++++!G$F*$!35+++$fD]1
*Ffn7$$\"3H++++++?LF*$!3%******R:F>F#Ffn7$$\"39++++++gLF*$\"3`++++\"HE
5(F\\p7$$\"#MF;$\"3w*****Rtl3s#Ffn7$$\"3')************RMF*$\"3u******)
*yOg%)Ffn7$$\"3W++++++![$F*$\"3;+++NQst\\Ffn7$$\"3H++++++?NF*$\"3))***
***GmJt;F-7$$\"39++++++gNF*$\"3'******pu*H*>\"F-7$$\"#OF;$\"3&******\\
E9[!>F-7$$\"3')************ROF*$\"3y*****fdMTD#F-7$$\"3W++++++!o$F*$\"
3!******>G/*H<F-7$$\"3H++++++?PF*$\"3)*******QhZo=F-7$$\"39++++++gPF*$
\"3)******\\@$>m:F-7$$\"#QF;$\"3/+++&*R,Y:F-7$$\"3d++++++SQF*$\"3\"***
***favgs\"F-7$$\"3W++++++!)QF*$\"3,+++rA4K:F-7$$\"3H++++++?RF*$\"3g***
**RKAb3)Ffn7$$\"39++++++gRF*$\"35+++p2(=$zFfn7$$\"#SF;$\"3z*****zau/i%
Ffn7$$\"3d++++++SSF*$!33+++]B()QCFfn7$$\"3W++++++!3%F*$!3O+++A=)p=(Ffn
7$$\"3H++++++?TF*$!3++++zQml6F-7$$\"39++++++gTF*$!3%******4)o\\x;F-7$$
\"#UF;$!37+++sKLrAF-7$$\"3d++++++SUF*$!3%)*****R!G\\YAF-7$$\"3W++++++!
G%F*$!3@+++))[\"GT#F-7$$\"3H++++++?VF*$!3))*******p%HnFF-7$$\"39++++++
gVF*$!3?+++mx'\\o#F-7$$\"#WF;$!3')*****pB4*3FF-7$$\"3d++++++SWF*$!33++
+T(=mz#F-7$$\"3W++++++![%F*$!37+++@)=tY#F-7$$\"3H++++++?XF*$!3++++ovC&
4#F-7$$\"39++++++gXF*$!3-+++!zm&p>F-7$$\"#YF;$!3\")*****RMSn-#F-7$$\"3
d++++++SYF*$!3(******>oH%=>F-7$$\"3W++++++!o%F*$!3M+++hI2L$)Ffn7$$\"3H
++++++?ZF*$!3-+++s*oB1\"F-7$$\"39++++++gZF*$!3))******ejh6>Ffn7$$\"#[F
;$\"3D+++YoI3QFfn7$$\"3d++++++S[F*$\"3#******H^#\\vbFfn7$$\"3W++++++!)
[F*$\"3k*****pVcyG)Ffn7$$\"3H++++++?\\F*$\"35+++HK`j7F-7$$\"39++++++g
\\F*$\"3-+++=0D*y\"F-7$$\"#]F;$\"32+++%z8r?#F-7$$\"3d++++++S]F*$\"3!**
*****Hm4[=F-7$$\"3W++++++!3&F*$\"3%******paCVt\"F--%'COLOURG6&%$RGBG$F
b[n!\"\"FecmFecm-%&STYLEG6#%%LINEG-%+AXESLABELSG6$Q!6\"F`\\p-%%VIEWG6$
%(DEFAULTGFe\\p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalhf(FFT
(n, var(yR), var(yI)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$c#\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot([seq([pv[i],yR[i]]
,i=1..N)],style=line);" }}{PARA 13 "" 1 "" {GLPLOT2D 824 208 208 
{PLOTDATA 2 "6&-%'CURVESG6$7\\[l7$$\"\"!F)$\"31++#**R\")*\\@!#;7$$\"3o
kDa^J#f8'!#>$\"3\\Q1FQz<;w!#<7$$\"3%H^3.j%=F7!#=$\"3'*yZ&yp!H,5F,7$$\"
3EpFYXpxS=F7$\"3[nSt?MS5()F37$$\"3)e-<1EpVX#F7$\"3'[5eG**[2\\\"F,7$$\"
3[#Grddhz1$F7$\"3-,sn&o73Y\"F,7$$\"3_Qb#4*Qb\"o$F7$\"3KKY[M&zim\"F,7$$
\"37&zzg?Y^H%F7$\"3l/ZO<g-BKF,7$$\"3u^SB@&Q(3\\F7$\"33%G)\\==2cC!#:7$$
\"3N3$)QO3LAbF7$!3w!p))z1)o&H%F,7$$\"3'\\cU::Bf8'F7$!3Wp]AFTwf8F,7$$\"
3c@opma^\\nF7$!3Q=ATzEr(3\"F,7$$\"31x5&=y2JO(F7$!3yf=l!\\8`3'F37$$\"3m
L`+(4+n(zF7$!3+!=B(=Q$3D%F37$$\"3E!ff@T#H!f)F7$!3WZ.8+a*Hc\"F37$$\"3)o
%QJFZ)Q?*F7$!3#zg;A^GG$[F37$$\"3\\.\"oC/xu\")*F7$!3'QM=`g(=GNF37$$\"3,
OAwNp5V5F3$!3i&prxTj)=JF77$$\"3nhwFnhY/6F3$!3k&HsI0e6B#F37$$\"3K(3$z)R
De;\"F3$\"3O,8u>JO&z$F37$$\"3*H^3.j%=F7F3$!3sxf5[ZX^:F37$$\"3mQR#='Qa)
G\"F3$!3I$fFrb%R<5F07$$\"3Kk$RL4.*\\8F3$!3;k3qPJr&H&F37$$\"3(**ya[Ki7T
\"F3$!3@&f.@f3kA$F37$$\"3U:-Pc:is9F3$!3'f[&o\"**p@n$F37$$\"32Tc)yy!)R`
\"F3$!3bXNI)4KRs\"F37$$\"3sm5S>+M&f\"F3$!3!f$\\*G1bsr\"F37$$\"3S#\\;4D
*pc;F3$\"3M>ER'\\12w#F37$$\"30=>V#[e!=<F3$\"3BDA!y\\y;W#F37$$\"3sVt%Rr
<%z<F3$!3U*)))*=%\\M?iF37$$\"3PpFYXpxS=F3$!3_Sy19:^\\*)F77$$\"3/&>yp<O
@!>F3$!3bK/a!Rt<g$F37$$\"3q?O\\3a\\j>F3$\"3y4f0\"\\n)yyF77$$\"39Y!4+ka
[-#F3$!3\"))Qi\\7oC-&F37$$\"3-sW_rQ@'3#F3$\"3O%\\Az!o(*)z#F37$$\"3Y(*)
RI5tv9#F3$\"3SV],`!)[\\AF37$$\"3MB`bMB$*3AF3$\"3g14@V&)*pK#F37$$\"3y[2
2m:HqAF3$!3Cr2W&eK`=$F37$$\"3muhe(z];L#F3$!3D#R+LR]J,#F37$$\"35+;5H+,$
R#F3$!3\\8qhEGkJ[F37$$\"3)f-<1EpVX#F3$\"37dH,:hVfCF37$$\"3U^C8#\\Gd^#F
3$\"3#*y5t2d(oh(F77$$\"3IxykBx3xDF3$!3kuC\">(*f!R<F37$$\"3u-L;bpWQEF3$
\"3YMxnSdt&=#F77$$\"3iG(ym=1)*p#F3$!3)[&[^TP;^KF37$$\"31aT>=a;hFF3$\"3
k(y-Uj#3q=F77$$\"3&*z&4(\\Y_AGF3$!3\"=Jko3&yKvF37$$\"3Q0]A\")Q)Q)GF3$!
3//X;8Kh(*=F37$$\"3#3VSF6V_%HF3$\"3'*4E^(4yLO$F37$$\"3rceDWBg1IF3$\"3e
A^0'3\\9Y*F77$$\"39#Grddhz1$F3$!3*Q&4DdcEBRF37$$\"3.3nG23KHJF3$!3G2yTl
6M[6F37$$\"3YL@!)Q+o!>$F3$!3]ws<tg_$Q%F37$$\"3OfvJq#R?D$F3$\"3\"3#e337
:@JF37$$\"3z%)H$=])R8LF3$\"333(yg(G.0>F37$$\"3m5%[LtdZP$F3$\"3yYM)HyA_
'\\F37$$\"36OQ'['p6OMF3$!3if'*=h\"fH3$F37$$\"3+i#zj>wu\\$F3$!3![liz&y[
Z))F07$$\"3W(o%*yUN)eNF3$\"3\"3&H^Ux@UiF37$$\"3J8,TfY>?OF3$!3<X+hD@rDK
F37$$\"3uQb#4*Qb\"o$F3$\"3vH]3.L]+JF37$$\"3kk4WAJ\"Hu$F3$!3YN)o$eQ'yp'
F37$$\"32!RcRNsU!QF3$\"3C#*H35\\\\W>F37$$\"3'f\"=Z&eJc'QF3$\"3mW%ey*\\
DyJF37$$\"3RTs)p\"3*p#RF3$!37&)****oYXVGF77$$\"3%om-&[+N))RF3$\"35CAM,
fk!R\"F37$$\"3G#4=+G4(\\SF3$!3mk![41F'[EF37$$\"3:=N`6&o56%F3$!3O[R$*\\
R3&z$F77$$\"3/W*[IuFC<%F3$\"3OQ,*yma'pKF77$$\"3#*pVcupyLUF3$!3)oU-A!yD
_AF37$$\"3#\\zzg?Y^H%F3$\"3gZ?Mx(=x%GF37$$\"3!3A&fPa]cVF3$\"3')*[wMG!
\\e6F37$$\"3oY16pY'yT%F3$!3?z;jzHE?mF77$$\"3csgi+RAzWF3$!3kCNqHS?`JF77
$$\"3c(\\T@8$eSXF3$\"3WMb^Fd\"[=#F77$$\"3WBpljB%>g%F3$\"3O\\D$*3bkvAF3
7$$\"3K\\B<&f,Lm%F3$\"3**\\7_@<q.PF37$$\"3@vxoE3mCZF3$!3UUzXey/+DF37$$
\"3?+K?e+-'y%F3$!3erJ'zH/gC#F37$$\"33E'=(*Gzt%[F3$!3Ka4GHee@&)F77$$\"3
(>0M7_Q(3\\F3$\"35sRXvwy3aF77$$\"3'pZ\\Fv(4q\\F3$!3a[!H9w:'p;F37$$\"3%
G!\\E%)pXJ]F3$!3w_Kz1E$)yIF37$$\"3sG.y:i\"G4&F3$\"3'\\E0'=dl[]F77$$\"3
iadHZa<a^F3$!3kSSj*Q%HYDF37$$\"3gz6\")yY`:_F3$\"3c\\!R\"p!*3-FF37$$\"3
[0mK5R*oF&F3$!3=\")f^N$HA#>F37$$\"3PJ?%=9`#Q`F3$!3V\"Rd5\\\\=.%F37$$\"
3CduNtBh*R&F3$\"379Se\"[*oDBF37$$\"3C#)G([gr4Y&F3$\"3+UM/p'*R!)>F37$$
\"383$)QO3LAbF3$!3)yce*G\"*GKBF77$$\"3-MP!z1!p$e&F3$\"3j#4o%e-w7=F37$$
\"3!*f\">%*H\\]k&F3$!3E%*Q-?0*\\b#F77$$\"3*[eM4`3kq&F3$!3)z]%Rqb/k:F37
$$\"3y5+XixwndF3$!3+#)**H1hlffF07$$\"3lOa'R*p7HeF3$\"3rp9i`\"3L%QF37$$
\"3kh3[Di[!*eF3$!3^#f0b_>3,%F37$$\"3a(G'*pXX=&fF3$\"3mqm.fzE^7F37$$\"3
T8<^)o/K,'F3$\"3il2m$><-K$F37$$\"3IRr-?RcugF3$!3!R#>e'*4pQWF37$$\"3HkD
a^J#f8'F3$!3&4lC@DF:?#F77$$\"3;!*z0$Q#G(>'F3$\"3Y>pW>V$fC\"F37$$\"31;M
d9;keiF3$\"33*3kZ8Mxq#F37$$\"3%>%))3Y3+?jF3$\"3-7TV\"yK*z=F37$$\"3%pE/
w2g8Q'F3$!3Ro9B21]HOF77$$\"3\"Gp>\"4$>FW'F3$!3kN!R1=kGo\"F37$$\"3q=^jS
&yS]'F3$!3i]p3-^C2SF37$$\"3eW0:sxVllF3$\"3[qzfW&o#*>\"F37$$\"3epfm.qzE
mF3$!3Yk'p&o)3pm#F37$$\"3Y&R\"=Ni:)o'F3$\"3geE-0_hdqF37$$\"3M@opma^\\n
F3$!3!3;oeh]R\\%F77$$\"3BZA@)pu3\"oF3$!3U-*>^iGg*\\F37$$\"3AswsHRBsoF3
$!3#pmR\\f@Tx$F37$$\"35)4V7;$fLpF3$\"3ABdc5#oI2\"F37$$\"3)R_eFR_\\*pF3
$\"3#y_z&H;vGSF37$$\"3)*[RFC;JcqF3$!3`uy$>PK'==F37$$\"3'[P*yb3n<rF3$\"
3Q\")==D2+'p*F77$$\"3u+[I(3I!zrF3$\"3Q=8]FiZJmF77$$\"3jE-#)=$*QSsF3$!3
q+Wz.1F3?F37$$\"3i^cL]&[<I(F3$!3tA!><olO!>F37$$\"3]x5&=y2JO(F3$!3#GRRY
c%\\drF77$$\"3R.lO8qYCuF3$!3,R&\\6ZhB_&F37$$\"3GH>)[CEe[(F3$\"3'G7-*pt
w$f\"F37$$\"3EatRwa=ZvF3$\"3)4TihJul)HF37$$\"3:!y7zqW&3wF3$!3q?\"R'o5(
e9$F37$$\"3-1#G%RR!*pwF3$\"3v)[s?O!yD7F37$$\"3#>jV4<j7t(F3$!39+)fXAe[F
\"F37$$\"3!p0fCSAEz(F3$\"3=N_a$)R*H*QF37$$!3y#[uRj\")R&yF3$\"3O2++5gsz
**F77$$!3!p0fCSAEz(F3$\"3GQ_a$)R*H*QF37$$!3#>jV4<j7t(F3$!3-,)fXAe[F\"F
37$$!3-1#G%RR!*pwF3Fjbm7$$!3:!y7zqW&3wF3$!3-E\"R'o5(e9$F37$$!3EatRwa=Z
vF3$\"3)=TihJul)HF37$$!3GH>)[CEe[(F3$\"3U=@!*ptw$f\"F37$$!3R.lO8qYCuF3
Ffam7$$!3]x5&=y2JO(F3$!3s]#RYc%\\drF77$$!3i^cL]&[<I(F3$!3i:!><olO!>F37
$$!3jE-#)=$*QSsF3$!3D/Wz.1F3?F37$$!3u+[I(3I!zrF3$\"39O8]FiZJmF77$$!3'[
P*yb3n<rF3$\"3E!*==D2+'p*F77$$!3)*[RFC;JcqF3$!3'y'y$>PK'==F37$$!3)R_eF
R_\\*pF3$\"3\\A&z&H;vGSF37$$!35)4V7;$fLpF3$\"3)=sl0@oI2\"F37$$!3AswsHR
BsoF3$!3Qn'R\\f@Tx$F37$$!3BZA@)pu3\"oF3$!330*>^iGg*\\F37$$!3M@opma^\\n
F3$!37a\"oeh]R\\%F77$$!3Y&R\"=Ni:)o'F3$\"3OwE-0_hdqF37$$!3epfm.qzEmF3$
!3Ol'p&o)3pm#F37$$!3eW0:sxVllF3$\"3/ozfW&o#*>\"F37$$!3q=^jS&yS]'F3$!3t
Tp3-^C2SF37$$!3\"Gp>\"4$>FW'F3$!33O!R1=kGo\"F37$$!3%pE/w2g8Q'F3$!33N9B
21]HOF77$$!3%>%))3Y3+?jF3F\\\\m7$$!31;Md9;keiF3$\"3v\"4kZ8Mxq#F37$$!3;
!*z0$Q#G(>'F3$\"3Y@pW>V$fC\"F37$$!3HkDa^J#f8'F3$!3p)oC@DF:?#F77$$!3IRr
-?RcugF3$!3N7>e'*4pQWF37$$!3T8<^)o/K,'F3$\"3ug2m$><-K$F37$$!3a(G'*pXX=
&fF3$\"3Klm.fzE^7F37$$!3kh3[Di[!*eF3Fiil7$$!3lOa'R*p7HeF3$\"3Et9i`\"3L
%QF37$$!3y5+XixwndF3$!3-$4+j5c'ffF07$$!3*[eM4`3kq&F3$!3k3XRqb/k:F37$$!
3!*f\">%*H\\]k&F3$!3inQ-?0*\\b#F77$$!3-MP!z1!p$e&F3$\"3J&3o%e-w7=F37$$
!383$)QO3LAbF3$!3@r&e*G\"*GKBF77$$!3C#)G([gr4Y&F3$\"3MPM/p'*R!)>F37$$!
3CduNtBh*R&F3$\"38-Se\"[*oDBF37$$!3PJ?%=9`#Q`F3$!3x)Qd5\\\\=.%F37$$!3[
0mK5R*oF&F3$!3=$)f^N$HA#>F37$$!3gz6\")yY`:_F3$\"3XY!R\"p!*3-FF37$$!3ia
dHZa<a^F3$!3)[.M'*Q%HYDF37$$!3sG.y:i\"G4&F3$\"3)QA0'=dl[]F77$$!3%G!\\E
%)pXJ]F3$!3K[Kz1E$)yIF37$$!3'pZ\\Fv(4q\\F3$!3mZ!H9w:'p;F37$$!3(>0M7_Q(
3\\F3$\"3mnRXvwy3aF77$$!33E'=(*Gzt%[F3$!3()f4GHee@&)F77$$!3?+K?e+-'y%F
3$!3-wJ'zH/gC#F37$$!3@vxoE3mCZF3$!3>[zXey/+DF37$$!3K\\B<&f,Lm%F3$\"35`
7_@<q.PF37$$!3WBpljB%>g%F3$\"3-WD$*3bkvAF37$$!3c(\\T@8$eSXF3$\"3e&\\:v
s:[=#F77$$!3csgi+RAzWF3$!3K6NqHS?`JF77$$!3oY16pY'yT%F3$!3v79jzHE?mF77$
$!3!3A&fPa]cVF3$\"3I%\\wMG!\\e6F37$$!3#\\zzg?Y^H%F3$\"3[[?Mx(=x%GF37$$
!3#*pVcupyLUF3$!3+MC?-yD_AF37$$!3/W*[IuFC<%F3$\"3[\\,*yma'pKF77$$!3:=N
`6&o56%F3$!3a5S$*\\R3&z$F77$$!3G#4=+G4(\\SF3$!3ch![41F'[EF37$$!3%om-&[
+N))RF3$\"3a=AM,fk!R\"F37$$!3RTs)p\"3*p#RF3$!3c*)****oYXVGF77$$!3'f\"=
Z&eJc'QF3$\"3+]%ey*\\DyJF37$$!32!RcRNsU!QF3$\"3e$*H35\\\\W>F37$$!3kk4W
AJ\"Hu$F3$!3K_)o$eQ'yp'F37$$!3uQb#4*Qb\"o$F3$\"3(3-&3.L]+JF37$$!3J8,Tf
Y>?OF3F`^l7$$!3W(o%*yUN)eNF3$\"3EZH^Ux@UiF37$$!3+i#zj>wu\\$F3$!3i*>iz&
y[Z))F07$$!36OQ'['p6OMF3$!3=Z'*=h\"fH3$F37$$!3m5%[LtdZP$F3$\"3mZM)HyA_
'\\F37$$!3z%)H$=])R8LF3$\"3>6(yg(G.0>F37$$!3OfvJq#R?D$F3$\"3OCe337:@JF
37$$!3YL@!)Q+o!>$F3$!3%4GxJ2ENQ%F37$$!3.3nG23KHJF3$!3u4yTl6M[6F37$$!39
#Grddhz1$F3$!3*e%4DdcEBRF37$$!3rceDWBg1IF3$\"3Q!4bg3\\9Y*F77$$!3#3VSF6
V_%HF3$\"3_4E^(4yLO$F37$$!3Q0]A\")Q)Q)GF3$!392X;8Kh(*=F37$$!3&*z&4(\\Y
_AGF3$!3-EV'o3&yKvF37$$!31aT>=a;hFF3$\"3!=y-Uj#3q=F77$$!3iG(ym=1)*p#F3
$!3?k[^TP;^KF37$$!3u-L;bpWQEF3$\"3#yqx1uNd=#F77$$!3IxykBx3xDF3$!3>sC\"
>(*f!R<F37$$!3U^C8#\\Gd^#F3$\"3Pj5t2d(oh(F77$$!3)f-<1EpVX#F3$\"3cTH,:h
VfCF37$$!35+;5H+,$R#F3$!3r>qhEGkJ[F37$$!3muhe(z];L#F3$!3-)R+LR]J,#F37$
$!3y[22m:HqAF3$!3yy2W&eK`=$F37$$!3MB`bMB$*3AF3$\"3s,4@V&)*pK#F37$$!3Y(
*)RI5tv9#F3$\"3SN],`!)[\\AF37$$!3-sW_rQ@'3#F3$\"3q([Az!o(*)z#F37$$!39Y
!4+ka[-#F3$!3/(Qi\\7oC-&F37$$!3q?O\\3a\\j>F3$\"3A9f0\"\\n)yyF77$$!3/&>
yp<O@!>F3$!3*pVS0Rt<g$F37$$!3PpFYXpxS=F3$!3-'*y19:^\\*)F77$$!3sVt%Rr<%
z<F3$!3I1*)*=%\\M?iF37$$!30=>V#[e!=<F3$\"3MGA!y\\y;W#F37$$!3S#\\;4D*pc
;F3$\"3y:ER'\\12w#F37$$!3sm5S>+M&f\"F3$!3dM\\*G1bsr\"F37$$!32Tc)yy!)R`
\"F3$!3l[NI)4KRs\"F37$$!3U:-Pc:is9F3$!3wrao\"**p@n$F37$$!3(**ya[Ki7T\"
F3$!3*Hf.@f3kA$F37$$!3Kk$RL4.*\\8F3$!3Er3qPJr&H&F37$$!3mQR#='Qa)G\"F3$
!3L/x7dXR<5F07$$!3*H^3.j%=F7F3$!3%**)f5[ZX^:F37$$!3K(3$z)RDe;\"F3$\"3!
QHT(>JO&z$F37$$!3nhwFnhY/6F3$!3k*HsI0e6B#F37$$!3,OAwNp5V5F3$!35g;x<M')
=JF77$$!3\\.\"oC/xu\")*F7$!3IW$=`g(=GNF37$$!3)o%QJFZ)Q?*F7$!3Y>m@7&GG$
[F37$$!3E!ff@T#H!f)F7$!3)QNI,S&*Hc\"F37$$!3mL`+(4+n(zF7$!3K,Ks=Q$3D%F3
7$$!31x5&=y2JO(F7$!3k%)=l!\\8`3'F37$$!3c@opma^\\nF7$!3-=ATzEr(3\"F,7$$
!3'\\cU::Bf8'F7$!3so]AFTwf8F,7$$!3N3$)QO3LAbF7$!3]'o))z1)o&H%F,7$$!3u^
SB@&Q(3\\F7FV7$$!37&zzg?Y^H%F7$\"3y1ZO<g-BKF,7$$!3_Qb#4*Qb\"o$F7$\"3uL
Y[M&zim\"F,7$$!3[#Grddhz1$F7$\"3W-sn&o73Y\"F,7$$!3)e-<1EpVX#F7$\"3k4\"
eG**[2\\\"F,7$$!3EpFYXpxS=F7$\"3;YSt?MS5()F37$$!3%H^3.j%=F7F7$\"3lwZ&y
p!H,5F,7$$!3okDa^J#f8'F0$\"3%yiq#Qz<;wF3-%'COLOURG6&%$RGBG$\"#5!\"\"F(
F(-%+AXESLABELSG6$Q!6\"Fd[p-%&STYLEG6#%%LINEG-%%VIEWG6$%(DEFAULTGF]\\p
" 1 6 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot([seq([pv[i],yI[i]],i=
1..N)],style=line);" }}{PARA 13 "" 1 "" {GLPLOT2D 824 208 208 
{PLOTDATA 2 "6&-%'CURVESG6$7\\[l7$$\"\"!F)F(7$$\"3okDa^J#f8'!#>$!3%y1p
@'=Mve!#=7$$\"3%H^3.j%=F7F0$!3%G\"*=sb><x&!#<7$$\"3EpFYXpxS=F0$!3?xpG5
<3(e)F67$$\"3)e-<1EpVX#F0$!3qu,i\"H?,F\"!#:7$$\"3[#Grddhz1$F0$\"3qfJ\\
*=%3m5!#;7$$\"3_Qb#4*Qb\"o$F0$\"32F8;!yd79\"FG7$$\"37&zzg?Y^H%F0$!3dSG
8*)=c0<F07$$\"3u^SB@&Q(3\\F0$\"3kXf[/#4T)RF67$$\"3N3$)QO3LAbF0$\"3m!='
)HDN'z@F67$$\"3'\\cU::Bf8'F0$!3NqLK,f_QQF67$$\"3c@opma^\\nF0$!3+H;laR(
R<$F67$$\"31x5&=y2JO(F0$\"3Ji)pVh3uS\"F67$$\"3mL`+(4+n(zF0$!3EAC'>Z@V*
HF67$$\"3E!ff@T#H!f)F0$\"3\")\\LRR%)y3;F67$$\"3)o%QJFZ)Q?*F0$!3_:/252&
=H\"F67$$\"3\\.\"oC/xu\")*F0$\"3IezK'G\"p`HF67$$\"3,OAwNp5V5F6$!3mQ%4%
=5>XQF67$$\"3nhwFnhY/6F6$\"3q!z!zQ3f[6FG7$$\"3K(3$z)RDe;\"F6$\"3k@[jA
\"eVU$F67$$\"3*H^3.j%=F7F6$!35t)=LyJ\"GOF67$$\"3mQR#='Qa)G\"F6$\"3EiNo
><@aAF67$$\"3Kk$RL4.*\\8F6$\"3O;k0xR&*flF67$$\"3(**ya[Ki7T\"F6$!3ZPD<?
(*z/YF67$$\"3U:-Pc:is9F6$!3yzBF!f5mm*F07$$\"32Tc)yy!)R`\"F6$!3/+nr)*\\
NuIF67$$\"3sm5S>+M&f\"F6$\"3g$p,e(o#3\\#F67$$\"3S#\\;4D*pc;F6$!3W#eJj!
=-lGF07$$\"30=>V#[e!=<F6$!3aqF1+m<F;F67$$\"3sVt%Rr<%z<F6$!3'>?CQ?4>c#F
07$$\"3PpFYXpxS=F6$!34G!RO\\>J8*F07$$\"3/&>yp<O@!>F6$\"3n$fZLG')Q]$F07
$$\"3q?O\\3a\\j>F6$\"3uu!GC'f?wcF07$$\"39Y!4+ka[-#F6$\"3N<d))**\\+Q@F6
7$$\"3-sW_rQ@'3#F6$\"3v:/]A))>3BF07$$\"3Y(*)RI5tv9#F6$!3IdKA+j(G\"=F67
$$\"3MB`bMB$*3AF6$\"3-u\"=7<Y&)\\'F67$$\"3y[22m:HqAF6$!3)*[!fN/,;*RF67
$$\"3muhe(z];L#F6$\"3'Qv8_]*)fj\"F67$$\"35+;5H+,$R#F6$\"3U5'zah_Jv#F07
$$\"3)f-<1EpVX#F6$\"3G!Hxv]jxR\"F67$$\"3U^C8#\\Gd^#F6$\"3'\\m\"GCMeHCF
67$$\"3IxykBx3xDF6$\"3;mLZGI&zq$F67$$\"3u-L;bpWQEF6$!3%z=wWA')Hm$F67$$
\"3iG(ym=1)*p#F6$!3!))>O\"p!3Bc%F67$$\"31aT>=a;hFF6$\"39EBk'\\)G*z\"F0
7$$\"3&*z&4(\\Y_AGF6$!3%R]TTvZDq#F67$$\"3Q0]A\")Q)Q)GF6$!3E43!yv?PP\"F
67$$\"3#3VSF6V_%HF6$!3JOa4L962GF67$$\"3rceDWBg1IF6$!33k!f_E:*\\EF67$$
\"39#Grddhz1$F6$\"3S<'=B9+nm#F67$$\"3.3nG23KHJF6$!3:Bi9a+j1AF67$$\"3YL
@!)Q+o!>$F6$\"3!Hm(pz0s%4&F67$$\"3OfvJq#R?D$F6$!3uHM,Q%HP)=F67$$\"3z%)
H$=])R8LF6$!3QIJ5%px!*f\"F07$$\"3m5%[LtdZP$F6$!3-/&=:Ql@<\"F67$$\"36OQ
'['p6OMF6$!3u8;D'*zh+NF07$$\"3+i#zj>wu\\$F6$\"3g$y_jk%>e5F67$$\"3W(o%*
yUN)eNF6$!3-Q:$[>C[K*F07$$\"3J8,TfY>?OF6$!3o-y')[ePWMF67$$\"3uQb#4*Qb
\"o$F6$!3O%Hq(fhg^qF67$$\"3kk4WAJ\"Hu$F6$\"32JB<6j`7AF67$$\"32!RcRNsU!
QF6$\"3?z3M\"3C0'QF07$$\"3'f\"=Z&eJc'QF6$!3;U*opX**z1'F67$$\"3RTs)p\"3
*p#RF6$!3s1+]jLS)G$F67$$\"3%om-&[+N))RF6$!3wrG2`)*4*Q$F67$$\"3G#4=+G4(
\\SF6$!3([BY%ps!>7\"F67$$\"3:=N`6&o56%F6$\"3g*3#=tsO(f$F67$$\"3/W*[IuF
C<%F6$!3v&fK$=0G3KF67$$\"3#*pVcupyLUF6$!3oUp6k*QBv&F67$$\"3#\\zzg?Y^H%
F6$!3))efxb!G&e:F67$$\"3!3A&fPa]cVF6$\"3TK<[^IaASF67$$\"3oY16pY'yT%F6$
\"3#HfFmq*f\"G%F07$$\"3csgi+RAzWF6$\"3\")>4r4<@QNF67$$\"3c(\\T@8$eSXF6
$\"3C:A?Fl4#>\"F67$$\"3WBpljB%>g%F6$\"3sK;aBzBB8F67$$\"3K\\B<&f,Lm%F6$
\"3vwE\"><N78#F67$$\"3@vxoE3mCZF6$\"3<Cb(\\T]oB#F67$$\"3?+K?e+-'y%F6$!
3?;J5PTHr8F67$$\"33E'=(*Gzt%[F6$!3728CEBb`:F67$$\"3(>0M7_Q(3\\F6$!348:
oc\"Ry#))F67$$\"3'pZ\\Fv(4q\\F6$!3\"4rhEb,+#>F67$$\"3%G!\\E%)pXJ]F6$\"
3LtPAi\"oFT\"F67$$\"3sG.y:i\"G4&F6$\"3)G?kNXz>d$F67$$\"3iadHZa<a^F6$!3
<\"z>dB5eJ&F07$$\"3gz6\")yY`:_F6$!3/s\"40;ca?&F67$$\"3[0mK5R*oF&F6$\"3
4\"*QO7?AMOF67$$\"3PJ?%=9`#Q`F6$\"3-%[NYBF`E&F07$$\"3CduNtBh*R&F6$\"3W
_(>yU1xV(F07$$\"3C#)G([gr4Y&F6$!3oX95E)3FJ&F67$$\"383$)QO3LAbF6$!34!G3
2$=[VBF67$$\"3-MP!z1!p$e&F6$\"3!Qpgz_&)*z5F67$$\"3!*f\">%*H\\]k&F6$\"3
S\\Rj$H'QysF67$$\"3*[eM4`3kq&F6$!3#o\\05VRl*GF67$$\"3y5+XixwndF6$!3]*3
R1?P)*e$F67$$\"3lOa'R*p7HeF6$\"3=i5Nm3iC;F67$$\"3kh3[Di[!*eF6$!35(>d\"
>LU99F67$$\"3a(G'*pXX=&fF6$\"3s_9?iki'G#F67$$\"3T8<^)o/K,'F6$\"3_Rs@**
)G#y]F67$$\"3IRr-?RcugF6$\"3u1x)*=?l:lF67$$\"3HkDa^J#f8'F6$\"3]3Ylh..)
G\"F67$$\"3;!*z0$Q#G(>'F6$\"31\")Qr$*>lc;F67$$\"31;Md9;keiF6$!3Q<&4+z5
%[sF07$$\"3%>%))3Y3+?jF6$!3Dl_\\Ay$GI\"F67$$\"3%pE/w2g8Q'F6$!3MV9A5g,M
$)F07$$\"3\"Gp>\"4$>FW'F6$\"3an[xYsNb?F67$$\"3q=^jS&yS]'F6$\"3!)\\X*HP
[?1&F07$$\"3eW0:sxVllF6$!3iN/)zBQ`A$F67$$\"3epfm.qzEmF6$!3$[B#feuEJ=F6
7$$\"3Y&R\"=Ni:)o'F6$!38@h@X'zK9%F67$$\"3M@opma^\\nF6$\"3qT7ZQ2WUAF67$
$\"3BZA@)pu3\"oF6$\"3'e$p\"=Bx>N\"F67$$\"3AswsHRBsoF6$!3BE\"eK8%o=NF67
$$\"35)4V7;$fLpF6$\"33_CFl.!*)*=F67$$\"3)R_eFR_\\*pF6$!3x%z>:`u5S'F67$
$\"3)*[RFC;JcqF6$\"3;5W1u.j]^F67$$\"3'[P*yb3n<rF6$!3'**[dsKQM:%F67$$\"
3u+[I(3I!zrF6$\"3!3N\"**p(os2\"F07$$\"3jE-#)=$*QSsF6$!3yaj\"GWOw'yF07$
$\"3i^cL]&[<I(F6$\"3bt%)zJ.#HU&F67$$\"3]x5&=y2JO(F6$!3CBd#3E'QF;F67$$
\"3R.lO8qYCuF6$\"39wX!\\\">9\"[$F67$$\"3GH>)[CEe[(F6$\"3o)QI$*eH&R=F67
$$\"3EatRwa=ZvF6$\"3:Q(pN\\\\0I\"F67$$\"3:!y7zqW&3wF6$\"3'G\"p\"\\%G)y
/'F67$$\"3-1#G%RR!*pwF6$\"39P?qpiMlmF67$$\"3#>jV4<j7t(F6$!31?Y?q34'\\%
F67$$\"3!p0fCSAEz(F6$\"3#zrc\"Rpj?@F67$$!3y#[uRj\")R&yF6F(7$$!3!p0fCSA
Ez(F6$!3e?n:Rpj?@F67$$!3#>jV4<j7t(F6$\"3#)HY?q34'\\%F67$$!3-1#G%RR!*pw
F6$!3e\\?qpiMlmF67$$!3:!y7zqW&3wF6$!3'G\"p\"\\%G)y/'F67$$!3EatRwa=ZvF6
$!3.R(pN\\\\0I\"F67$$!3GH>)[CEe[(F6$!3!pQI$*eH&R=F67$$!3R.lO8qYCuF6$!3
[xX!\\\">9\"[$F67$$!3]x5&=y2JO(F6$\"3!)=d#3E'QF;F67$$!3i^cL]&[<I(F6$!3
Kv%)zJ.#HU&F67$$!3jE-#)=$*QSsF6$\"3'eQ;GWOw'yF07$$!3u+[I(3I!zrF6$!3Wx8
**p(os2\"F07$$!3'[P*yb3n<rF6$\"3S'[dsKQM:%F67$$!3)*[RFC;JcqF6$!3;-W1u.
j]^F67$$!3)R_eFR_\\*pF6$\"3B$y>:`u5S'F67$$!35)4V7;$fLpF6$!3kZCFl.!*)*=
F67$$!3AswsHRBsoF6$\"3BE\"eK8%o=NF67$$!3BZA@)pu3\"oF6$!3IQp\"=Bx>N\"F6
7$$!3M@opma^\\nF6$!3.Z7ZQ2WUAF67$$!3Y&R\"=Ni:)o'F6$\"3oKh@X'zK9%F67$$!
3epfm.qzEmF6$\"3%zB#feuEJ=F67$$!3eW0:sxVllF6$\"3iJ/)zBQ`A$F67$$!3q=^jS
&yS]'F6$!3!)\\X*HP[?1&F07$$!3\"Gp>\"4$>FW'F6$!3?m[xYsNb?F67$$!3%pE/w2g
8Q'F6$\"3\\%R@-,;SL)F07$$!3%>%))3Y3+?jF6$\"3Eh_\\Ay$GI\"F67$$!31;Md9;k
eiF6$\"3qI&4+z5%[sF07$$!3;!*z0$Q#G(>'F6$!3;!)Qr$*>lc;F67$$!3HkDa^J#f8'
F6$!3Q,Ylh..)G\"F67$$!3IRr-?RcugF6$!3>&p()*=?l:lF67$$!3T8<^)o/K,'F6$!3
_Js@**)G#y]F67$$!3a(G'*pXX=&fF6$!3G[9?iki'G#F67$$!3kh3[Di[!*eF6$\"3W'>
d\">LU99F67$$!3lOa'R*p7HeF6$!3Hf5Nm3iC;F67$$!3y5+XixwndF6$\"3](4R1?P)*
e$F67$$!3*[eM4`3kq&F6$\"3q0b+J%Rl*GF67$$!3!*f\">%*H\\]k&F6$!3S\\Rj$H'Q
ysF67$$!3-MP!z1!p$e&F6$!3q!pgz_&)*z5F67$$!383$)QO3LAbF6$\"3lr#32$=[VBF
67$$!3C#)G([gr4Y&F6$\"3\"RW,h#)3FJ&F67$$!3CduNtBh*R&F6$!3lu(>yU1xV(F07
$$!3PJ?%=9`#Q`F6$!3-%[NYBF`E&F07$$!3[0mK5R*oF&F6$!3k%*QO7?AMOF67$$!3gz
6\")yY`:_F6$\"3e$=40;ca?&F67$$!3iadHZa<a^F6$\"3YW)>dB5eJ&F07$$!3sG.y:i
\"G4&F6$!3y&>kNXz>d$F67$$!3%G!\\E%)pXJ]F6$!3+sPAi\"oFT\"F67$$!3'pZ\\Fv
(4q\\F6$\"3p1<m_:+?>F67$$!3(>0M7_Q(3\\F6$\"3J6:oc\"Ry#))F67$$!33E'=(*G
zt%[F6$\"3Y/8CEBb`:F67$$!3?+K?e+-'y%F6$\"3a<J5PTHr8F67$$!3@vxoE3mCZF6$
!3%G_v\\T]oB#F67$$!3K\\B<&f,Lm%F6$!3UvE\"><N78#F67$$!3WBpljB%>g%F6$!3%
yiTN#zBB8F67$$!3c(\\T@8$eSXF6$!3-8A?Fl4#>\"F67$$!3csgi+RAzWF6$!3#\\\"4
r4<@QNF67$$!3oY16pY'yT%F6$!3;_yi1(*f\"G%F07$$!3!3A&fPa]cVF6$!33N<[^IaA
SF67$$!3#\\zzg?Y^H%F6$\"3Kffxb!G&e:F67$$!3#*pVcupyLUF6$\"3Y_p6k*QBv&F6
7$$!3/W*[IuFC<%F6$\"34$fK$=0G3KF67$$!3:=N`6&o56%F6$!3['3#=tsO(f$F67$$!
3G#4=+G4(\\SF6$\"3lMiWps!>7\"F67$$!3%om-&[+N))RF6$\"3UmG2`)*4*Q$F67$$!
3RTs)p\"3*p#RF6$\"3s1+]jLS)G$F67$$!3'f\"=Z&eJc'QF6$\"3/^*opX**z1'F67$$
!32!RcRNsU!QF6$!3ax3M\"3C0'QF07$$!3kk4WAJ\"Hu$F6$!3%oLs6JOD@#F67$$!3uQ
b#4*Qb\"o$F6$\"3#GGq(fhg^qF67$$!3J8,TfY>?OF6$\"3Y'zn)[ePWMF67$$!3W(o%*
yUN)eNF6$\"3qC:$[>C[K*F07$$!3+i#zj>wu\\$F6$!3Q*y_jk%>e5F67$$!36OQ'['p6
OMF6$\"3hP;D'*zh+NF07$$!3m5%[LtdZP$F6$\"374&=:Ql@<\"F67$$!3z%)H$=])R8L
F6$\"3CzJ5%px!*f\"F07$$!3OfvJq#R?D$F6$\"3'>V8!Q%HP)=F67$$!3YL@!)Q+o!>$
F6$!3cdwpz0s%4&F67$$!3.3nG23KHJF6$\"3/?i9a+j1AF67$$!39#Grddhz1$F6$!3=:
'=B9+nm#F67$$!3rceDWBg1IF6$\"3if!f_E:*\\EF67$$!3#3VSF6V_%HF6$\"3wOa4L9
62GF67$$!3Q0]A\")Q)Q)GF6$\"39;3!yv?PP\"F67$$!3&*z&4(\\Y_AGF6$\"3h5:9ax
a-FF67$$!31aT>=a;hFF6$!3!GLUm\\)G*z\"F07$$!3iG(ym=1)*p#F6$\"3e+i8p!3Bc
%F67$$!3u-L;bpWQEF6$\"3E&=wWA')Hm$F67$$!3IxykBx3xDF6$!3&*fLZGI&zq$F67$
$!3U^C8#\\Gd^#F6$!3in;GCMeHCF67$$!3)f-<1EpVX#F6$!3i(Gxv]jxR\"F67$$!35+
;5H+,$R#F6$!3)fgzah_Jv#F07$$!3muhe(z];L#F6$!3>bP@0&*)fj\"F67$$!3y[22m:
HqAF6$\"3we!fN/,;*RF67$$!3MB`bMB$*3AF6$!3-u\"=7<Y&)\\'F67$$!3Y(*)RI5tv
9#F6$\"3T_KA+j(G\"=F67$$!3-sW_rQ@'3#F6$!32H/]A))>3BF07$$!39Y!4+ka[-#F6
$!3d>d))**\\+Q@F67$$!3q?O\\3a\\j>F6$!3'p1GC'f?wcF07$$!3/&>yp<O@!>F6$!3
X\"eZLG')Q]$F07$$!3PpFYXpxS=F6$\"3lB!RO\\>J8*F07$$!3sVt%Rr<%z<F6$\"3uz
T#Q?4>c#F07$$!30=>V#[e!=<F6$\"34yF1+m<F;F67$$!3S#\\;4D*pc;F6$\"3U7;L1=
-lGF07$$!3sm5S>+M&f\"F6$!3[!p,e(o#3\\#F67$$!32Tc)yy!)R`\"F6$\"3/+nr)*
\\NuIF67$$!3U:-Pc:is9F6$\"358CF!f5mm*F07$$!3(**ya[Ki7T\"F6$\"3eOD<?(*z
/YF67$$!3Kk$RL4.*\\8F6$!39Ek0xR&*flF67$$!3mQR#='Qa)G\"F6$!3[sNo><@aAF6
7$$!3*H^3.j%=F7F6$\"3)3()=LyJ\"GOF67$$!3K(3$z)RDe;\"F6$!3l8[jA\"eVU$F6
7$$!3nhwFnhY/6F6$!35*y!zQ3f[6FG7$$!3,OAwNp5V5F6$\"3oE%4%=5>XQF67$$!3\\
.\"oC/xu\")*F0$!3UdzK'G\"p`HF67$$!3)o%QJFZ)Q?*F0$\"3I,/252&=H\"F67$$!3
E!ff@T#H!f)F0$!38jLRR%)y3;F67$$!3mL`+(4+n(zF0$\"3Q@C'>Z@V*HF67$$!31x5&
=y2JO(F0$!3(\\')pVh3uS\"F67$$!3c@opma^\\nF0$\"3!*H;laR(R<$F67$$!3'\\cU
::Bf8'F0$\"3NqLK,f_QQF67$$!3N3$)QO3LAbF0$!3w(=')HDN'z@F67$$!3u^SB@&Q(3
\\F0$!3A<f[/#4T)RF67$$!37&zzg?Y^H%F0$\"3erH8*)=c0<F07$$!3_Qb#4*Qb\"o$F
0$!3%)G8;!yd79\"FG7$$!3[#Grddhz1$F0$!31jJ\\*=%3m5FG7$$!3)e-<1EpVX#F0$
\"3Uu,i\"H?,F\"FA7$$!3EpFYXpxS=F0$\"3wkpG5<3(e)F67$$!3%H^3.j%=F7F0$\"3
s8*=sb><x&F67$$!3okDa^J#f8'F-$\"3?8$p@'=MveF0-%'COLOURG6&%$RGBG$\"#5!
\"\"F(F(-%+AXESLABELSG6$Q!6\"F\\\\p-%&STYLEG6#%%LINEG-%%VIEWG6$%(DEFAU
LTGFe\\p" 1 6 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "p_c:=1.5;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$p_cG$\"#:!\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 72 "for i from 1 to N do: if abs(pv[i])>p_c then yR[i]
:=0: yI[i]:=0: fi: od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Now go \+
back to the time domain using the inverse Fourier transform:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalhf(iFFT(n, var(yR), var(
yI)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$c#\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "P2:=plot([seq([xv[i],yR[i]],i=1..N)
],style=line,color=blue): display(P1,P2);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 824 208 208 {PLOTDATA 2 "6&-%'CURVESG6%7\\[l7$$!3H++++++?^!#
;$\"34+++&p`j!=!#<7$$!3W++++++!3&F*$\"3,+++J3^2AF-7$$!3d++++++S]F*$\"3
1+++\"Gm&*>#F-7$$!#]\"\"!$\"3$******>@c_+#F-7$$!39++++++g\\F*$\"3>+++U
Q1NAF-7$$!3H++++++?\\F*$\"3:+++3\">4B#F-7$$!3W++++++!)[F*$\"3'********
3Ho'=F-7$$!3d++++++S[F*$\"3)*******H%>5b\"F-7$$!#[F;$\"3)******f,e'f:F
-7$$!39++++++gZF*$\"3a******pb*Ql*!#=7$$!3H++++++?ZF*$\"33+++`:a'>\"F-
7$$!3W++++++!o%F*$!3e*******4#o%>'!#?7$$!3d++++++SYF*$\"3%******zN3%pQ
Ffn7$$!#YF;$\"3m******4d<8Q!#>7$$!39++++++gXF*$!3=+++!y0X\"[F\\p7$$!3H
++++++?XF*$!3!******\\6+2,\"F-7$$!3W++++++![%F*$!3'******4J)[h5F-7$$!3
d++++++SWF*$!3#)******z%R&o#*Ffn7$$!#WF;$!3O+++L)Hu\"zFfn7$$!39++++++g
VF*$!3#*******\\3\"HD\"F-7$$!3H++++++?VF*$!3y******\\e+vZFfn7$$!3W++++
++!G%F*$!3%)*****fZ:hR*Ffn7$$!3d++++++SUF*$!3y*****4lo/e%Ffn7$$!#UF;$
\"31+++`<jf=Ffn7$$!39++++++gTF*$\"3&)*****fC]Ei$Ffn7$$!3H++++++?TF*$\"
39+++)f))zf%Ffn7$$!3W++++++!3%F*$\"3A+++%Gx[x(Ffn7$$!3d++++++SSF*$\"3!
******f&)=@L\"F-7$$!#SF;$\"35+++UZ!**e\"F-7$$!39++++++gRF*$\"33+++I'zk
v\"F-7$$!3H++++++?RF*$\"3@+++u:M/@F-7$$!3W++++++!)QF*$\"38+++1GNR?F-7$
$!3d++++++SQF*$\"3-+++2D!*4<F-7$$!#QF;$\"3))*****4\\J'y?F-7$$!39++++++
gPF*$\"35+++^tl=:F-7$$!3H++++++?PF*$\"37+++'o$po;F-7$$!3W++++++!o$F*$
\"3!******>[5d*=F-7$$!3')************ROF*$\"3-+++Wi)y[\"F-7$$!#OF;$\"3
3+++1V&QJ\"F-7$$!39++++++gNF*$\"3-+++Es2#*HFfn7$$!3H++++++?NF*$\"3O+++
IIh9XF\\p7$$!3W++++++![$F*$!35+++u5^`XFfn7$$!3')************RMF*$!3_**
***RCr%p%*Ffn7$$!#MF;$!35+++m!=c3\"F-7$$!39++++++gLF*$!3%******RhB`x\"
F-7$$!3H++++++?LF*$!3A+++uu=_@F-7$$!3W++++++!G$F*$!3y*****pqAiq#F-7$$!
3')************RKF*$!39+++.['*)*GF-7$$!#KF;$!3&)******oWOFFF-7$$!39+++
+++gJF*$!3%******H)*[&*\\#F-7$$!3H++++++?JF*$!3z*****4q'[EKF-7$$!33+++
+++!3$F*$!3/+++&3UFN$F-7$$!3A++++++SIF*$!32+++?A(QA#F-7$$!#IF;$!3++++c
TCw?F-7$$!39++++++gHF*$!3*)******\\z[6@F-7$$!3H++++++?HF*$!3)******>&o
4/:F-7$$!33++++++!)GF*$!3'*******Q)R]G\"F-7$$!3A++++++SGF*$!3B+++YmuGW
Ffn7$$!#GF;$\"3O+++!)==\")[F\\p7$$!39++++++gFF*$!3B+++G'[^@$Ffn7$$!3H+
+++++?FF*$\"3++++y1gJ6Ffn7$$!33++++++!o#F*$\"33+++c@6#=\"F-7$$!3A+++++
+SEF*$\"3]*****fW*y#\\*Ffn7$$!#EF;$\"3&******p)zS<8F-7$$!39++++++gDF*$
\"3)******\\p6Q(>F-7$$!3H++++++?DF*$\"3A+++!pS3E#F-7$$!33++++++![#F*$
\"39+++oD!R=#F-7$$!3A++++++SCF*$\"3=+++-W9@AF-7$$!#CF;$\"3$*******zu*o
.#F-7$$!39++++++gBF*$\"3/+++_1C'y\"F-7$$!3H++++++?BF*$\"3(******p**GUq
\"F-7$$!33++++++!G#F*$\"3#******R)3*GC\"F-7$$!3A++++++SAF*$\"3]******e
=2\"*yFfn7$$!#AF;$\"3%)*****HZ2mA'Ffn7$$!39++++++g@F*$\"3%******4j?fH#
Ffn7$$!3H++++++?@F*$\"3'******R@fd#[Ffn7$$!33++++++!3#F*$!3()*****R4CD
]#Ffn7$$!3A++++++S?F*$!3%*******zW!\\q#Ffn7$$FaoF;$!37+++i&ouv%Ffn7$$!
39++++++g>F*$!3e*****RNndX&Ffn7$$!3H++++++?>F*$!3,+++l\\**4yFfn7$$!33+
+++++!)=F*$!3w*****Rev>E)Ffn7$$!3A++++++S=F*$!3]+++v5NyvFfn7$$FfnF;$!3
j*****Hb#e?nFfn7$$!39++++++g<F*$!3)******R4I:$HFfn7$$!3$*************>
<F*$!3++++Jtov\")Ffn7$$!33++++++!o\"F*$\"39++++Q![l$Fao7$$!3A++++++S;F
*$!3!******zp_$)f#Ffn7$$F*F;$\"3!*******G$3ma\"Ffn7$$!39++++++g:F*$\"3
=+++iZM,KFfn7$$!36++++++?:F*$\"3-+++EX-![\"F-7$$!33++++++![\"F*$\"3)**
****4T(3\"Q\"F-7$$!3/++++++S9F*$\"3++++;)4\"G=F-7$$!#9F;$\"3/+++\\Xgl9
F-7$$!39++++++g8F*$\"3#)*****H4]lK#F-7$$!36++++++?8F*$\"3$******f8x.#>
F-7$$!33++++++!G\"F*$\"3-+++&*Q$)[?F-7$$!3/++++++S7F*$\"3/+++iX*)))>F-
7$$!#7F;$\"3#******Rm\"3*o\"F-7$$!39++++++g6F*$\"3/+++)=qfC\"F-7$$!36+
+++++?6F*$\"3))*****fuvFS(Ffn7$$!33++++++!3\"F*$\"3y*****\\C7?a&Ffn7$$
!3/++++++S5F*$\"3\"******\\qN%=!)Ffn7$$!#5F;$\"3-+++h`'*>6Ffn7$$!3U,++
++++'*F-$!3_+++i)[iu'Ffn7$$!31,++++++#*F-$!33+++WFgY5F-7$$!3q+++++++))
F-$!3)*******H_7y)*Ffn7$$!3M+++++++%)F-$!3$******p@y)49F-7$$!\")F;$!3#
******fRLa*>F-7$$!3_+++++++wF-$!3?+++*fqB%HF-7$$!3;+++++++sF-$!3-+++,*
zzj#F-7$$!3q+++++++oF-$!3))******H%y:Y#F-7$$!3M+++++++kF-$!32+++V%z-!H
F-7$$!\"'F;$!3;+++Ge-*\\#F-7$$!3_+++++++cF-$!3%******pI.&*G$F-7$$!3;++
+++++_F-$!3;+++4[FUCF-7$$!3q+++++++[F-$!3%)*****4FNJq#F-7$$!3M+++++++W
F-$!3!*******G#ooX#F-7$$!\"%F;$!3++++i\\he<F-7$$!33+++++++OF-$!3/+++IP
'y4\"F-7$$!3;+++++++KF-$!3#)*****zY[]R$Ffn7$$!3E+++++++GF-$\"3++++!\\'
*yX#F\\p7$$!3M+++++++CF-$\"3Y*****\\O>4D'Ffn7$$!\"#F;$\"3y*****\\Fopb$
Ffn7$$!33+++++++;F-$\"3(*******4rS=9F-7$$!3;+++++++7F-$\"3#******p0CTf
\"F-7$$!3U+++++++!)Ffn$\"3)******H&Qs2@F-7$$!3A+++++++SFfn$\"3++++cKXJ
AF-7$$F;F;$\"3)*******>)3O.#F-7$$\"3A+++++++SFfn$\"39+++onYj@F-7$$\"3U
+++++++!)Ffn$\"3/+++)=E\"*e\"F-7$$\"3;+++++++7F-$\"3-+++mz%e&=F-7$$\"3
3+++++++;F-$\"33+++*3V$e:F-7$$\"\"#F;$\"31+++n$omu\"F-7$$\"3M+++++++CF
-$\"3()*****pQ()4s)Ffn7$$\"3E+++++++GF-$\"3++++.1r[8F-7$$\"3;+++++++KF
-$\"32+++1V/F5F-7$$\"33+++++++OF-$\"3i*****z!f^4jFfn7$$\"\"%F;$!3-+++9
o%yI$Ffn7$$\"3M+++++++WF-$!3%******H!=0%y\"Ffn7$$\"3q+++++++[F-$!33+++
kuMg()Ffn7$$\"3;+++++++_F-$!3G+++)RtvI%Ffn7$$\"3_+++++++cF-$!3i*****HX
;_E(Ffn7$$\"\"'F;$!34+++7UT<8F-7$$\"3M+++++++kF-$!3#******z(*y/2\"F-7$
$\"3q+++++++oF-$!36+++/f>?xFfn7$$\"3;+++++++sF-$!3;+++I<etVFfn7$$\"3_+
++++++wF-$!3#*******Hu5\\cFfn7$$\"\")F;$!31+++QQ\"y<#Ffn7$$\"3M+++++++
%)F-$!3a+++q,R8vF\\p7$$\"3q+++++++))F-$\"3D++++S/VTFao7$$\"31,++++++#*
F-$\"37+++Led@%)Ffn7$$\"3U,++++++'*F-$\"3?+++)y>O-%Ffn7$$\"#5F;$\"33++
+sE%\\Z\"F-7$$\"3/++++++S5F*$\"3/+++BBt$)=F-7$$\"33++++++!3\"F*$\"3?++
+<4)o1#F-7$$\"36++++++?6F*$\"3(******RU'p6=F-7$$\"39++++++g6F*$\"3?+++
Duw#=#F-7$$\"#7F;$\"37+++*3+BQ#F-7$$\"3/++++++S7F*$\"3/+++W)R9'>F-7$$
\"33++++++!G\"F*$\"3))*****R[xa4#F-7$$\"36++++++?8F*$\"3'******>;`s#>F
-7$$\"39++++++g8F*$\"3$******>H!zQ;F-7$$\"#9F;$\"36+++\"fWoB\"F-7$$\"3
/++++++S9F*$\"3:++++x$3B(Ffn7$$\"33++++++![\"F*$\"3.+++t^uB9Ffn7$$\"36
++++++?:F*$!3)*******[4Hg:Ffn7$$\"39++++++g:F*$!3++++m=f#o\"Ffn7$$\"#;
F;$!35+++ra;\\6F-7$$\"3A++++++S;F*$!3%******4P*4n6F-7$$\"33++++++!o\"F
*$!3.+++1\\\"H/#F-7$$\"3$*************><F*$!3')*****4\\H)\\AF-7$$\"39+
+++++g<F*$!3))*****pYj))H#F-7$$\"#=F;$!33+++9\"z!)H$F-7$$\"3A++++++S=F
*$!32+++1%>BI$F-7$$\"33++++++!)=F*$!3%*******>\\xXDF-7$$\"3H++++++?>F*
$!3A+++?D@6LF-7$$\"39++++++g>F*$!3)******RSv^#HF-7$$\"#?F;$!3!******Ho
$prIF-7$$\"3A++++++S?F*$!3>+++FPrEDF-7$$\"33++++++!3#F*$!3/+++#R!>b<F-
7$$\"3H++++++?@F*$!3/+++*RC]A\"F-7$$\"39++++++g@F*$!3a******3Rb\\')Ffn
7$$\"#AF;$!3-+++ko$Q@#Ffn7$$\"3A++++++SAF*$!3&******pa15/#Ffn7$$\"33++
++++!G#F*$\"3t******z^].KFfn7$$\"3H++++++?BF*$\"3;+++%)yO$=)Ffn7$$\"39
++++++gBF*$\"3]+++g\\j]$*Ffn7$$\"#CF;$\"3(******\\)4B1:F-7$$\"3A++++++
SCF*$\"3-+++Yw(*Q8F-7$$\"33++++++![#F*$\"3%******>[o'3>F-7$$\"3H++++++
?DF*$\"3\")******QhjSAF-7$$\"39++++++gDF*$\"3-+++3t$f*>F-7$$\"#EF;$\"3
)******Rx&GCDF-7$$\"3A++++++SEF*$\"3)******H[P@#>F-7$$\"33++++++!o#F*$
\"3/+++\\`5a<F-7$$\"3H++++++?FF*$\"3++++u;f$e\"F-7$$\"39++++++gFF*$\"3
#******>CkNY\"F-7$$\"#GF;$\"3++++Ev?$H\"F-7$$\"3A++++++SGF*$\"3Q+++#RR
$e&*Ffn7$$\"33++++++!)GF*$\"31+++Y@+!G$Ffn7$$\"3H++++++?HF*$!3'******f
3z3N&Ffn7$$\"39++++++gHF*$\"3L+++]]tS]F\\p7$$\"#IF;$!35+++0pJ!3\"F-7$$
\"3A++++++SIF*$!3<+++'49^!QFfn7$$\"33++++++!3$F*$!3\"******>TKR%[Ffn7$
$\"3H++++++?JF*$!3u*****\\[G=%yFfn7$$\"39++++++gJF*$!3)******p)>.-9F-7
$$\"#KF;$!3]+++8d8z_Ffn7$$\"3')************RKF*$!3,+++ka)Q8%Ffn7$$\"3W
++++++!G$F*$!35+++$fD]1*Ffn7$$\"3H++++++?LF*$!3%******R:F>F#Ffn7$$\"39
++++++gLF*$\"3`++++\"HE5(F\\p7$$\"#MF;$\"3w*****Rtl3s#Ffn7$$\"3')*****
*******RMF*$\"3u******)*yOg%)Ffn7$$\"3W++++++![$F*$\"3;+++NQst\\Ffn7$$
\"3H++++++?NF*$\"3))******GmJt;F-7$$\"39++++++gNF*$\"3'******pu*H*>\"F
-7$$\"#OF;$\"3&******\\E9[!>F-7$$\"3')************ROF*$\"3y*****fdMTD#
F-7$$\"3W++++++!o$F*$\"3!******>G/*H<F-7$$\"3H++++++?PF*$\"3)*******Qh
Zo=F-7$$\"39++++++gPF*$\"3)******\\@$>m:F-7$$\"#QF;$\"3/+++&*R,Y:F-7$$
\"3d++++++SQF*$\"3\"******favgs\"F-7$$\"3W++++++!)QF*$\"3,+++rA4K:F-7$
$\"3H++++++?RF*$\"3g*****RKAb3)Ffn7$$\"39++++++gRF*$\"35+++p2(=$zFfn7$
$\"#SF;$\"3z*****zau/i%Ffn7$$\"3d++++++SSF*$!33+++]B()QCFfn7$$\"3W++++
++!3%F*$!3O+++A=)p=(Ffn7$$\"3H++++++?TF*$!3++++zQml6F-7$$\"39++++++gTF
*$!3%******4)o\\x;F-7$$\"#UF;$!37+++sKLrAF-7$$\"3d++++++SUF*$!3%)*****
R!G\\YAF-7$$\"3W++++++!G%F*$!3@+++))[\"GT#F-7$$\"3H++++++?VF*$!3))****
***p%HnFF-7$$\"39++++++gVF*$!3?+++mx'\\o#F-7$$\"#WF;$!3')*****pB4*3FF-
7$$\"3d++++++SWF*$!33+++T(=mz#F-7$$\"3W++++++![%F*$!37+++@)=tY#F-7$$\"
3H++++++?XF*$!3++++ovC&4#F-7$$\"39++++++gXF*$!3-+++!zm&p>F-7$$\"#YF;$!
3\")*****RMSn-#F-7$$\"3d++++++SYF*$!3(******>oH%=>F-7$$\"3W++++++!o%F*
$!3M+++hI2L$)Ffn7$$\"3H++++++?ZF*$!3-+++s*oB1\"F-7$$\"39++++++gZF*$!3)
)******ejh6>Ffn7$$\"#[F;$\"3D+++YoI3QFfn7$$\"3d++++++S[F*$\"3#******H^
#\\vbFfn7$$\"3W++++++!)[F*$\"3k*****pVcyG)Ffn7$$\"3H++++++?\\F*$\"35++
+HK`j7F-7$$\"39++++++g\\F*$\"3-+++=0D*y\"F-7$$\"#]F;$\"32+++%z8r?#F-7$
$\"3d++++++S]F*$\"3!*******Hm4[=F-7$$\"3W++++++!3&F*$\"3%******paCVt\"
F--%'COLOURG6&%$RGBG$Fb[n!\"\"FecmFecm-%&STYLEG6#%%LINEG-F$6%7\\[l7$F(
$\"3&\\Svn7e$Q?F-7$F/$\"3Y!*plvZww?F-7$F4$\"3')[DZ=^N1@F-7$F9$\"3n%3[
\"4$H'>@F-7$F?$\"361G!=%R_-@F-7$FD$\"3yOl61w9R?F-7$FI$\"3JDZe\\]!o\">F
-7$FN$\"3NAQMyA))H<F-7$FS$\"3#4t[&4ei\"[\"F-7$FX$\"3Gc.(GugN=\"F-7$Fhn
$\"3[7jQw1;I&)Ffn7$F]o$\"3'[mG$3FM(4&Ffn7$Fco$\"30p0$)f>\"ps\"Ffn7$Fho
$!3u1k\\8_y>9Ffn7$F^p$!3\\%3C8G+c@%Ffn7$Fcp$!3CX,1D**RglFfn7$Fhp$!3'*o
V/#\\tvO)Ffn7$F]q$!3'zB0,'4x_&*Ffn7$Fbq$!3$)[&\\iK(>.5F-7$Fgq$!3C=7f`r
'=t*Ffn7$F\\r$!3'=y$Q_@A5')Ffn7$Far$!3A3\"HO6h%zmFfn7$Ffr$!3M7O#y_+X-%
Ffn7$F[s$!3#\\(p.UU%*p!)F\\p7$F`s$\"3+GOl'Qlzu#Ffn7$Fes$\"3-.&fQK^6Q'F
fn7$Fjs$\"3<zuPRJtP)*Ffn7$F_t$\"3!zL'p%400H\"F-7$Fdt$\"3;Z)H#z<&Ra\"F-
7$Fit$\"3I5Rx$yfvt\"F-7$F^u$\"3K-@)H.3:(=F-7$Fcu$\"3Y5Gc\"Gp+&>F-7$Fhu
$\"3I@r`YSEy>F-7$F]v$\"3m(o2mU_)e>F-7$Fbv$\"3!zR+2Kv0*=F-7$Fgv$\"3o,kA
J<:o<F-7$F\\w$\"3kTrCrb*Re\"F-7$Faw$\"3S451PZ5J8F-7$Ffw$\"3o)e0b&3D15F
-7$F[x$\"3<y0^BM[DhFfn7$F`x$\"3!4fZgdTqg\"Ffn7$Fex$!3j6k)))Q>MJ$Ffn7$F
jx$!3&R\"=zF]@-%)Ffn7$F_y$!3NN=ZU5')R8F-7$Fdy$!3;MSw@eV/=F-7$Fiy$!3`P%
fPKA4@#F-7$F^z$!3;,rK5&\\3a#F-7$Fcz$!3'z(=z,I:\"y#F-7$Fhz$!39v^aV\")HC
HF-7$F][l$!3G`Be5?(z'HF-7$Fb[l$!3t$\\4V4BY\"HF-7$Fg[l$!3#z+0PE()4x#F-7
$F\\\\l$!3SAX/4@cZDF-7$Fa\\l$!3s0]H$=syD#F-7$Ff\\l$!331ds!4qt\">F-7$F[
]l$!3%oIp%4B&>a\"F-7$F`]l$!3#)RfKi@GY6F-7$Fe]l$!3!H27X+4JU(Ffn7$Fj]l$!
3QRfpE$4ZQ$Ffn7$F_^l$\"3!>ew+nG#3gF\\p7$Fd^l$\"3%)>x1Oh!3]%Ffn7$Fi^l$
\"33nJ?hA!*y#)Ffn7$F^_l$\"3mA3W6C/(=\"F-7$Fc_l$\"3V!)H1M*)R;:F-7$Fh_l$
\"3O%*>,^xu*z\"F-7$F]`l$\"3/6mmD.H<?F-7$Fb`l$\"3'Hn$p$)>%*[@F-7$Fg`l$
\"3c!)eJR[xy@F-7$F\\al$\"3qx(y7Vw%*4#F-7$Faal$\"3h6LCS&Qb\">F-7$Ffal$
\"3h2gaZ06W;F-7$F[bl$\"3mp()\\rm,88F-7$F`bl$\"3/GuZu1,i&*Ffn7$Febl$\"3
]#\\<'=h5wgFfn7$Fjbl$\"3c-MedFD]HFfn7$F_cl$\"3@gIz;(pNa$F\\p7$Fdcl$!3%
*Q)3LE5!o;Ffn7$Ficl$!3-O7'z1V`=$Ffn7$F^dl$!3$)z.DQUKNVFfn7$Fbdl$!3INH
\"*\\'QbE&Ffn7$Fgdl$!3KEQ8_?sugFfn7$F\\el$!3!G#4DDsWrnFfn7$Fael$!3MV;2
MkDisFfn7$Ffel$!3M%p\"p@Y)GP(Ffn7$Fjel$!35#pSzy%z'*oFfn7$F_fl$!3?V:A^i
\\dcFfn7$Fdfl$!3'\\8(Gq/FnNFfn7$Fifl$!3%[:*oj<dgmF\\p7$F^gl$\"3]2HZ(oZ
+(GFfn7$Fbgl$\"3W_T!4())p_nFfn7$Fggl$\"3_*ph/hzI1\"F-7$F\\hl$\"37wiWVj
@:9F-7$Fahl$\"3%4'eT(Q[@q\"F-7$Ffhl$\"3*fy\"*pUjS!>F-7$F[il$\"3W()Hp\"
4SB,#F-7$F`il$\"3NxD9,KUG?F-7$Feil$\"3+-4SCCyg>F-7$Fjil$\"3,W7)4a'3@=F
-7$F_jl$\"3!*R5\\L**z?;F-7$Fdjl$\"310tt2:6p8F-7$Fijl$\"3yh`n46cs5F-7$F
^[m$\"3oI<Z8VFhtFfn7$Fc[m$\"3%)[lA^^#3l$Ffn7$Fh[m$!3[eE]w9%fK$F\\p7$F]
\\m$!3iUU&=*4H'[%Ffn7$Fb\\m$!39#\\)=r$e1o)Ffn7$Fg\\m$!3+;Qf0E@x7F-7$F
\\]m$!3!QrB%)f&4i;F-7$Fa]m$!31A$[lZl1,#F-7$Ff]m$!3&ov-!)>)[8BF-7$F[^m$
!3G*=gpfELc#F-7$F`^m$!3As;jU%>Qv#F-7$Fe^m$!3/$3&=n0(y(GF-7$Fj^m$!3jjoU
DHbEHF-7$F__m$!3!p#QGs*G#*)GF-7$Fd_m$!3?0(o=#)3_v#F-7$Fi_m$!3r26rDzs;D
F-7$F^`m$!3_Y<*=g.D<#F-7$Fc`m$!3&e]m5h/3t\"F-7$Fh`m$!32#)QS!=j4@\"F-7$
F]am$!3,QBV3N)fU'Ffn7$Fbam$!3Sb<+0VrTiF\\p7$Fgam$\"3yM!G$H2c3\\Ffn7$F
\\bm$\"3$QT<9)*pI#)*Ffn7$Fabm$\"3m<??\")3.'Q\"F-7$Ffbm$\"3_!Ru;6N%)o\"
F-7$F[cm$\"33rM7lO&)))=F-7$F`cm$\"3e`/&H]*Q(*>F-7$Fecm$\"3Zl%))QOO1.#F
-7$Ficm$\"3\"*z^zDO[1?F-7$F^dm$\"32bql$f>%R>F-7$Fcdm$\"3a&fbaCiv$=F-7$
Fhdm$\"3Cqa)\\#37-<F-7$F]em$\"3rY&*z%y#GH:F-7$Fbem$\"31V>BmX$QJ\"F-7$F
gem$\"3C!=XzWcL0\"F-7$F\\fm$\"3@wfW3G3<vFfn7$Fafm$\"3)e#yZZUo2UFfn7$Ff
fm$\"3%*>w8&eq>*zF\\p7$F[gm$!3=MQbEkolCFfn7$F`gm$!3B&RhP\"HFN`Ffn7$Feg
m$!3g&oTc6?:f(Ffn7$Fjgm$!3IF^C#\\!H%3*Ffn7$F_hm$!3'=9Gmh2*[(*Ffn7$Fdhm
$!3T$)*42(3\"Qg*Ffn7$Fihm$!3aAQ6)=R3t)Ffn7$F^im$!3AvajK\"p`C(Ffn7$Fcim
$!3GES3p@)eE&Ffn7$Fhim$!3cwmQE5(>*GFfn7$F]jm$!3=cn'oZ?h'>F\\p7$Fbjm$\"
3?X`<b;BmFFfn7$Fgjm$\"3x*p$4.5]UfFfn7$F\\[n$\"31qq#4MN[D*Ffn7$Fa[n$\"3
1.)4GR&Re7F-7$Ff[n$\"3M-`Ege>w:F-7$F[\\n$\"3K8?iWi8e=F-7$F`\\n$\"3_b%Q
l&))*=3#F-7$Fe\\n$\"3Ze]$yEVpA#F-7$Fj\\n$\"3))G,a;*zzF#F-7$F_]n$\"30)e
2X>twA#F-7$Fd]n$\"3IBW,[d&y2#F-7$Fi]n$\"3U8IDo[\"*Q=F-7$F^^n$\"3Fi'>aD
)[F:F-7$Fc^n$\"3q'fPrA_I;\"F-7$Fh^n$\"3&3swpF4Gk(Ffn7$F]_n$\"3%QAqi_)>
hMFfn7$Fb_n$!3m+(e+nXp;)F\\p7$Fg_n$!3H=#G8%\\HT^Ffn7$F\\`n$!3YrZsZUF!
\\*Ffn7$Fa`n$!3A9&>>u*o$Q\"F-7$Ff`n$!3yuh9zJp6=F-7$F[an$!3xPb3+>W?AF-7
$F`an$!3'R*=5b\\Z!f#F-7$Fean$!3!po\\&[]#p*GF-7$Fjan$!3&*o%H_!**)H6$F-7
$F_bn$!3=005L9s9KF-7$Fdbn$!3=)eHh.Kg=$F-7$Fibn$!3f$)*3Z!onAIF-7$F^cn$!
3I=2Fvk9MFF-7$Fccn$!3$)Q3f))*pGM#F-7$Fhcn$!3E_Vg60x!)=F-7$F]dn$!3UX-&G
M%)QQ\"F-7$Fbdn$!3Io@-?[ji))Ffn7$Fgdn$!3qW]4*\\rf9%Ffn7$F\\en$\"3VU[H`
Q\\;:F\\p7$Faen$\"3Mpx%en')R)RFfn7$Ffen$\"33$)=$eD!G&R(Ffn7$F[fn$\"31t
oP$pjv/\"F-7$F`fn$\"3gwk8\"eB2L\"F-7$Fefn$\"3U<v&*>dl\"f\"F-7$Fjfn$\"3
_Jg>([FY#=F-7$F_gn$\"3.vg5F.^:?F-7$Fdgn$\"3()y$pFm:Y9#F-7$Fign$\"3U!e!
)zvO;>#F-7$F^hn$\"3@BRV<wMT@F-7$Fchn$\"3M<X&4yx%))>F-7$Fhhn$\"3s4#H>'G
RS<F-7$F]in$\"3.#e%[L1\"oT\"F-7$Fbin$\"3G2+([l.m/\"F-7$Fgin$\"3Xe.wC;Z
DmFfn7$F\\jn$\"33FcvLBJaHFfn7$Fajn$!3)G?jCf(\\*3$F\\p7$Ffjn$!3#QvOtreG
-$Ffn7$F[[o$!3#eH``$>zU^Ffn7$F`[o$!3AfV(f@#Q&p'Ffn7$Fe[o$!3cljEM?\\PxF
fn7$Fj[o$!3W(=9n)Q+9$)Ffn7$F_\\o$!3Bg,)R;q&G%)Ffn7$Fd\\o$!37\"HgE*>$p.
)Ffn7$Fi\\o$!3M[E/Rj([1(Ffn7$F^]o$!35%*Qf[sqWaFfn7$Fc]o$!3?\"R#3&3Ry:$
Ffn7$Fh]o$!3OiaX`=z(o#F\\p7$F]^o$\"37$[6(yc&=1$Ffn7$Fb^o$\"3g!\\G=RR7f
'Ffn7$Fg^o$\"3iI65yWE.5F-7$F\\_o$\"3L[n*p#)z2J\"F-7$Fa_o$\"33I^z'\\=)f
:F-7$Ff_o$\"3;?^`[k,Q<F-7$F[`o$\"3]Caqa;NV=F-7$F``o$\"3-&*4^2'=@)=F-7$
Fe`o$\"3a'RXBF-Z'=F-7$Fj`o$\"3\"\\e&[Yh`+=F-7$F_ao$\"33YE%QX3Rp\"F-7$F
dao$\"3)p%)y)zb.U:F-7$Fiao$\"35d:+Az6O8F-7$F^bo$\"3m%=+=IU^1\"F-7$Fcbo
$\"3t!48/_;H@(Ffn7$Fhbo$\"3g3([n\">`_IFfn7$F]co$!35%))=c@:Lq\"Ffn7$Fbc
o$!3xx[&Re\"H0oFfn7$Fgco$!3o/$\\!e`B\">\"F-7$F\\do$!3Koc_`i0l;F-7$Fado
$!3[%GEo7'Ro?F-7$Ffdo$!3]kvM\"o#zxBF-7$F[eo$!3qhZ$>&Gs$e#F-7$F`eo$!3)*
HsHW@#3p#F-7$Feeo$!3jt>8lj]9FF-7$Fjeo$!3EvIA(HT]n#F-7$F_fo$!3IQo1CD&3f
#F-7$Fdfo$!3[PmP\">wIZ#F-7$Fifo$!3Wd**R&QdHK#F-7$F^go$!3\\ER'\\;DG8#F-
7$Fcgo$!3];0Tu(y-*=F-7$Fhgo$!35bmcM1C%e\"F-7$F]ho$!36GK\\&ok4@\"F-7$Fb
ho$!3)HJTaRK5y(Ffn7$Fgho$!3+FT6)*eBbIFfn7$F\\io$\"3Qf36/'\\Xx\"Ffn7$Fa
io$\"3u(Q(fBJhwjFfn7$Ffio$\"3<G\\9Kt#\\/\"F-7$F[jo$\"3yV'>Qba#y8F-7$F`
jo$\"3XaM8?3dH;F-7$Fejo$\"3YlvdU!zS!=F-7$Fjjo$\"3<$[;F/,s\">F-7$F_[p$
\"3EaqWn?4*)>F--Fd[p6&Ff[pFecmFecm$\"*++++\"Fb]mFi[p-%+AXESLABELSG6%Q!
6\"Fg\\s-%%FONTG6#%(DEFAULTG-%%VIEWG6$F\\]sF\\]s" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 122 "Thus, we have demonstrated low-pass filt
ering using a sequence of FFT/iFFT with a truncation step in the frequ
ency domain." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
267 11 "Exercise 5:" }}{PARA 0 "" 0 "" {TEXT -1 83 "Explore the result
s of the above filtering step by changing the cut-off frequency. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 262 27 "A quantum mechanics example" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 201 "We proceed with the example of fi
nding the DFT of a Gaussian wavepacket. The example is interesting, be
cause we have a detailed understanding of how the wavepacket is made u
p of plane waves of momenta " }{TEXT 19 1 "p" }{TEXT -1 60 " using a G
aussian weight function defined by a center value " }{TEXT 19 2 "p0" }
{TEXT -1 13 " and a width " }{TEXT 19 2 "dp" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x0:=0
; p0:=1/2; sigma:=1/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p0G#\"\"\"\"\"#" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&sigmaG#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 62 "fx:=exp(-sigma^2*(x-x0)^2/2)*exp(I*p0*x)/(Pi/(sigma
)^2)^(1/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxG,$*&#\"\"\"\"#;F(
**-%$expG6#,$*&\"#K!\"\"%\"xG\"\"#F1F(-F,6#*&^##F(F3F(F2F(F(F)#\"\"$\"
\"%%#PiG#F1F;F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "simpli
fy(int(abs(fx)^2,x=-infinity..infinity));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The anal
ytical FT to p-space:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Fp
:=1/sqrt(2*Pi)*int(fx*exp(-I*p*x), x=-infinity..infinity);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#FpG,$*(\"\"#\"\"\"%#PiG#!\"\"\"\"%-%$expG
6#,$*&F'F(),&F(F+*&F'F(%#p|irGF(F(F'F(F+F(F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 151 "The transform is real-valued, because the original fun
ction has symmetry in the real part, and anti-symmetry in the imaginar
y part. These combine with " }{TEXT 19 10 "exp(I*p*x)" }{TEXT -1 154 "
 in such a way as to cancel the imaginary part of the transform, since
 one is integrating an odd function in the imaginary part of the combi
ned integrand." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int(Fp^2,
p=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 523 "Now we calculate the DFT. It is a
dvantageous to re-define the position mesh as also cut up in such a wa
y that the array starts with small x-values followed by the wrap-aroun
d to the large negative values. This takes care of the symmetry betwee
n the coordinate and momentum representations. If one keeps the mesh u
sed for the time-to-frequency domain conversion, an annoying phase (al
ternating sign) appears between adjacent points, and a wrong imaginary
 part is calculated. This makes it difficult to interpret the results.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dx:=0.4; n:=8; N:=2^n;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dxG$\"\"%!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"nG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"NG\"$c#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "dp:=evalhf(2*P
i/(N*dx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dpG$\"3okDa^J#f8'!#>
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "p_max:=N*dp/2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&p_maxG$\"+N;)R&y!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "xv:=Vector(N): pv:=Vector(N):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "for i from 1 to N do:\nif i < N/2+
1 then\nxv[i]:=evalhf((i-1)*dx): xv[N-i+1]:=evalhf(-i*dx):\npv[i]:=eva
lhf((i-1)*dp): pv[N-i+1]:=evalhf(-i*dp): end if:\nod:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "pv[20];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"3K(3$z)RDe;\"!#<" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "pointplot(\{seq([i,xv[i]],i=1..N)\});" }}{PARA 13 "" 
1 "" {GLPLOT2D 608 236 236 {PLOTDATA 2 "6#-%'POINTSG6\\[l7$$\"\"\"\"\"
!$F)F)7$$\"\"#F)$\"+++++S!#57$$\"\"$F)$\"+++++!)F07$$\"\"%F)$\"+++++7!
\"*7$$\"\"&F)$\"+++++;F;7$$\"\"'F)F,7$$\"$`#F)$!+++++;F;7$$\"$a#F)$!++
+++7F;7$$\"$b#F)$!+++++!)F07$$\"$c#F)$!+++++SF07$$\"$_#F)$!\"#F)7$$\"$
X#F)$!+++++[F;7$$\"$Y#F)$!+++++WF;7$$\"$Z#F)$!\"%F)7$$\"$[#F)$!+++++OF
;7$$\"$\\#F)$!+++++KF;7$$\"$]#F)$!+++++GF;7$$\"$^#F)$!+++++CF;7$$\"$R#
F)$!+++++sF;7$$\"$S#F)$!+++++oF;7$$\"$T#F)$!+++++kF;7$$\"$U#F)$!\"'F)7
$$\"$V#F)$!+++++cF;7$$\"$W#F)$!+++++_F;7$$\"$@#F)$!++++S9!\")7$$\"$A#F
)$!#9F)7$$\"$B#F)$!++++g8F]s7$$\"$C#F)$!++++?8F]s7$$\"$D#F)$!++++!G\"F
]s7$$\"$E#F)$!++++S7F]s7$$\"$F#F)$!#7F)7$$\"$G#F)$!++++g6F]s7$$\"$H#F)
$!++++?6F]s7$$\"$I#F)$!++++!3\"F]s7$$\"$J#F)$!++++S5F]s7$$\"$K#F)$F0F)
7$$\"$L#F)$!+++++'*F;7$$\"$M#F)$!+++++#*F;7$$\"$N#F)$!+++++))F;7$$\"$O
#F)$!+++++%)F;7$$\"$P#F)$F]sF)7$$\"$Q#F)$!+++++wF;7$$\"$3#F)$!++++g>F]
s7$$\"$4#F)$!++++?>F]s7$$\"$5#F)$!++++!)=F]s7$$\"$6#F)$!++++S=F]s7$$\"
$7#F)$!#=F)7$$\"$8#F)$!++++g<F]s7$$\"$9#F)$!++++?<F]s7$$\"$:#F)$!++++!
o\"F]s7$$\"$;#F)$!++++S;F]s7$$\"$<#F)$!#;F)7$$\"$=#F)$!++++g:F]s7$$\"$
>#F)$!++++?:F]s7$$\"$?#F)$!++++![\"F]s7$$\"$2#F)$!#?F)7$$\"$+#F)$!++++
!G#F]s7$$\"$,#F)$!++++SAF]s7$$\"$-#F)$!#AF)7$$\"$.#F)$!++++g@F]s7$$\"$
/#F)$!++++?@F]s7$$\"$0#F)$!++++!3#F]s7$$\"$1#F)$!++++S?F]s7$$\"$*=F)$!
++++?FF]s7$$\"$!>F)$!++++!o#F]s7$$\"$\">F)$!++++SEF]s7$$\"$#>F)$!#EF)7
$$\"$$>F)$!++++gDF]s7$$\"$%>F)$!++++?DF]s7$$\"$&>F)$!++++![#F]s7$$\"$'
>F)$!++++SCF]s7$$\"$(>F)$!#CF)7$$\"$)>F)$!++++gBF]s7$$\"$*>F)$!++++?BF
]s7$$\"$\"=F)$!++++SIF]s7$$\"$#=F)$!#IF)7$$\"$$=F)$!++++gHF]s7$$\"$%=F
)$!++++?HF]s7$$\"$&=F)$!++++!)GF]s7$$\"$'=F)$!++++SGF]s7$$\"$(=F)$!#GF
)7$$\"$)=F)$!++++gFF]s7$$\"$k\"F)$!++++?PF]s7$$\"$l\"F)$!++++!o$F]s7$$
\"$m\"F)$!++++SOF]s7$$\"$n\"F)$!#OF)7$$\"$o\"F)$!++++gNF]s7$$\"$p\"F)$
!++++?NF]s7$$\"$q\"F)$!++++![$F]s7$$\"$r\"F)$!++++SMF]s7$$\"$s\"F)$!#M
F)7$$\"$t\"F)$!++++gLF]s7$$\"$u\"F)$!++++?LF]s7$$\"$v\"F)$!++++!G$F]s7
$$\"$w\"F)$!++++SKF]s7$$\"$x\"F)$!#KF)7$$\"$y\"F)$!++++gJF]s7$$\"$z\"F
)$!++++?JF]s7$$\"$!=F)$!++++!3$F]s7$$\"$e\"F)$!++++gRF]s7$$\"$f\"F)$!+
+++?RF]s7$$\"$g\"F)$!++++!)QF]s7$$\"$h\"F)$!++++SQF]s7$$\"$i\"F)$!#QF)
7$$\"$j\"F)$!++++gPF]s7$$\"$^\"F)$!++++SUF]s7$$\"$_\"F)$!#UF)7$$\"$`\"
F)$!++++gTF]s7$$\"$a\"F)$!++++?TF]s7$$\"$b\"F)$!++++!3%F]s7$$\"$c\"F)$
!++++SSF]s7$$\"$d\"F)$!#SF)7$$\"$Q\"F)$!++++gZF]s7$$\"$R\"F)$!++++?ZF]
s7$$\"$S\"F)$!++++!o%F]s7$$\"$T\"F)$!++++SYF]s7$$\"$U\"F)$!#YF)7$$\"$V
\"F)$!++++gXF]s7$$\"$W\"F)$!++++?XF]s7$$\"$X\"F)$!++++![%F]s7$$\"$Y\"F
)$!++++SWF]s7$$\"$Z\"F)$!#WF)7$$\"$[\"F)$!++++gVF]s7$$\"$\\\"F)$!++++?
VF]s7$$\"$]\"F)$!++++!G%F]s7$$\"$;\"F)$\"#YF)7$$\"$<\"F)$\"++++SYF]s7$
$\"$=\"F)$\"++++!o%F]s7$$\"$>\"F)$\"++++?ZF]s7$$\"$?\"F)$\"++++gZF]s7$
$\"$@\"F)$\"#[F)7$$\"$A\"F)$\"++++S[F]s7$$\"$B\"F)$\"++++!)[F]s7$$\"$C
\"F)$\"++++?\\F]s7$$\"$D\"F)$\"++++g\\F]s7$$\"$E\"F)$\"#]F)7$$\"$F\"F)
$\"++++S]F]s7$$\"$G\"F)$\"++++!3&F]s7$$\"$H\"F)$!++++?^F]s7$$\"$I\"F)$
!++++!3&F]s7$$\"$J\"F)$!++++S]F]s7$$\"$K\"F)$!#]F)7$$\"$L\"F)$!++++g\\
F]s7$$\"$M\"F)$!++++?\\F]s7$$\"$N\"F)$!++++!)[F]s7$$\"$O\"F)$!++++S[F]
s7$$\"$P\"F)$!#[F)7$$\"$3\"F)$\"++++!G%F]s7$$\"$4\"F)$\"++++?VF]s7$$\"
$5\"F)$\"++++gVF]s7$$\"$6\"F)$\"#WF)7$$\"$7\"F)$\"++++SWF]s7$$\"$8\"F)
$\"++++![%F]s7$$\"$9\"F)$\"++++?XF]s7$$\"$:\"F)$\"++++gXF]s7$$\"#%*F)$
\"++++?PF]s7$$\"#&*F)$\"++++gPF]s7$$\"#'*F)$\"#QF)7$$\"#(*F)$\"++++SQF
]s7$$\"#)*F)$\"++++!)QF]s7$$\"#**F)$\"++++?RF]s7$$\"$+\"F)$\"++++gRF]s
7$$\"$,\"F)$\"#SF)7$$\"$-\"F)$\"++++SSF]s7$$\"$.\"F)$\"++++!3%F]s7$$\"
$/\"F)$\"++++?TF]s7$$\"$0\"F)$\"++++gTF]s7$$\"$1\"F)$\"#UF)7$$\"$2\"F)
$\"++++SUF]s7$$\"#&)F)$\"++++gLF]s7$$\"#')F)$\"#MF)7$$\"#()F)$\"++++SM
F]s7$$\"#))F)$\"++++![$F]s7$$\"#*)F)$\"++++?NF]s7$$\"#!*F)$\"++++gNF]s
7$$\"#\"*F)$\"#OF)7$$\"##*F)$\"++++SOF]s7$$\"#$*F)$\"++++!o$F]s7$$\"#x
F)$\"++++SIF]s7$$\"#yF)$\"++++!3$F]s7$$\"#zF)$\"++++?JF]s7$$\"#!)F)$\"
++++gJF]s7$$\"#\")F)$\"#KF)7$$\"##)F)$\"++++SKF]s7$$\"#$)F)$\"++++!G$F
]s7$$\"#%)F)$\"++++?LF]s7$$\"#kF)$\"++++?DF]s7$$\"#lF)$\"++++gDF]s7$$
\"#mF)$\"#EF)7$$\"#nF)$\"++++SEF]s7$$\"#oF)$\"++++!o#F]s7$$\"#pF)$\"++
++?FF]s7$$\"#qF)$\"++++gFF]s7$$\"#rF)$\"#GF)7$$\"#sF)$\"++++SGF]s7$$\"
#tF)$\"++++!)GF]s7$$\"#uF)$\"++++?HF]s7$$\"#vF)$\"++++gHF]s7$$\"#wF)$
\"#IF)7$$\"#jF)$\"++++![#F]s7$$\"#_F)$\"++++S?F]s7$$\"#`F)$\"++++!3#F]
s7$$\"#aF)$\"++++?@F]s7$$\"#bF)$\"++++g@F]s7$$\"#cF)$\"#AF)7$$\"#dF)$
\"++++SAF]s7$$\"#eF)$\"++++!G#F]s7$$\"#fF)$\"++++?BF]s7$$\"#gF)$\"++++
gBF]s7$$\"#hF)$\"#CF)7$$\"#iF)$\"++++SCF]s7$$\"#LF)$\"++++!G\"F]s7$Fd`
n$\"++++?8F]s7$$\"#NF)$\"++++g8F]s7$F]bn$\"#9F)7$$\"#PF)$\"++++S9F]s7$
Fc\\n$\"++++![\"F]s7$$\"#RF)$\"++++?:F]s7$F\\^n$\"++++g:F]s7$$\"#TF)$
\"#;F)7$Fe_n$\"++++S;F]s7$$\"#VF)$\"++++!o\"F]s7$F`jm$\"++++?<F]s7$$\"
#XF)$\"++++g<F]s7$Fcbm$\"#=F)7$$\"#ZF)$\"++++S=F]s7$F\\dm$\"++++!)=F]s
7$$\"#\\F)$\"++++?>F]s7$Feem$\"++++g>F]s7$$\"#^F)$\"#?F)7$$\"#FF)$\"++
++S5F]s7$Fggn$\"++++!3\"F]s7$$\"#HF)$\"++++?6F]s7$F`in$\"++++g6F]s7$$
\"#JF)$\"#7F)7$F`dn$\"++++S7F]s7$$\"#:F)$\"+++++cF;7$Fa_oFB7$$\"#<F)$
\"+++++kF;7$Fd`o$\"+++++oF;7$$\"#>F)$\"+++++sF;7$Fiao$\"+++++wF;7$$\"#
@F)$\"\")F)7$F^[o$\"+++++%)F;7$$\"#BF)$\"+++++))F;7$Fg\\o$\"+++++#*F;7
$$\"#DF)$\"+++++'*F;7$F^fn$\"#5F)7$F\\^o$\"+++++_F;7$$\"\"(F)$\"+++++C
F;7$F\\eo$\"+++++GF;7$$\"\"*F)$\"+++++KF;7$F_fo$\"+++++OF;7$$\"#6F)F77
$F^co$\"+++++WF;7$$\"#8F)$\"+++++[F;" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "pointplot(\{seq([i,pv[i]],i=1..N)\});" }}{PARA 13 "" 
0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 640 222 222 {PLOTDATA 
2 "6#-%'POINTSG6\\[l7$$\"\"\"\"\"!$F)F)7$$\"#dF)$\"+lp6OM!\"*7$$\"#eF)
$\"+'>wu\\$F07$$\"#fF)$\"+Ga$)eNF07$$\"#gF)$\"+fY>?OF07$$\"#hF)$\"+\"*
Qb\"o$F07$$\"#iF)$\"+AJ\"Hu$F07$$\"#jF)$\"+aBF/QF07$$\"#kF)$\"+&eJc'QF
07$$\"#lF)$\"+<3*p#RF07$$\"#mF)$\"+\\+N))RF07$$\"#nF)$\"+!G4(\\SF07$$
\"#oF)$\"+7&o56%F07$$\"#pF)$\"+VxUsTF07$$\"#qF)$\"+vpyLUF07$$\"#rF)$\"
+1i9&H%F07$$\"#sF)$\"+Qa]cVF07$$\"#TF)$\"+h#pVX#F07$$\"#UF)$\"+#\\Gd^#
F07$$\"#VF)$\"+Cx3xDF07$$\"#WF)$\"+bpWQEF07$$\"#XF)$\"+(=1)*p#F07$$\"#
YF)$\"+=a;hFF07$$\"#ZF)$\"+]Y_AGF07$$\"#[F)$\"+\")Q)Q)GF07$$\"#\\F)$\"
+8JCXHF07$$\"#]F)$\"+WBg1IF07$$\"#^F)$\"+w:'z1$F07$$\"#_F)$\"+23KHJF07
$$\"#`F)$\"+R+o!>$F07$$\"#aF)$\"+q#R?D$F07$$\"#bF)$\"+-&)R8LF07$$\"#cF
)$\"+LxvuLF07$$\"#SF)$\"+H+,$R#F07$$\"#HF)$\"+#[e!=<F07$$\"#IF)$\"+9xT
z<F07$$\"#JF)$\"+XpxS=F07$$\"#KF)$\"+xh8->F07$$\"#LF)$\"+3a\\j>F07$$\"
#MF)$\"+SY&[-#F07$$\"#NF)$\"+sQ@'3#F07$$\"#OF)$\"+.JdZ@F07$$\"#PF)$\"+
NB$*3AF07$$\"#QF)$\"+m:HqAF07$$\"#RF)$\"+)z];L#F07$$\"#GF)$\"+^#*pc;F0
7$$\"#:F)$\"+7CH!f)!#57$$\"#;F)$\"+FZ)Q?*F\\z7$$\"#<F)$\"+UqZ<)*F\\z7$
$\"#=F)$\"+Op5V5F07$$\"#>F)$\"+nhY/6F07$$\"#?F)$\"+*RDe;\"F07$$\"#@F)$
\"+IY=F7F07$$\"#AF)$\"+iQa)G\"F07$$\"#BF)$\"+$4.*\\8F07$$\"#CF)$\"+DBE
69F07$$\"#DF)$\"+c:is9F07$$\"#EF)$\"+)y!)R`\"F07$$\"#FF)$\"+>+M&f\"F07
$$\"\")F)$F`pF\\z7$$\"\"*F)$\"+@&Q(3\\F\\z7$$\"#5F)$\"+O3LAbF\\z7$$\"#
6F)$\"+_J#f8'F\\z7$$\"#7F)$\"+na^\\nF\\z7$$\"#8F)$\"+#y2JO(F\\z7$$\"#9
F)$\"+(4+n(zF\\z7$$\"\"#F)$F[_l!#67$$\"\"$F)$Fj[lF\\z7$$\"\"%F)$FivF\\
z7$$\"\"&F)$FjpF\\z7$$\"\"'F)$F\\tF\\z7$$\"\"(F)$FDF\\z7$$\"$.#F)$!+-&
)R8LF07$$\"$/#F)$!+q#R?D$F07$$\"$0#F)$!+R+o!>$F07$$\"$1#F)$!+23KHJF07$
$\"$2#F)$!+w:'z1$F07$$\"$3#F)$!+WBg1IF07$$\"$4#F)$!+8JCXHF07$$\"$5#F)$
!+\")Q)Q)GF07$$\"$6#F)$!+]Y_AGF07$$\"$7#F)$!+=a;hFF07$$\"$8#F)$!+(=1)*
p#F07$$\"$9#F)$!+bpWQEF07$$\"$:#F)$!+Cx3xDF07$$\"$;#F)$!+#\\Gd^#F07$$
\"$<#F)$!+h#pVX#F07$$\"$=#F)$!+H+,$R#F07$$\"$>#F)$!+)z];L#F07$$\"$?#F)
$!+m:HqAF07$$\"$(>F)$!+\"*Qb\"o$F07$$\"$)>F)$!+fY>?OF07$$\"$*>F)$!+Ga$
)eNF07$$\"$+#F)$!+'>wu\\$F07$$\"$,#F)$!+lp6OMF07$$\"$-#F)$!+LxvuLF07$$
\"$'=F)$!+Qa]cVF07$$\"$(=F)$!+1i9&H%F07$$\"$)=F)$!+vpyLUF07$$\"$*=F)$!
+VxUsTF07$$\"$!>F)$!+7&o56%F07$$\"$\">F)$!+!G4(\\SF07$$\"$#>F)$!+\\+N)
)RF07$$\"$$>F)$!+<3*p#RF07$$\"$%>F)$!+&eJc'QF07$$\"$&>F)$!+aBF/QF07$$
\"$'>F)$!+AJ\"Hu$F07$$\"$z\"F)$!+e+-'y%F07$$\"$!=F)$!+F3mCZF07$$\"$\"=
F)$!+&f,Lm%F07$$\"$#=F)$!+kB%>g%F07$$\"$$=F)$!+KJeSXF07$$\"$%=F)$!+,RA
zWF07$$\"$&=F)$!+pY'yT%F07$$\"$f\"F)$!+*o/K,'F07$$\"$g\"F)$!+da%=&fF07
$$\"$h\"F)$!+Di[!*eF07$$\"$i\"F)$!+%*p7HeF07$$\"$j\"F)$!+ixwndF07$$\"$
k\"F)$!+J&3kq&F07$$\"$l\"F)$!+*H\\]k&F07$$\"$m\"F)$!+o+p$e&F07$$\"$n\"
F)$!+O3LAbF07$$\"$o\"F)$!+0;(4Y&F07$$\"$p\"F)$!+tBh*R&F07$$\"$q\"F)$!+
UJDQ`F07$$\"$r\"F)$!+5R*oF&F07$$\"$s\"F)$!+zY`:_F07$$\"$t\"F)$!+Za<a^F
07$$\"$u\"F)$!+;i\"G4&F07$$\"$v\"F)$!+%)pXJ]F07$$\"$w\"F)$!+`x4q\\F07$
$\"$x\"F)$!+@&Q(3\\F07$$\"$y\"F)$!+!Hzt%[F07$$\"$R\"F)$!+>$*QSsF07$$\"
$S\"F)$!+(3I!zrF07$$\"$T\"F)$!+c3n<rF07$$\"$U\"F)$!+C;JcqF07$$\"$V\"F)
$!+$R_\\*pF07$$\"$W\"F)$!+hJfLpF07$$\"$X\"F)$!+IRBsoF07$$\"$Y\"F)$!+)p
u3\"oF07$$\"$Z\"F)$!+na^\\nF07$$\"$[\"F)$!+Ni:)o'F07$$\"$\\\"F)$!+/qzE
mF07$$\"$]\"F)$!+sxVllF07$$\"$^\"F)$!+T&yS]'F07$$\"$_\"F)$!+4$>FW'F07$
$\"$`\"F)$!+y+O\"Q'F07$$\"$a\"F)$!+Y3+?jF07$$\"$b\"F)$!+:;keiF07$$\"$c
\"F)$!+$Q#G(>'F07$$\"$d\"F)$!+_J#f8'F07$$\"$e\"F)$!+?RcugF07$$\"$K\"F)
$!+RR!*pwF07$$\"$L\"F)$!+3Za3wF07$$\"$M\"F)$!+wa=ZvF07$$\"$N\"F)$!+Xi#
e[(F07$$\"$O\"F)$!+8qYCuF07$$\"$P\"F)$!+#y2JO(F07$$\"$Q\"F)$!+]&[<I(F0
7$$\"$6\"F)$F`_lF07$$\"$7\"F)$\"+)pu3\"oF07$$\"$8\"F)$\"+IRBsoF07$$\"$
9\"F)$\"+hJfLpF07$$\"$:\"F)$\"+$R_\\*pF07$$\"$;\"F)$\"+C;JcqF07$$\"$<
\"F)$\"+c3n<rF07$$\"$=\"F)$\"+(3I!zrF07$$\"$>\"F)$\"+>$*QSsF07$$\"$?\"
F)$\"+]&[<I(F07$$\"$@\"F)$Fe_lF07$$\"$A\"F)$\"+8qYCuF07$$\"$B\"F)$\"+X
i#e[(F07$$\"$C\"F)$\"+wa=ZvF07$$\"$D\"F)$\"+3Za3wF07$$\"$E\"F)$\"+RR!*
pwF07$$\"$F\"F)$\"+rJEJxF07$$\"$G\"F)$\"+-Ci#z(F07$$\"$H\"F)$!+M;)R&yF
07$$\"$I\"F)$!+-Ci#z(F07$$\"$J\"F)$!+rJEJxF07$$\"#*)F)$\"+tBh*R&F07$$
\"#!*F)$\"+0;(4Y&F07$$\"#\"*F)$Ff^lF07$$\"##*F)$\"+o+p$e&F07$$\"#$*F)$
\"+*H\\]k&F07$$\"#%*F)$\"+J&3kq&F07$$\"#&*F)$\"+ixwndF07$$\"#'*F)$\"+%
*p7HeF07$$\"#(*F)$\"+Di[!*eF07$$\"#)*F)$\"+da%=&fF07$$\"#**F)$\"+*o/K,
'F07$$\"$+\"F)$\"+?RcugF07$$\"$,\"F)$F[_lF07$$\"$-\"F)$\"+$Q#G(>'F07$$
\"$.\"F)$\"+:;keiF07$$\"$/\"F)$\"+Y3+?jF07$$\"$0\"F)$\"+y+O\"Q'F07$$\"
$1\"F)$\"+4$>FW'F07$$\"$2\"F)$\"+T&yS]'F07$$\"$3\"F)$\"+sxVllF07$$\"$4
\"F)$\"+/qzEmF07$$\"$5\"F)$\"+Ni:)o'F07$$\"#\")F)$Fa^lF07$$\"##)F)$\"+
`x4q\\F07$$\"#$)F)$\"+%)pXJ]F07$$\"#%)F)$\"+;i\"G4&F07$$\"#&)F)$\"+Za<
a^F07$$\"#')F)$\"+zY`:_F07$$\"#()F)$\"+5R*oF&F07$$\"#))F)$\"+UJDQ`F07$
$\"#tF)$\"+pY'yT%F07$$\"#uF)$\"+,RAzWF07$$\"#vF)$\"+KJeSXF07$$\"#wF)$
\"+kB%>g%F07$$\"#xF)$\"+&f,Lm%F07$$\"#yF)$\"+F3mCZF07$$\"#zF)$\"+e+-'y
%F07$$\"#!)F)$\"+!Hzt%[F07$$\"$^#F)$FbglF\\z7$$\"$_#F)$F\\clF\\z7$$\"$
`#F)$F^flF\\z7$$\"$a#F)$!+XpxS=F\\z7$$\"$b#F)$!+IY=F7F\\z7$$\"$c#F)$Fh
jmF_`l7$$\"$Y#F)$FfgmF\\z7$$\"$Z#F)$FhjmF\\z7$$\"$[#F)$FbamF\\z7$$\"$
\\#F)$FddmF\\z7$$\"$]#F)$FeilF\\z7$$\"$X#F)$F[]nF\\z7$$\"$N#F)$!+$4.*
\\8F07$$\"$O#F)$!+iQa)G\"F07$$\"$P#F)$Fh`oF07$$\"$Q#F)$!+*RDe;\"F07$$
\"$R#F)$!+nhY/6F07$$\"$S#F)$!+Op5V5F07$$\"$T#F)$!+UqZ<)*F\\z7$$\"$U#F)
$!+FZ)Q?*F\\z7$$\"$V#F)$!+7CH!f)F\\z7$$\"$W#F)$!+(4+n(zF\\z7$$\"$@#F)$
!+NB$*3AF07$$\"$A#F)$!+.JdZ@F07$$\"$B#F)$!+sQ@'3#F07$$\"$C#F)$!+SY&[-#
F07$$\"$D#F)$!+3a\\j>F07$$\"$E#F)$!+xh8->F07$$\"$F#F)$Fc`oF07$$\"$G#F)
$!+9xTz<F07$$\"$H#F)$!+#[e!=<F07$$\"$I#F)$!+^#*pc;F07$$\"$J#F)$!+>+M&f
\"F07$$\"$K#F)$!+)y!)R`\"F07$$\"$L#F)$!+c:is9F07$$\"$M#F)$!+DBE69F0" 
1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "yR:=Vector(N): yI:=Vector(N)
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "for i from 1 to N do: yR[i]:=e
valf(subs(x=xv[i],Re(fx))); yI[i]:=evalf(subs(x=xv[i],Im(fx))); od:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([seq([xv[i],yR[i]],i=
1..N)],style=point);" }}{PARA 13 "" 1 "" {GLPLOT2D 824 208 208 
{PLOTDATA 2 "6&-%'CURVESG6$7\\[l7$$\"\"!F)$\"31+++AxibP!#=7$$\"3A+++++
++SF,$\"3!)*****fJ2Cm$F,7$$\"3U+++++++!)F,$\"3#******4?m1R$F,7$$\"3;++
+++++7!#<$\"3E+++k2EjHF,7$$\"33+++++++;F:$\"37+++\"\\*R:CF,7$$\"\"#F)$
\"3))*****40S2z\"F,7$$\"3M+++++++CF:$\"3$******p@.n8\"F,7$$\"3E+++++++
GF:$\"3I+++;QE'*\\!#>7$$\"3;+++++++KF:$!3E******\\W8jz!#?7$$\"33++++++
+OF:$!3!*******))HA\"p&FQ7$$\"\"%F)$!3k******oFUz%*FQ7$$\"3M+++++++WF:
$!3(******H%)Gp?\"F,7$$\"3q+++++++[F:$!37+++_,+[8F,7$$\"3;+++++++_F:$!
31+++GAQ#Q\"F,7$$\"3_+++++++cF:$!3()*****R\\*3G8F,7$$\"\"'F)$!3(******
*=>227F,7$$\"3M+++++++kF:$!3$******4*RUU5F,7$$\"3q+++++++oF:$!3o+++pYy
f&)FQ7$$\"3;+++++++sF:$!3_******=?,lmFQ7$$\"3_+++++++wF:$!33+++]d%e)[F
Q7$$\"\")F)$!3J+++7vEALFQ7$$\"3M+++++++%)F:$!3$*******3O(*H?FQ7$$\"3q+
++++++))F:$!3.+++q!ej-\"FQ7$$\"31,++++++#*F:$!3'*******)yn2*HFW7$$\"3U
,++++++'*F:$\"3/+++%pkY%=FW7$$\"#5F)$\"3U+++[9t!o%FW7$$\"3/++++++S5!#;
$\"3O+++*G-4*fFW7$$\"33++++++!3\"Fjs$\"3I+++>D[EiFW7$$\"36++++++?6Fjs$
\"3#******z])=zdFW7$$\"39++++++g6Fjs$\"3!)*****pVy@'\\FW7$$\"#7F)$\"3!
******>(4&f+%FW7$$\"3/++++++S7Fjs$\"3-+++=ptkIFW7$$\"33++++++!G\"Fjs$
\"3#)******[h2HAFW7$$\"36++++++?8Fjs$\"3\"******49;4a\"FW7$$\"39++++++
g8Fjs$\"3&******zv1&35FW7$$\"#9F)$\"3))*****4I5O>'!#@7$$\"3/++++++S9Fj
s$\"36+++R-O/NFjv7$$\"33++++++![\"Fjs$\"3!******4#=p`<Fjv7$$\"36++++++
?:Fjs$\"36+++c)ob!p!#A7$$\"39++++++g:Fjs$\"3$******RIR!45Fjw7$$\"#;F)$
!3;+++q_6L=Fjw7$$\"3A++++++S;Fjs$!3#******fKQ*\\GFjw7$$\"33++++++!o\"F
js$!3#******>ep9)GFjw7$$\"3$*************><Fjs$!3'******fE&=iCFjw7$$\"
39++++++g<Fjs$!3,+++rr]/>Fjw7$$\"#=F)$!3%******p\"3)4P\"Fjw7$$\"3A++++
++S=Fjs$!3]+++iLS1$*!#B7$$\"33++++++!)=Fjs$!3I+++U*)o%*fFcz7$$\"3H++++
++?>Fjs$!3)******z%30sOFcz7$$\"39++++++g>Fjs$!3-+++p_GO@Fcz7$$\"#?F)$!
3#******R'zNu6Fcz7$$\"3A++++++S?Fjs$!30+++c$GJ.'!#C7$$\"33++++++!3#Fjs
$!3<+++\\#37$GF]\\l7$$\"3H++++++?@Fjs$!3/+++WdU]6F]\\l7$$\"39++++++g@F
js$!3++++^Y)yR$!#D7$$\"#AF)$\"3C+++7M/([%!#F7$$\"3A++++++SAFjs$\"3++++
k$[:=\"F]]l7$$\"33++++++!G#Fjs$\"3$******pIx;I\"F]]l7$$\"3H++++++?BFjs
$\"3/+++*zPy0\"F]]l7$$\"39++++++gBFjs$\"3_+++pz-ru!#E7$$\"#CF)$\"3P+++
*=(oE[Fh^l7$$\"3A++++++SCFjs$\"3#)*****fC_h\"HFh^l7$$\"33++++++![#Fjs$
\"3%******>%\\Ml;Fh^l7$$\"3H++++++?DFjs$\"3S+++cGkL!*Fc]l7$$\"39++++++
gDFjs$\"3_******Ha#)fYFc]l7$$\"#EF)$\"37+++YT^!G#Fc]l7$$\"3A++++++SEFj
s$\"3)******4I8?0\"Fc]l7$$\"33++++++!o#Fjs$\"3p*****p%[q7X!#G7$$\"3H++
++++?FFjs$\"3!******R2$4\\<Faal7$$\"39++++++gFFjs$\"3:+++/];+d!#H7$$\"
#GF)$\"3/+++%*o&e<\"F\\bl7$$\"3A++++++SGFjs$!3))*****R7T!oE!#I7$$\"33+
+++++!)GFjs$!3')*****RoY0S&Fgbl7$$\"3H++++++?HFjs$!3=+++ra#[\\%Fgbl7$$
\"39++++++gHFjs$!3#)*****H6f/(HFgbl7$$\"#IF)$!3++++'fY4u\"Fgbl7$$\"3A+
+++++SIFjs$!3.+++J;16%*!#J7$$\"33++++++!3$Fjs$!35+++GrKwZFadl7$$\"3H++
++++?JFjs$!3#******RIkcH#Fadl7$$\"39++++++gJFjs$!3%******fLs([5Fadl7$$
\"#KF)$!3E+++-'4[b%!#K7$$\"3')************RKFjs$!39+++w5=u=Ffel7$$\"3W
++++++!G$Fjs$!3i*****f?'3Zs!#L7$$\"3H++++++?LFjs$!3%******z6-(*e#Fafl7
$$\"39++++++gLFjs$!3#)*****RMJYC)!#M7$$\"#MF)$!3:+++4KO:@F\\gl7$$\"3')
************RMFjs$!33+++r<LsD!#N7$$\"3W++++++![$Fjs$\"3++++#**4\\m\"Fg
gl7$$\"3H++++++?NFjs$\"3&)*****>4.>\"=Fggl7$$\"39++++++gNFjs$\"3%*****
*4hW$z6Fggl7$$\"#OF)$\"3_*****\\d4,R'!#O7$$\"3')************ROFjs$\"3,
+++c>j<JF\\il7$$\"3W++++++!o$Fjs$\"31+++#H%=79F\\il7$$\"3H++++++?PFjs$
\"3I+++c9#\\-'!#P7$$\"39++++++gPFjs$\"3))*****4d;zV#F\\jl7$$\"#QF)$\"3
++++1x;!Q*!#Q7$$\"3d++++++SQFjs$\"3K+++)GV*GMFgjl7$$\"3W++++++!)QFjs$
\"3'*******Q3Y&=\"Fgjl7$$\"3H++++++?RFjs$\"3?+++M]9PQ!#R7$$\"39++++++g
RFjs$\"33+++=F$*Q6Fg[m7$$\"#SF)$\"39+++>.,cH!#S7$$\"3d++++++SSFjs$\"3E
+++fU[&z&!#T7$$\"3W++++++!3%Fjs$\"33+++Co`b>!#U7$$\"3H++++++?TFjs$!3q*
****zA'[ghF^]m7$$\"39++++++gTFjs$!3S+++%4K(QXF^]m7$$\"#UF)$!3#)*****>>
-#fBF^]m7$$\"3d++++++SUFjs$!3)******H)oOa5F^]m7$$\"3W++++++!G%Fjs$!39+
++\"***p#H%!#V7$$\"3H++++++?VFjs$!3'******4db7j\"Fh^m7$$\"39++++++gVFj
s$!3)******4CuP&e!#W7$$\"#WF)$!35+++\"RqX*>Fc_m7$$\"3d++++++SWFjs$!3o*
****\\#>$HY'!#X7$$\"3W++++++![%Fjs$!33+++`N\"y)>F^`m7$$\"3H++++++?XFjs
$!3a+++;2Npd!#Y7$$\"39++++++gXFjs$!3!*******e=dg:Fi`m7$$\"#YF)$!3,+++c
z5LQ!#Z7$$\"3d++++++SYFjs$!3t+++]iUD!)!#[7$$\"3W++++++!o%Fjs$!3#******
z%G*)R6Fjam7$$\"3H++++++?ZFjs$\"3j*****zB@jI)!#]7$$\"39++++++gZFjs$\"3
#******4))*3u:!#\\7$$\"#[F)$\"3O+++i4&4d)Febm7$$\"3d++++++S[Fjs$\"3w**
***f&\\!pg$Febm7$$\"3W++++++!)[Fjs$\"3,+++H6gN8Febm7$$\"3H++++++?\\Fjs
$\"3))*****zVQma%!#^7$$\"39++++++g\\Fjs$\"3)******H5a3X\"F`dm7$$\"#]F)
$\"3%*******[a:\"Q%!#_7$$\"3d++++++S]Fjs$\"3#******H[(>d7F[em7$$\"3W++
++++!3&Fjs$\"3&******RZ!4IM!#`7$$!3H++++++?^Fjs$\"3/+++vHHs))!#a7$$!3W
++++++!3&FjsFdem7$$!3d++++++S]FjsF_em7$$FebmF)Fidm7$$!39++++++g\\FjsFd
dm7$$!3H++++++?\\FjsF^dm7$$!3W++++++!)[FjsFicm7$$!3d++++++S[FjsFdcm7$$
FjamF)F_cm7$$!39++++++gZFjsFibm7$$!3H++++++?ZFjsFcbm7$$!3W++++++!o%Fjs
F^bm7$$!3d++++++SYFjsFham7$$Fi`mF)Fbam7$$!39++++++gXFjsF]am7$$!3H+++++
+?XFjsFg`m7$$!3W++++++![%FjsFb`m7$$!3d++++++SWFjsF\\`m7$$Fc_mF)Fg_m7$$
!39++++++gVFjsFa_m7$$!3H++++++?VFjsF\\_m7$$!3W++++++!G%FjsFf^m7$$!3d++
++++SUFjsFa^m7$$F^]mF)F\\^m7$$!39++++++gTFjsFg]m7$$!3H++++++?TFjsFb]m7
$$!3W++++++!3%FjsF\\]m7$$!3d++++++SSFjsFf\\m7$$Fb\\mF)F`\\m7$$!39+++++
+gRFjsF[\\m7$$!3H++++++?RFjsFe[m7$$!3W++++++!)QFjsF`[m7$$!3d++++++SQFj
sF[[m7$$FgjlF)Fejl7$$!39++++++gPFjsF`jl7$$!3H++++++?PFjsFjil7$$!3W++++
++!o$FjsFeil7$$!3')************ROFjsF`il7$$F\\ilF)Fjhl7$$!39++++++gNFj
sFehl7$$!3H++++++?NFjsF`hl7$$!3W++++++![$FjsF[hl7$$!3')************RMF
jsFegl7$$F\\glF)F`gl7$$!39++++++gLFjsFjfl7$$!3H++++++?LFjsFefl7$$!3W++
++++!G$FjsF_fl7$$!3')************RKFjsFjel7$$FfelF)Fdel7$$!39++++++gJF
jsF_el7$$!3H++++++?JFjsFjdl7$$!33++++++!3$FjsFedl7$$!3A++++++SIFjsF_dl
7$$FgblF)Fjcl7$$!39++++++gHFjsFecl7$$!3H++++++?HFjsF`cl7$$!33++++++!)G
FjsF[cl7$$!3A++++++SGFjsFebl7$$FaalF)F`bl7$$!39++++++gFFjsFjal7$$!3H++
++++?FFjsFeal7$$!33++++++!o#FjsF_al7$$!3A++++++SEFjsFj`l7$$Fh^lF)Fe`l7
$$!39++++++gDFjsF``l7$$!3H++++++?DFjsF[`l7$$!33++++++![#FjsFf_l7$$!3A+
+++++SCFjsFa_l7$$F]\\lF)F\\_l7$$!39++++++gBFjsFf^l7$$!3H++++++?BFjsFa^
l7$$!33++++++!G#FjsF\\^l7$$!3A++++++SAFjsFg]l7$$FjwF)Fa]l7$$!39++++++g
@FjsF[]l7$$!3H++++++?@FjsFf\\l7$$!33++++++!3#FjsFa\\l7$$!3A++++++S?Fjs
F[\\l7$$FWF)Ff[l7$$!39++++++g>FjsFa[l7$$!3H++++++?>FjsF\\[l7$$!33+++++
+!)=FjsFgz7$$!3A++++++S=FjsFaz7$$F,F)F\\z7$$!39++++++g<FjsFgy7$$!3$***
**********><FjsFby7$$!33++++++!o\"FjsF]y7$$!3A++++++S;FjsFhx7$$FjsF)Fc
x7$$!39++++++g:FjsF^x7$$!36++++++?:FjsFhw7$$!33++++++![\"FjsFcw7$$!3/+
+++++S9FjsF^w7$$!#9F)Fhv7$$!39++++++g8FjsFcv7$$!36++++++?8FjsF^v7$$!33
++++++!G\"FjsFiu7$$!3/++++++S7FjsFdu7$$!#7F)F_u7$$!39++++++g6FjsFjt7$$
!36++++++?6FjsFet7$$!33++++++!3\"FjsF`t7$$!3/++++++S5FjsF[t7$$!#5F)Fes
7$$!3U,++++++'*F:F`s7$$!31,++++++#*F:F[s7$$!3q+++++++))F:Ffr7$$!3M++++
+++%)F:Far7$$!\")F)F\\r7$$!3_+++++++wF:Fgq7$$!3;+++++++sF:Fbq7$$!3q+++
++++oF:F]q7$$!3M+++++++kF:Fhp7$$!\"'F)Fcp7$$!3_+++++++cF:F^p7$$!3;++++
+++_F:Fio7$$!3q+++++++[F:Fdo7$$!3M+++++++WF:F_o7$$!\"%F)Fjn7$$!33+++++
++OF:Fen7$$!3;+++++++KF:FU7$$!3E+++++++GF:FO7$$!3M+++++++CF:FJ7$$!\"#F
)FE7$$!33+++++++;F:F@7$$!3;+++++++7F:F;7$$!3U+++++++!)F,F57$$!3A++++++
+SF,F0-%'COLOURG6&%$RGBG$Fds!\"\"F(F(-%&STYLEG6#%&POINTG-%+AXESLABELSG
6$Q!6\"Fe]o-%%VIEWG6$%(DEFAULTGFj]o" 1 5 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "plot([seq([xv[i],yI[i]],i=1..N)],style=point,color=bl
ue);" }}{PARA 13 "" 1 "" {GLPLOT2D 824 208 208 {PLOTDATA 2 "6'-%'CURVE
SG6#7\\[l7$$\"\"!F)F(7$$\"3A+++++++S!#=$\"3e******or1Cu!#>7$$\"3U+++++
++!)F-$\"30+++x1bL9F-7$$\"3;+++++++7!#<$\"3,+++hdFF?F-7$$\"33+++++++;F
9$\"3'******fW))p[#F-7$$\"\"#F)$\"3'******zQ7*)y#F-7$$\"3M+++++++CF9$
\"3v*****H-tP#HF-7$$\"3E+++++++GF9$\"3/+++hcx'*GF-7$$\"3;+++++++KF9$\"
3#******zg#)fs#F-7$$\"33+++++++OF9$\"3*)*****z42%RCF-7$$\"\"%F)$\"3+++
+j;Hr?F-7$$\"3M+++++++WF9$\"37+++sm5e;F-7$$\"3q+++++++[F9$\"31+++-uyM7
F-7$$\"3;+++++++_F9$\"3')*****flkjJ)F07$$\"3_+++++++cF9$\"3s*****fMa<s
%F07$$\"\"'F)$\"3*)*****RHR1s\"F07$$\"3M+++++++kF9$!39+++s[X&4'!#?7$$
\"3q+++++++oF9$!3))*****fv&\\iAF07$$\"3;+++++++sF9$!3%)*****foh*)G$F07
$$\"3_+++++++wF9$!3/+++OdZzPF07$$\"\")F)$!3y******H?fYQF07$$\"3M++++++
+%)F9$!3N+++Eg%)3OF07$$\"3q+++++++))F9$!3=++++p$z<$F07$$\"31,++++++#*F
9$!3!*******oD()\\EF07$$\"3U,++++++'*F9$!3:+++%*o7+@F07$$\"#5F)$!3#***
*****GGBe\"F07$$\"3/++++++S5!#;$!3#******Rip'H6F07$$\"33++++++!3\"Fhs$
!3S+++im*4e(Fgp7$$\"36++++++?6Fhs$!3l*****>nJRq%Fgp7$$\"39++++++g6Fhs$
!3!******HT([.EFgp7$$\"#7F)$!3/+++Olvl6Fgp7$$\"3/++++++S7Fhs$!3y******
3wIbD!#@7$$\"33++++++!G\"Fhs$\"3.+++*R(z:EFdu7$$\"36++++++?8Fhs$\"3Y++
+Up,_]Fdu7$$\"39++++++g8Fhs$\"3b*****\\4Z<t&Fdu7$$\"#9F)$\"3++++-#4uR&
Fdu7$$\"3/++++++S9Fhs$\"3C+++\\<'=d%Fdu7$$\"33++++++![\"Fhs$\"3)******
f)G\"Qf$Fdu7$$\"36++++++?:Fhs$\"3;+++,'3-m#Fdu7$$\"39++++++g:Fhs$\"3#*
*****R/6u'=Fdu7$$\"#;F)$\"3))*******[lkC\"Fdu7$$\"3A++++++S;Fhs$\"3M++
+DW,0z!#A7$$\"33++++++!o\"Fhs$\"3!********Rl?u%Fgx7$$\"3$*************
><Fhs$\"3=+++\"HkTm#Fgx7$$\"39++++++g<Fhs$\"3#******fRHMP\"Fgx7$$\"#=F
)$\"3:+++C4;,i!#B7$$\"3A++++++S=Fhs$\"35+++$*>$y7#F\\z7$$\"33++++++!)=
Fhs$\"37+++Fdm&[\"!#C7$$\"3H++++++?>Fhs$!3v*****z$4\"4]'Fgz7$$\"39++++
++g>Fhs$!3%******\\PlWT)Fgz7$$\"#?F)$!3Z+++>q29wFgz7$$\"3A++++++S?Fhs$
!3@+++*Qt:\"fFgz7$$\"33++++++!3#Fhs$!3%******4<Bz<%Fgz7$$\"3H++++++?@F
hs$!3')******4E%\\v#Fgz7$$\"39++++++g@Fhs$!37+++U-=:<Fgz7$$\"#AF)$!3&*
*****zD^Q,\"Fgz7$$\"3A++++++SAFhs$!3y*****\\1/\"*p&!#D7$$\"33++++++!G#
Fhs$!39+++35;TIFe]l7$$\"3H++++++?BFhs$!3(******z9Z;`\"Fe]l7$$\"39+++++
+gBFhs$!39+++WH*>>(!#E7$$\"#CF)$!3!******>(p4pIFe^l7$$\"3A++++++SCFhs$
!3-+++T!4*=6Fe^l7$$\"33++++++![#Fhs$!3=+++,R\\'z#!#F7$$\"3H++++++?DFhs
$\"3++++eV5RI!#G7$$\"39++++++gDFhs$\"3(******R$>#*36Fe_l7$$\"#EF)$\"3+
+++Vi#f0\"Fe_l7$$\"3A++++++SEFhs$\"3%******4j=!HxF[`l7$$\"33++++++!o#F
hs$\"39+++$oV/(\\F[`l7$$\"3H++++++?FFhs$\"3%)*****fAkn$HF[`l7$$\"39+++
+++gFFhs$\"3.+++D<0E;F[`l7$$\"#GF)$\"3m+++Q1i=&)!#H7$$\"3A++++++SGFhs$
\"3h*****fL[1C%F_bl7$$\"33++++++!)GFhs$\"3:+++z*4s+#F_bl7$$\"3H++++++?
HFhs$\"3U+++>r)z+*!#I7$$\"39++++++gHFhs$\"39+++1$)30QF_cl7$$\"#IF)$\"3
%******zzQ-\\\"F_cl7$$\"3A++++++SIFhs$\"3F+++rG-R_!#J7$$\"33++++++!3$F
hs$\"3++++O'p#>:F_dl7$$\"3H++++++?JFhs$\"3')******o!\\\")[#!#K7$$\"39+
+++++gJFhs$!3!3+++N0*z'*!#L7$$\"#KF)$!3-+++BEKp8Fjdl7$$\"3')**********
**RKFhs$!30+++?Xd/5Fjdl7$$\"3W++++++!G$Fhs$!3w******e&Qh+'F`el7$$\"3H+
+++++?LFhs$!37+++rDf5KF`el7$$\"39++++++gLFhs$!3-+++JIX)e\"F`el7$$\"#MF
)$!3g*****>h+4R(!#M7$$\"3')************RMFhs$!33+++D\"*HfKF_gl7$$\"3W+
+++++![$Fhs$!3'******zA&\\m8F_gl7$$\"3H++++++?NFhs$!3&******\\&o'\\W&!
#N7$$\"39++++++gNFhs$!35+++jd&Q0#F_hl7$$\"#OF)$!3m*****RCfvE(!#O7$$\"3
')************ROFhs$!3'*******)o`yO#Fjhl7$$\"3W++++++!o$Fhs$!35+++v[*Q
\"o!#P7$$\"3H++++++?PFhs$!3*******4Olb`\"Feil7$$\"39++++++gPFhs$!30+++
o>747!#Q7$$\"#QF)$\"3'******flM>U\"F`jl7$$\"3d++++++SQFhs$\"3)******H!
zQ`7F`jl7$$\"3W++++++!)QFhs$\"3k*****\\Un`F(!#R7$$\"3H++++++?RFhs$\"3+
+++!*[&yd$F`[m7$$\"39++++++gRFhs$\"3#******zTfTf\"F`[m7$$\"#SF)$\"3***
****z$328m!#S7$$\"3d++++++SSFhs$\"31+++)>$R(e#F`\\m7$$\"3W++++++!3%Fhs
$\"3V*******HHrg*!#T7$$\"3H++++++?TFhs$\"3w*****z(GB#R$F[]m7$$\"39++++
++gTFhs$\"34+++yW^P6F[]m7$$\"#UF)$\"3*******\\(yn.O!#U7$$\"3d++++++SUF
hs$\"3)*******QKcm5F[^m7$$\"3W++++++!G%Fhs$\"3++++jyM\")G!#V7$$\"3H+++
+++?VFhs$\"3K+++Q&oss'!#W7$$\"39++++++gVFhs$\"3*******fjqF8\"F\\_m7$$
\"#WF)$!3/+++0^_l<!#Y7$$\"3d++++++SWFhs$!33+++4mwp8!#X7$$\"3W++++++![%
Fhs$!3^******=q_7')Fg_m7$$\"3H++++++?XFhs$!3u*****RV_C-%Fg_m7$$\"39+++
+++gXFhs$!3-+++q\\aN;Fg_m7$$\"#YF)$!3E+++@Ac(3'!#Z7$$\"3d++++++SYFhs$!
3,+++KVi>@Fbam7$$\"3W++++++!o%Fhs$!3w*****zVGr(p!#[7$$\"3H++++++?ZFhs$
!3')*****4Vb;=#F]bm7$$\"39++++++gZFhs$!3'******z36p['!#\\7$$\"#[F)$!3'
******4\\4)H=Fhbm7$$\"3d++++++S[Fhs$!3=+++G3<k[!#]7$$\"3W++++++!)[Fhs$
!3++++XC'=?\"Fccm7$$\"3H++++++?\\Fhs$!3))*****R#=s!o#!#^7$$\"39++++++g
\\Fhs$!3_******)G:S,&!#_7$$\"#]F)$!3q******f%***\\e!#`7$$\"3d++++++S]F
hs$\"3Q******o%H&o%)!#a7$$\"3W++++++!3&Fhs$\"3C+++QB!>R*F`em7$$!3H++++
++?^Fhs$!3m*****f@oiZ%F`em7$$!3W++++++!3&Fhs$!3C+++QB!>R*F`em7$$!3d+++
+++S]Fhs$!3Q******o%H&o%)F`em7$$FccmF)$\"3q******f%***\\eFjdm7$$!39+++
+++g\\Fhs$\"3_******)G:S,&Fddm7$$!3H++++++?\\Fhs$\"3))*****R#=s!o#F^dm
7$$!3W++++++!)[Fhs$\"3++++XC'=?\"Fccm7$$!3d++++++S[Fhs$\"3=+++G3<k[Fcc
m7$$F]bmF)$\"3'******4\\4)H=Fhbm7$$!39++++++gZFhs$\"3'******z36p['Fhbm
7$$!3H++++++?ZFhs$\"3')*****4Vb;=#F]bm7$$!3W++++++!o%Fhs$\"3w*****zVGr
(pF]bm7$$!3d++++++SYFhs$\"3,+++KVi>@Fbam7$$Fg_mF)$\"3E+++@Ac(3'Fbam7$$
!39++++++gXFhs$\"3-+++q\\aN;Fg_m7$$!3H++++++?XFhs$\"3u*****RV_C-%Fg_m7
$$!3W++++++![%Fhs$\"3^******=q_7')Fg_m7$$!3d++++++SWFhs$\"33+++4mwp8F]
`m7$$F\\_mF)$\"3/+++0^_l<Fg_m7$$!39++++++gVFhs$!3*******fjqF8\"F\\_m7$
$!3H++++++?VFhs$!3K+++Q&oss'F\\_m7$$!3W++++++!G%Fhs$!3++++jyM\")GFf^m7
$$!3d++++++SUFhs$!3)*******QKcm5F[^m7$$F[^mF)$!3*******\\(yn.OF[^m7$$!
39++++++gTFhs$!34+++yW^P6F[]m7$$!3H++++++?TFhs$!3w*****z(GB#R$F[]m7$$!
3W++++++!3%Fhs$!3V*******HHrg*F[]m7$$!3d++++++SSFhs$!31+++)>$R(e#F`\\m
7$$F`\\mF)$!3*******z$328mF`\\m7$$!39++++++gRFhs$!3#******zTfTf\"F`[m7
$$!3H++++++?RFhs$!3++++!*[&yd$F`[m7$$!3W++++++!)QFhs$!3k*****\\Un`F(F`
[m7$$!3d++++++SQFhs$!3)******H!zQ`7F`jl7$$F`jlF)$!3'******flM>U\"F`jl7
$$!39++++++gPFhs$\"30+++o>747F`jl7$$!3H++++++?PFhs$\"3*******4Olb`\"Fe
il7$$!3W++++++!o$Fhs$\"35+++v[*Q\"oFeil7$$!3')************ROFhs$\"3'**
*****)o`yO#Fjhl7$$FjhlF)$\"3m*****RCfvE(Fjhl7$$!39++++++gNFhs$\"35+++j
d&Q0#F_hl7$$!3H++++++?NFhs$\"3&******\\&o'\\W&F_hl7$$!3W++++++![$Fhs$
\"3'******zA&\\m8F_gl7$$!3')************RMFhs$\"33+++D\"*HfKF_gl7$$F_g
lF)$\"3g*****>h+4R(F_gl7$$!39++++++gLFhs$\"3-+++JIX)e\"F`el7$$!3H+++++
+?LFhs$\"37+++rDf5KF`el7$$!3W++++++!G$Fhs$\"3w******e&Qh+'F`el7$$!3')*
***********RKFhs$\"30+++?Xd/5Fjdl7$$FjdlF)$\"3-+++BEKp8Fjdl7$$!39+++++
+gJFhs$\"3!3+++N0*z'*F`el7$$!3H++++++?JFhs$!3')******o!\\\")[#Fjdl7$$!
33++++++!3$Fhs$!3++++O'p#>:F_dl7$$!3A++++++SIFhs$!3F+++rG-R_F_dl7$$F_c
lF)$!3%******zzQ-\\\"F_cl7$$!39++++++gHFhs$!39+++1$)30QF_cl7$$!3H+++++
+?HFhs$!3U+++>r)z+*F_cl7$$!33++++++!)GFhs$!3:+++z*4s+#F_bl7$$!3A++++++
SGFhs$!3h*****fL[1C%F_bl7$$F[`lF)$!3m+++Q1i=&)F_bl7$$!39++++++gFFhs$!3
.+++D<0E;F[`l7$$!3H++++++?FFhs$!3%)*****fAkn$HF[`l7$$!33++++++!o#Fhs$!
39+++$oV/(\\F[`l7$$!3A++++++SEFhs$!3%******4j=!HxF[`l7$$Fe^lF)$!3++++V
i#f0\"Fe_l7$$!39++++++gDFhs$!3(******R$>#*36Fe_l7$$!3H++++++?DFhs$!3++
++eV5RIF[`l7$$!33++++++![#Fhs$\"3=+++,R\\'z#Fe_l7$$!3A++++++SCFhs$\"3-
+++T!4*=6Fe^l7$$FgzF)$\"3!******>(p4pIFe^l7$$!39++++++gBFhs$\"39+++WH*
>>(Fe^l7$$!3H++++++?BFhs$\"3(******z9Z;`\"Fe]l7$$!33++++++!G#Fhs$\"39+
++35;TIFe]l7$$!3A++++++SAFhs$\"3y*****\\1/\"*p&Fe]l7$$FgxF)$\"3&******
zD^Q,\"Fgz7$$!39++++++g@Fhs$\"37+++U-=:<Fgz7$$!3H++++++?@Fhs$\"3')****
**4E%\\v#Fgz7$$!33++++++!3#Fhs$\"3%******4<Bz<%Fgz7$$!3A++++++S?Fhs$\"
3@+++*Qt:\"fFgz7$$FgpF)$\"3Z+++>q29wFgz7$$!39++++++g>Fhs$\"3%******\\P
lWT)Fgz7$$!3H++++++?>Fhs$\"3v*****z$4\"4]'Fgz7$$!33++++++!)=Fhs$!37+++
Fdm&[\"Fgz7$$!3A++++++S=Fhs$!35+++$*>$y7#F\\z7$$F-F)$!3:+++C4;,iF\\z7$
$!39++++++g<Fhs$!3#******fRHMP\"Fgx7$$!3$*************><Fhs$!3=+++\"Hk
Tm#Fgx7$$!33++++++!o\"Fhs$!3!********Rl?u%Fgx7$$!3A++++++S;Fhs$!3M+++D
W,0zFgx7$$FhsF)$!3))*******[lkC\"Fdu7$$!39++++++g:Fhs$!3#******R/6u'=F
du7$$!36++++++?:Fhs$!3;+++,'3-m#Fdu7$$!33++++++![\"Fhs$!3)******f)G\"Q
f$Fdu7$$!3/++++++S9Fhs$!3C+++\\<'=d%Fdu7$$!#9F)$!3++++-#4uR&Fdu7$$!39+
+++++g8Fhs$!3b*****\\4Z<t&Fdu7$$!36++++++?8Fhs$!3Y+++Up,_]Fdu7$$!33+++
+++!G\"Fhs$!3.+++*R(z:EFdu7$$!3/++++++S7Fhs$\"3y******3wIbDFdu7$$!#7F)
$\"3/+++Olvl6Fgp7$$!39++++++g6Fhs$\"3!******HT([.EFgp7$$!36++++++?6Fhs
$\"3l*****>nJRq%Fgp7$$!33++++++!3\"Fhs$\"3S+++im*4e(Fgp7$$!3/++++++S5F
hs$\"3#******Rip'H6F07$$!#5F)$\"3#********GGBe\"F07$$!3U,++++++'*F9$\"
3:+++%*o7+@F07$$!31,++++++#*F9$\"3!*******oD()\\EF07$$!3q+++++++))F9$
\"3=++++p$z<$F07$$!3M+++++++%)F9$\"3N+++Eg%)3OF07$$!\")F)$\"3y******H?
fYQF07$$!3_+++++++wF9$\"3/+++OdZzPF07$$!3;+++++++sF9$\"3%)*****foh*)G$
F07$$!3q+++++++oF9$\"3))*****fv&\\iAF07$$!3M+++++++kF9$\"39+++s[X&4'Fg
p7$$!\"'F)$!3*)*****RHR1s\"F07$$!3_+++++++cF9$!3s*****fMa<s%F07$$!3;++
+++++_F9$!3')*****flkjJ)F07$$!3q+++++++[F9$!31+++-uyM7F-7$$!3M+++++++W
F9$!37+++sm5e;F-7$$!\"%F)$!3++++j;Hr?F-7$$!33+++++++OF9$!3*)*****z42%R
CF-7$$!3;+++++++KF9$!3#******zg#)fs#F-7$$!3E+++++++GF9$!3/+++hcx'*GF-7
$$!3M+++++++CF9$!3v*****H-tP#HF-7$$!\"#F)$!3'******zQ7*)y#F-7$$!33++++
+++;F9$!3'******fW))p[#F-7$$!3;+++++++7F9$!3,+++hdFF?F-7$$!3U+++++++!)
F-$!30+++x1bL9F-7$$!3A+++++++SF-$!3e******or1CuF0-%&STYLEG6#%&POINTG-%
'COLOURG6&%$RGBGF(F($\"*++++\"Fbfo-%+AXESLABELSG6$Q!6\"Fa]p-%%VIEWG6$%
(DEFAULTGFf]p" 1 5 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "N_x:=dx*add(
yR[i]^2+yI[i]^2,i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N_xG$\"
+#*********!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalhf(FF
T(n, var(yR), var(yI)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$c#\"\"
!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "N_p:=dp*add(yR[i]^2+yI
[i]^2,i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N_pG$\"+;3*p#R!\"
)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "This is not normalized corre
ctly, so let us define a normalized DFT:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 124 "yRn:=Vector(N): yIn:=Vector(N): sq:=evalf(1/sqrt(2*P
i)): for i from 1 to N do: yRn[i]:=dx*sq*yR[i]: yIn[i]:=dx*sq*yI[i]: o
d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "N_p:=dp*add(yRn[i]^2+
yIn[i]^2,i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N_pG$\"+$*****
****!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot([seq([pv[i]
,yRn[i]],i=1..N)],style=point);" }}{PARA 13 "" 1 "" {GLPLOT2D 824 208 
208 {PLOTDATA 2 "6&-%'CURVESG6$7\\[l7$$\"\"!F)$\"3/+++ld2L?!#=7$$\"3ok
Da^J#f8'!#>$\"3<+++5B(HA$F,7$$\"3%H^3.j%=F7F,$\"3!******f9'e5[F,7$$\"3
EpFYXpxS=F,$\"3C+++@Y[gnF,7$$\"3)e-<1EpVX#F,$\"3m*****\\&RKX*)F,7$$\"3
[#Grddhz1$F,$\"3&*******H)HW6\"!#<7$$\"3_Qb#4*Qb\"o$F,$\"3$******HK<sI
\"FG7$$\"37&zzg?Y^H%F,$\"31+++XVrV9FG7$$\"3u^SB@&Q(3\\F,$\"3(******H[]
7]\"FG7$$\"3N3$)QO3LAbF,$\"35+++aw\")p9FG7$$\"3'\\cU::Bf8'F,$\"3++++s`
\"\\N\"FG7$$\"3c@opma^\\nF,$\"3/+++c#yf<\"FG7$$\"31x5&=y2JO(F,$\"3Z+++
%yJ+h*F,7$$\"3mL`+(4+n(zF,$\"3++++.(eTR(F,7$$\"3E!ff@T#H!f)F,$\"3))***
**R]BmN&F,7$$\"3)o%QJFZ)Q?*F,$\"3C+++8;p`OF,7$$\"3\\.\"oC/xu\")*F,$\"3
)*******3([kM#F,7$$\"3,OAwNp5V5FG$\"37+++:W#)=9F,7$$\"3nhwFnhY/6FG$\"3
Y+++VHkx!)F07$$\"3K(3$z)RDe;\"FG$\"31+++@>\"*HVF07$$\"3*H^3.j%=F7FG$\"
3))*****46/`=#F07$$\"3mQR#='Qa)G\"FG$\"3/+++ySWQ5F07$$\"3Kk$RL4.*\\8FG
$\"3=+++[\\9YY!#?7$$\"3(**ya[Ki7T\"FG$\"3'******4@Ds&>Fgr7$$\"3U:-Pc:i
s9FG$\"3]+++F\"eHw(!#@7$$\"32Tc)yy!)R`\"FG$\"3/+++\"eD!**GFbs7$$\"3sm5
S>+M&f\"FG$\"3&******poI$>5Fbs7$$\"3S#\\;4D*pc;FG$\"31+++S6buL!#A7$$\"
30=>V#[e!=<FG$\"31+++@#\\=0\"Fbt7$$\"3sVt%Rr<%z<FG$\"31+++VR!p3$!#B7$$
\"3PpFYXpxS=FG$\"3()*****f9P#H&)!#C7$$\"3/&>yp<O@!>FG$\"31+++k<a=AFcu7
$$\"3q?O\\3a\\j>FG$\"3&)*****>*HtIa!#D7$$\"39Y!4+ka[-#FG$\"3-+++v5A]7F
^v7$$\"3-sW_rQ@'3#FG$\"3,+++R!eyq#!#E7$$\"3Y(*)RI5tv9#FG$\"3$******R@4
pi&!#F7$$\"3MB`bMB$*3AFG$\"3'******4z/8K\"F_w7$$\"3y[22m:HqAFG$\"3w***
**RUDq*f!#G7$$\"3muhe(z];L#FG$\"3#******>4fI\\&Fjw7$$\"35+;5H+,$R#FG$
\"3t*****H!y(>!eFjw7$$\"3)f-<1EpVX#FG$\"3?+++ss?5eFjw7$$\"3U^C8#\\Gd^#
FG$\"3G+++Y$yFP&Fjw7$$\"3IxykBx3xDFG$\"3%)*****46\\8d%Fjw7$$\"3u-L;bpW
QEFG$\"3++++5()>oNFjw7$$\"3iG(ym=1)*p#FG$\"37+++'*=jcDFjw7$$\"31aT>=a;
hFFG$\"34+++e=(4s\"Fjw7$$\"3&*z&4(\\Y_AGFG$\"3'******fdwu>\"Fjw7$$\"3Q
0]A\")Q)Q)GFG$\"31+++9n_Y5Fjw7$$\"3#3VSF6V_%HFG$\"3(******>iM?C\"Fjw7$
$\"3rceDWBg1IFG$\"3-+++U2$>o\"Fjw7$$\"39#Grddhz1$FG$\"31+++L'[L@#Fjw7$
$\"3.3nG23KHJFG$\"39+++MO`mEFjw7$$\"3YL@!)Q+o!>$FG$\"33+++Q,!z)GFjw7$$
\"3OfvJq#R?D$FG$\"3)******HoW#pFFjw7$$\"3z%)H$=])R8LFG$\"3-+++xnGpAFjw
7$$\"3m5%[LtdZP$FG$\"3*******4BgxU\"Fjw7$$\"36OQ'['p6OMFG$\"3/+++!f:wn
$!#H7$$\"3+i#zj>wu\\$FG$!3#)*****H\\9C:(Fj]l7$$\"3W(o%*yUN)eNFG$!3;+++
\"y=<e\"Fjw7$$\"3J8,TfY>?OFG$!3;+++\\\\:*)>Fjw7$$\"3uQb#4*Qb\"o$FG$!3%
*******ee>S<Fjw7$$\"3kk4WAJ\"Hu$FG$!3Q+++)R0ZC(Fj]l7$$\"32!RcRNsU!QFG$
\"3%******RsaV0\"Fjw7$$\"3'f\"=Z&eJc'QFG$\"3;+++/$GSZ$Fjw7$$\"3RTs)p\"
3*p#RFG$\"3B+++(e!R4jFjw7$$\"3%om-&[+N))RFG$\"3m******e?([E*Fjw7$$\"3G
#4=+G4(\\SFG$\"3!******H;j9?\"F_w7$$\"3:=N`6&o56%FG$\"3++++VmbC9F_w7$$
\"3/W*[IuFC<%FG$\"3%******p/r&p:F_w7$$\"3#*pVcupyLUFG$\"3'*******3^c=;
F_w7$$\"3#\\zzg?Y^H%FG$\"3#******>P1Pc\"F_w7$$\"3!3A&fPa]cVFG$\"3/+++O
ss29F_w7$$\"3oY16pY'yT%FG$\"3-+++&ysF;\"F_w7$$\"3csgi+RAzWFG$\"37+++,B
#>[)Fjw7$$\"3c(\\T@8$eSXFG$\"3G+++.\"4z([Fjw7$$\"3WBpljB%>g%FG$\"3)***
***H_l^2\"Fjw7$$\"3K\\B<&f,Lm%FG$!3!*******R'e@m#Fjw7$$\"3@vxoE3mCZFG$
!3=+++eGpugFjw7$$\"3?+K?e+-'y%FG$!3`+++4+\"*=*)Fjw7$$\"33E'=(*Gzt%[FG$
!3-+++96U)4\"F_w7$$\"3(>0M7_Q(3\\FG$!3$******H+w>@\"F_w7$$\"3'pZ\\Fv(4
q\\FG$!37+++)HCiA\"F_w7$$\"3%G!\\E%)pXJ]FG$!3'*******H@ZX6F_w7$$\"3sG.
y:i\"G4&FG$!3[*****R[BO%)*Fjw7$$\"3iadHZa<a^FG$!3y*****p)=p`wFjw7$$\"3
gz6\")yY`:_FG$!3K+++S3%o9&Fjw7$$\"3[0mK5R*oF&FG$!3')*****H,E9e#Fjw7$$
\"3PJ?%=9`#Q`FG$!3,+++%H&fG=Fj]l7$$\"3CduNtBh*R&FG$\"31+++u22o=Fjw7$$
\"3C#)G([gr4Y&FG$\"3'******\\:njV$Fjw7$$\"383$)QO3LAbFG$\"35+++560JWFj
w7$$\"3-MP!z1!p$e&FG$\"3B+++;dr4[Fjw7$$\"3!*f\">%*H\\]k&FG$\"3C+++0=)p
e%Fjw7$$\"3*[eM4`3kq&FG$\"3&******HZK@%QFjw7$$\"3y5+XixwndFG$\"3%*****
**3?a:FFjw7$$\"3lOa'R*p7HeFG$\"3++++\"[9<R\"Fjw7$$\"3kh3[Di[!*eFG$\"3E
+++1Vjkp!#I7$$\"3a(G'*pXX=&fFG$!3/+++Zl\\p5Fjw7$$\"3T8<^)o/K,'FG$!3)**
****Ho51*=Fjw7$$\"3IRr-?RcugFG$!3:+++PaP?BFjw7$$\"3HkDa^J#f8'FG$!3!***
***HrW3N#Fjw7$$\"3;!*z0$Q#G(>'FG$!3')*****fpv:.#Fjw7$$\"31;Md9;keiFG$!
35+++!\\$)fX\"Fjw7$$\"3%>%))3Y3+?jFG$!3J*****4^oIW(Fj]l7$$\"3%pE/w2g8Q
'FG$!3++++jLcYEFcjl7$$\"3\"Gp>\"4$>FW'FG$\"35+++v3!pw&Fj]l7$$\"3q=^jS&
yS]'FG$\"3,+++yfT<(*Fj]l7$$\"3eW0:sxVllFG$\"3.+++o?#36\"Fjw7$$\"3epfm.
qzEmFG$\"33+++W?q-5Fjw7$$\"3Y&R\"=Ni:)o'FG$\"3A+++4OQLrFj]l7$$\"3M@opm
a^\\nFG$\"3!)*****pjbG]$Fj]l7$$\"3BZA@)pu3\"oFG$\"3x*******49<V$Fcjl7$
$\"3AswsHRBsoFG$!3#******pJRUO\"Fj]l7$$\"35)4V7;$fLpFG$!3'******HB0i;
\"Fj]l7$$\"3)R_eFR_\\*pFG$\"3G+++gF\\#*pFcjl7$$\"3)*[RFC;JcqFG$\"3=+++
9>7fLFj]l7$$\"3'[P*yb3n<rFG$\"3r*****pZ(*>a&Fj]l7$$\"3u+[I(3I!zrFG$\"3
J+++zkSNfFj]l7$$\"3jE-#)=$*QSsFG$\"33+++A@)4`$Fj]l7$$\"3i^cL]&[<I(FG$!
3%******\\<\\97#Fj]l7$$\"3]x5&=y2JO(FG$!3/+++q*GB3\"Fjw7$$\"3R.lO8qYCu
FG$!3/+++\"=&Hx@Fjw7$$\"3GH>)[CEe[(FG$!3K+++N,IsLFjw7$$\"3EatRwa=ZvFG$
!3q******=\"=p^%Fjw7$$\"3:!y7zqW&3wFG$!3C+++(fIbX&Fjw7$$\"3-1#G%RR!*pw
FG$!3C+++go0YgFjw7$$\"3#>jV4<j7t(FG$!3Y+++bp()zhFjw7$$\"3!p0fCSAEz(FG$
!3H+++Mk8,eFjw7$$!3y#[uRj\")R&yFG$!3q*****H#*R[#\\Fjw7$$!3!p0fCSAEz(FG
$!3'******fwLfk$Fjw7$$!3#>jV4<j7t(FG$!31+++YYKN@Fjw7$$!3-1#G%RR!*pwFG$
!3%)*****4_'=rhFj]l7$$!3:!y7zqW&3wFG$\"3')******3D<xmFj]l7$$!3EatRwa=Z
vFG$\"37+++vV!\\]\"Fjw7$$!3GH>)[CEe[(FG$\"3*******\\A`Ou\"Fjw7$$!3R.lO
8qYCuFG$\"3%*******[2C?8Fjw7$$!3]x5&=y2JO(FG$\"3')*****R9,!HEFj]l7$$!3
i^cL]&[<I(FG$!3%*******)*y\"fJ\"Fjw7$$!3jE-#)=$*QSsFG$!3&)*****4nvsB$F
jw7$$!3u+[I(3I!zrFG$!3))******4Zqv_Fjw7$$!3'[P*yb3n<rFG$!3y*****p$4.$=
(Fjw7$$!3)*[RFC;JcqFG$!3c+++[+<9()Fjw7$$!3)R_eFR_\\*pFG$!3')*****H;=rl
*Fjw7$$!35)4V7;$fLpFG$!3]*****pnI<')*Fjw7$$!3AswsHRBsoFG$!3;+++'zUKE*F
jw7$$!3BZA@)pu3\"oFG$!3q*****f\\DB*yFjw7$$!3M@opma^\\nFG$!3o******H2Kr
eFjw7$$!3Y&R\"=Ni:)o'FG$!31+++\"f@oR$Fjw7$$!3epfm.qzEmFG$!3)******z,r3
;(Fj]l7$$!3eW0:sxVllFG$\"39++++G]**=Fjw7$$!3q=^jS&yS]'FG$\"3)******pPS
*zTFjw7$$!3\"Gp>\"4$>FW'FG$\"3o******3JR$)eFjw7$$!3%pE/w2g8Q'FG$\"3z**
***f)fNBoFjw7$$!3%>%))3Y3+?jFG$\"3c*****z`Mh*oFjw7$$!31;Md9;keiFG$\"36
+++At9,hFjw7$$!3;!*z0$Q#G(>'FG$\"3!*******)*\\J\\XFjw7$$!3HkDa^J#f8'FG
$\"33+++#*==`CFjw7$$!3IRr-?RcugFG$\"34+++k5N65Fj]l7$$!3T8<^)o/K,'FG$!3
%******>m2:=#Fjw7$$!3a(G'*pXX=&fFG$!3)******R\")=b2%Fjw7$$!3kh3[Di[!*e
FG$!3:+++>\\p5`Fjw7$$!3lOa'R*p7HeFG$!3E+++#*zW,dFjw7$$!3y5+XixwndFG$!3
-+++Q.xt^Fjw7$$!3*[eM4`3kq&FG$!3G+++Am'yx$Fjw7$$!3!*f\">%*H\\]k&FG$!3'
)*****HGzao\"Fjw7$$!3-MP!z1!p$e&FG$\"3o+++)>uBK)Fj]l7$$!383$)QO3LAbFG$
\"3H+++\\B(QW$Fjw7$$!3C#)G([gr4Y&FG$\"33+++\"o[7\"eFjw7$$!3CduNtBh*R&F
G$\"3M+++)Qdlk(Fjw7$$!3PJ?%=9`#Q`FG$\"3#)*****>bvFw)Fjw7$$!3[0mK5R*oF&
FG$\"3t*****4-'H.\"*Fjw7$$!3gz6\")yY`:_FG$\"3X+++s`YW()Fjw7$$!3iadHZa<
a^FG$\"3=+++GY2oyFjw7$$!3sG.y:i\"G4&FG$\"3k+++u8k8nFjw7$$!3%G!\\E%)pXJ
]FG$\"3B+++rArAbFjw7$$!3'pZ\\Fv(4q\\FG$\"3;+++H'f>\\%Fjw7$$!3(>0M7_Q(3
\\FG$\"3F+++9(RTu$Fjw7$$!33E'=(*Gzt%[FG$\"3y*****pYe:K$Fjw7$$!3?+K?e+-
'y%FG$\"37+++!>\")e>$Fjw7$$!3@vxoE3mCZFG$\"3-+++wrD)G$Fjw7$$!3K\\B<&f,
Lm%FG$\"3-+++<#\\)*[$Fjw7$$!3WBpljB%>g%FG$\"3#******z#*f+o$Fjw7$$!3c(
\\T@8$eSXFG$\"36+++[ssRPFjw7$$!3csgi+RAzWFG$\"3?+++EdsiNFjw7$$!3oY16pY
'yT%FG$\"3q******>L(p1$Fjw7$$!3!3A&fPa]cVFG$\"3.++++,P3AFjw7$$!3#\\zzg
?Y^H%FG$\"3r*****R!*oS%**Fj]l7$$!3#*pVcupyLUFG$!31+++\\mD]]Fj]l7$$!3/W
*[IuFC<%FG$!3#******z8%pe@Fjw7$$!3:=N`6&o56%FG$!3)******>*Gq*y$Fjw7$$!
3G#4=+G4(\\SFG$!3F+++z3Q2_Fjw7$$!3%om-&[+N))RFG$!3)******fH2ZC'Fjw7$$!
3RTs)p\"3*p#RFG$!3=+++%zp'*y'Fjw7$$!3'f\"=Z&eJc'QFG$!3S*****zMmiz'Fjw7
$$!32!RcRNsU!QFG$!3k*****45*otiFjw7$$!3kk4WAJ\"Hu$FG$!3')*****fG=%f_Fj
w7$$!3uQb#4*Qb\"o$FG$!3/+++?\\w$z$Fjw7$$!3J8,TfY>?OFG$!34+++SoI3>Fjw7$
$!3W(o%*yUN)eNFG$\"3&******H,9qj$Fj]l7$$!3+i#zj>wu\\$FG$\"3A+++b1XiHFj
w7$$!36OQ'['p6OMFG$\"3'******z*yUtdFjw7$$!3m5%[LtdZP$FG$\"3o*****pQ&\\
7')Fjw7$$!3z%)H$=])R8LFG$\"3********)*o5B6F_w7$$!3OfvJq#R?D$FG$\"3$***
***fksWL\"F_w7$$!3YL@!)Q+o!>$FG$\"3-+++D()pn9F_w7$$!3.3nG23KHJFG$\"3.+
++!p$4,:F_w7$$!39#Grddhz1$FG$\"3/+++()Q_B9F_w7$$!3rceDWBg1IFG$\"3()***
**z3)oP7F_w7$$!3#3VSF6V_%HFG$\"30+++,jH5'*Fjw7$$!3Q0]A\")Q)Q)GFG$\"3A+
++:%z2C'Fjw7$$!3&*z&4(\\Y_AGFG$\"3!*******HnrgEFjw7$$!31aT>=a;hFFG$!3w
*****z*\\BArFj]l7$$!3iG(ym=1)*p#FG$!3*)*****pn(o/NFjw7$$!3u-L;bpWQEFG$
!3K+++]ZzZaFjw7$$!3IxykBx3xDFG$!3T+++d+c;kFjw7$$!3U^C8#\\Gd^#FG$!3C+++
yO3RkFjw7$$!3)f-<1EpVX#FG$!39+++nwnxcFjw7$$!35+;5H+,$R#FG$!3!)*****4P>
nQ%Fjw7$$!3muhe(z];L#FG$!3%******HNw,'GFjw7$$!3y[22m:HqAFG$!3++++EDex8
Fjw7$$!3MB`bMB$*3AFG$!3-+++.8_#f\"Fj]l7$$!3Y(*)RI5tv9#FG$\"3w*****Rz<k
l'Fj]l7$$!3-sW_rQ@'3#FG$\"3-+++KlZs5Fjw7$$!39Y!4+ka[-#FG$\"3*******\\f
tj8\"Fjw7$$!3q?O\\3a\\j>FG$\"3/+++h\"pq+\"Fjw7$$!3/&>yp<O@!>FG$\"3M***
***>9y(o)Fj]l7$$!3PpFYXpxS=FG$\"3k*****f_r:!*)Fj]l7$$!3sVt%Rr<%z<FG$\"
3%******>&>3y6Fjw7$$!30=>V#[e!=<FG$\"3/+++uMaX<Fjw7$$!3S#\\;4D*pc;FG$
\"3!*******yJ`0DFjw7$$!3sm5S>+M&f\"FG$\"3))*****zx_7H$Fjw7$$!32Tc)yy!)
R`\"FG$\"37+++akE**QFjw7$$!3U:-Pc:is9FG$\"3q*****\\A]F9%Fjw7$$!3(**ya[
Ki7T\"FG$\"3%)******=s.7RFjw7$$!3Kk$RL4.*\\8FG$\"33+++/Gi%G$Fjw7$$!3mQ
R#='Qa)G\"FG$\"3-+++GFo%)HFjw7$$!3*H^3.j%=F7FG$\"3[+++MuMOnFjw7$$!3K(3
$z)RDe;\"FG$\"3;+++?kJ(G$F_w7$$!3nhwFnhY/6FG$\"33+++!)ply;Fiv7$$!3,OAw
Np5V5FG$\"3e+++jmoozFiv7$$!3\\.\"oC/xu\")*F,$\"34+++(*zAJNF^v7$$!3)o%Q
JFZ)Q?*F,$\"33+++bK;p9Fcu7$$!3E!ff@T#H!f)F,$\"3!)*****>`n0v&Fcu7$$!3mL
`+(4+n(zF,$\"3()*****47i)=@F]u7$$!31x5&=y2JO(F,$\"3?+++!3@/N(F]u7$$!3c
@opma^\\nF,$\"3&******z*pz+CFbt7$$!3'\\cU::Bf8'F,$\"3a*****f([0$Q(Fbt7
$$!3N3$)QO3LAbF,$\"3.+++`0uP@Fbs7$$!3u^SB@&Q(3\\F,$\"3))*****>p-z#eFbs
7$$!37&zzg?Y^H%F,$\"3++++Z(=f\\\"Fgr7$$!3_Qb#4*Qb\"o$F,$\"3\")*****Ri
\"G:OFgr7$$!3[#Grddhz1$F,$\"3c*****z$>\\E#)Fgr7$$!3)e-<1EpVX#F,$\"3')*
*****=e[i<F07$$!3EpFYXpxS=F,$\"3#)*****Hs*GbNF07$$!3%H^3.j%=F7F,$\"3++
++O<Z_nF07$$!3okDa^J#f8'F0$\"3'******>20v?\"F,-%'COLOURG6&%$RGBG$\"#5!
\"\"F(F(-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q!6\"F]]p-%%VIEWG6$%(DEFAUL
TGFb]p" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve
 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot([seq([pv[i],yIn
[i]],i=1..N)],style=point,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 
824 208 208 {PLOTDATA 2 "6'-%'CURVESG6#7\\[l7$$\"\"!F)$!3u*****ff!4Vr!
#b7$$\"3okDa^J#f8'!#>$\"39+++36@\\S!#K7$$\"3%H^3.j%=F7!#=$!3%)*****z/V
$*3$!#L7$$\"3EpFYXpxS=F7$\"3w******fx*o)RF37$$\"3)e-<1EpVX#F7$\"31+++-
ZYJ?F:7$$\"3[#Grddhz1$F7$\"3=+++&yD$[NF37$$\"3_Qb#4*Qb\"o$F7$!34+++l(*
)4?#F:7$$\"37&zzg?Y^H%F7$\"3++++HEz\\8F37$$\"3u^SB@&Q(3\\F7$\"3K+++xc
\"o0'!#M7$$\"3N3$)QO3LAbF7$!3\"******4x$44()F:7$$\"3'\\cU::Bf8'F7$\"37
+++B]-]BF:7$$\"3c@opma^\\nF7$!3&)*****R,q!GFF37$$\"31x5&=y2JO(F7$!3i**
****ftHnYF:7$$\"3mL`+(4+n(zF7$!3m*****zq+Jf%F37$$\"3E!ff@T#H!f)F7$!35+
++&QS#[HF:7$$\"3)o%QJFZ)Q?*F7$!3C+++EK>\"Q%F37$$\"3\\.\"oC/xu\")*F7$!3
7+++(>_(=BFY7$$\"3,OAwNp5V5!#<$!3$******z\"4GBNF37$$\"3nhwFnhY/6F_q$\"
31+++opn[>F:7$$\"3K(3$z)RDe;\"F_q$!3++++au!pk#F37$$\"3*H^3.j%=F7F_q$!3
m*****f)R\"R,&F:7$$\"3mQR#='Qa)G\"F_q$!3))*****\\I%[5GF37$$\"3Kk$RL4.*
\\8F_q$!3#)*****>np\")\\#F:7$$\"3(**ya[Ki7T\"F_q$!3\"*******o**oLAF37$
$\"3U:-Pc:is9F_q$!34+++M9^Q:F:7$$\"32Tc)yy!)R`\"F_q$!39+++nkY)y\"F37$$
\"3sm5S>+M&f\"F_q$\"3%******ROBV.%FY7$$\"3S#\\;4D*pc;F_q$!30+++q3.D9F3
7$$\"30=>V#[e!=<F_q$!33+++h:e$y$F:7$$\"3sVt%Rr<%z<F_q$!32+++FpuF;F37$$
\"3PpFYXpxS=F_q$!39+++.`O<DF:7$$\"3/&>yp<O@!>F_q$!3'******zO:FP\"F37$$
\"3q?O\\3a\\j>F_q$\"3`*****HP#3Ns!#T7$$\"39Y!4+ka[-#F_q$!31+++u$)*[/\"
F37$$\"3-sW_rQ@'3#F_q$\"3-+++BBJw^FY7$$\"3Y(*)RI5tv9#F_q$!3g+++a)*zc$*
F:7$$\"3MB`bMB$*3AF_q$!3*******fQ\\y7#F:7$$\"3y[22m:HqAF_q$!33+++f;a'3
\"F37$$\"3muhe(z];L#F_q$!35+++Ur,)\\\"F:7$$\"35+;5H+,$R#F_q$!31+++KoxB
5F37$$\"3)f-<1EpVX#F_q$!3!******\\7)G$H\"F:7$$\"3U^C8#\\Gd^#F_q$!3v***
**HDCFT)F:7$$\"3IxykBx3xDF_q$!3q*****ppfE#HFY7$$\"3u-L;bpWQEF_q$!37+++
(Ry!\\vF:7$$\"3iG(ym=1)*p#F_q$!3(******4&R\"R\"=F:7$$\"31aT>=a;hFF_q$!
37+++]W\\o&)F:7$$\"3&*z&4(\\Y_AGF_q$!35+++^^x[6F:7$$\"3Q0]A\")Q)Q)GF_q
$!3N+++w*y<?(F:7$$\"3#3VSF6V_%HF_q$\"3H*****\\N7,_*!#N7$$\"3rceDWBg1IF
_q$!3H+++jD73eF:7$$\"39#Grddhz1$F_q$\"3()*****pkX\\5&FY7$$\"3.3nG23KHJ
F_q$!3?+++.QgI]F:7$$\"3YL@!)Q+o!>$F_q$!31++++8#>+\"F:7$$\"3OfvJq#R?D$F
_q$!3/+++'y(4*e'F:7$$\"3z%)H$=])R8LF_q$!38+++2sjOBFY7$$\"3m5%[LtdZP$F_
q$!3++++(\\B\"QcF:7$$\"36OQ'['p6OMF_q$!3A+++W#RU'QFY7$$\"3+i#zj>wu\\$F
_q$!31+++dJ(*QcF:7$$\"3W(o%*yUN)eNF_q$\"3:+++W;4H^FY7$$\"3J8,TfY>?OF_q
$!3c*****>:&)>h%F:7$$\"3uQb#4*Qb\"o$F_q$!3!******4F%)[B\"F:7$$\"3kk4WA
J\"Hu$F_q$!3]+++5A%)p`F:7$$\"32!RcRNsU!QF_q$!3!******H%fgN(*FY7$$\"3'f
\"=Z&eJc'QF_q$!3[+++f99EYF:7$$\"3RTs)p\"3*p#RF_q$\"3!)*****>fI/,%!#W7$
$\"3%om-&[+N))RF_q$!31+++%p]')o$F:7$$\"3G#4=+G4(\\SF_q$\"3q*****\\+lDk
'F^[l7$$\"3:=N`6&o56%F_q$!36+++H0H<FF:7$$\"3/W*[IuFC<%F_q$!3y******>2s
*p(FY7$$\"3#*pVcupyLUF_q$!37+++fy$pN%F:7$$\"3#\\zzg?Y^H%F_q$!3o*****4w
;Vi$F^[l7$$\"3!3A&fPa]cVF_q$!33+++8=*3x#F:7$$\"3oY16pY'yT%F_q$!3;+++F*
)3_uFY7$$\"3csgi+RAzWF_q$!31+++yd>9SF:7$$\"3c(\\T@8$eSXF_q$!31+++0L'zh
'FY7$$\"3WBpljB%>g%F_q$!37+++QntmBF:7$$\"3K\\B<&f,Lm%F_q$!3;+++![O5\"*
*FY7$$\"3@vxoE3mCZF_q$!30+++Y$G0#RF:7$$\"3?+K?e+-'y%F_q$!3'******4B-.*
yFY7$$\"33E'=(*Gzt%[F_q$!3#)*****pEKI&HF:7$$\"3(>0M7_Q(3\\F_q$\"3%)***
**RLstQ$F^[l7$$\"3'pZ\\Fv(4q\\F_q$!31+++d.XdBF:7$$\"3%G!\\E%)pXJ]F_q$
\"3-+++$H()zV\"FY7$$\"3sG.y:i\"G4&F_q$!3))*****z5-FD#F:7$$\"3iadHZa<a^
F_q$!3=+++_%Q#4dFY7$$\"3gz6\")yY`:_F_q$!3*)*****4yRs$GF:7$$\"3[0mK5R*o
F&F_q$!3C+++nBD;TFY7$$\"3PJ?%=9`#Q`F_q$!3!******prAOk#F:7$$\"3CduNtBh*
R&F_q$!3(******z\\-k#HFY7$$\"3C#)G([gr4Y&F_q$!3?+++-XyJFF:7$$\"383$)QO
3LAbF_q$\"3m******4./OuF^[l7$$\"3-MP!z1!p$e&F_q$!3#******\\<(Q\\?F:7$$
\"3!*f\">%*H\\]k&F_q$\"3m*****H=$f4UF^[l7$$\"3*[eM4`3kq&F_q$!31+++Blrj
DF:7$$\"3y5+XixwndF_q$!32+++d!zr\"HFY7$$\"3lOa'R*p7HeF_q$!3++++cY2n@F:
7$$\"3kh3[Di[!*eF_q$!37+++$o.9,$F]v7$$\"3a(G'*pXX=&fF_q$!33+++*4!yc?F:
7$$\"3T8<^)o/K,'F_q$\"34+++sL'fT#FY7$$\"3IRr-?RcugF_q$!31+++D'y%p:F:7$
$\"3HkDa^J#f8'F_q$\"3$)*****p!y.7XF^[l7$$\"3;!*z0$Q#G(>'F_q$!31+++4?\"
G$>F:7$$\"31;Md9;keiF_q$\"3!*******oIPR**F^[l7$$\"3%>%))3Y3+?jF_q$!3))
*****RzMzE\"F:7$$\"3%pE/w2g8Q'F_q$\"37+++2,'*z@FY7$$\"3\"Gp>\"4$>FW'F_
q$!3'******pENb2\"F:7$$\"3q=^jS&yS]'F_q$!3%)*****zpfE#HFY7$$\"3eW0:sxV
llF_q$!3*******H`.g.\"F:7$$\"3epfm.qzEmF_q$!3)******>kCWb$FY7$$\"3Y&R
\"=Ni:)o'F_q$!3l*****H%pnU**FY7$$\"3M@opma^\\nF_q$!3,+++?iKH7FY7$$\"3B
ZA@)pu3\"oF_q$!3%******p\"yAcaFY7$$\"3AswsHRBsoF_q$\"3v+++??6?&*F^[l7$
$\"35)4V7;$fLpF_q$!3Y*****>nrEl(FY7$$\"3)R_eFR_\\*pF_q$!37+++8-W/9FY7$
$\"3)*[RFC;JcqF_q$!3y*****pV!GCwFY7$$\"3'[P*yb3n<rF_q$\"3%******4(=jY:
FY7$$\"3u+[I(3I!zrF_q$!3S+++t)RLX)FY7$$\"3jE-#)=$*QSsF_q$!3%4++]AX=-*!
#O7$$\"3i^cL]&[<I(F_q$!3Q+++ek%3p&FY7$$\"3]x5&=y2JO(F_q$\"3u*****z_i[[
%F^[l7$$\"3R.lO8qYCuF_q$!3#*******pn-u!)FY7$$\"3GH>)[CEe[(F_q$!3!*****
**eERrBFY7$$\"3EatRwa=ZvF_q$!3z*****4$)\\R`(FY7$$\"3:!y7zqW&3wF_q$\"3!
******>ltBx\"FY7$$\"3-1#G%RR!*pwF_q$!3-+++Xl,naFY7$$\"3#>jV4<j7t(F_q$!
3s*****4Jk4X%F^[l7$$\"3!p0fCSAEz(F_q$!3')******\\EzjZFY7$$!3y#[uRj\")R
&yF_qF*7$$!3!p0fCSAEz(F_q$!3S+++Hh+hcFY7$$!3#>jV4<j7t(F_q$\"3*)*****H@
.2K$F^[l7$$!3-1#G%RR!*pwF_q$!3)*******yDq0[FY7$$!3:!y7zqW&3wF_q$\"39++
+!\\P$\\gF^[l7$$!3EatRwa=ZvF_q$!3E+++#\\49P%FY7$$!3GH>)[CEe[(F_q$\"31+
++&[ynz\"FY7$$!3R.lO8qYCuF_q$!37+++4=blKFY7$$!3]x5&=y2JO(F_q$\"37+++5
\">#*)HF^[l7$$!3i^cL]&[<I(F_q$!37+++PfF^?FY7$$!3jE-#)=$*QSsF_q$\"3))**
***\\zM.C#FY7$$!3u+[I(3I!zrF_q$!31+++rY;+8FY7$$!3'[P*yb3n<rF_q$\"3-+++
oL1gFF^[l7$$!3)*[RFC;JcqF_q$\"3[*****HOa^y)F^[l7$$!3)R_eFR_\\*pF_q$\"3
9+++f=,'=#FY7$$!35)4V7;$fLpF_q$\"3'******4@MVm$F^[l7$$!3AswsHRBsoF_q$
\"3++++SBP(Q$F^[l7$$!3BZA@)pu3\"oF_q$\"3?+++q#\\p*RFY7$$!3M@opma^\\nF_
q$\"3!******z/8(*>#F^[l7$$!3Y&R\"=Ni:)o'F_q$\"3!******R&>geTFY7$$!3epf
m.qzEmF_q$\"3))*****f6(*z%=FY7$$!3eW0:sxVllF_q$\"3N+++xc_;#)FY7$$!3q=^
jS&yS]'F_q$\"37+++Y3)fR#FY7$$!3\"Gp>\"4$>FW'F_q$\"3(******HxpV+\"F:7$$
!3%pE/w2g8Q'F_q$!3>+++*eEsj#F^[l7$$!3%>%))3Y3+?jF_q$\"3w******H\\?%3)F
Y7$$!31;Md9;keiF_q$\"3'******R3B=9#F^[l7$$!3;!*z0$Q#G(>'F_q$\"3[+++==c
StFY7$$!3HkDa^J#f8'F_q$\"32+++D]PT9F^[l7$$!3IRr-?RcugF_q$\"3-+++_2%p5
\"F:7$$!3T8<^)o/K,'F_q$\"3%)*****p)3!H<$FY7$$!3a(G'*pXX=&fF_q$\"3Q+++
\"z'HL$*FY7$$!3kh3[Di[!*eF_q$!3++++8!\\G@\"F]v7$$!3lOa'R*p7HeF_q$\"37+
++x$y5d)FY7$$!3y5+XixwndF_q$!3>+++J0![/$F^[l7$$!3*[eM4`3kq&F_q$\"3,+++
FY9=#)FY7$$!3!*f\">%*H\\]k&F_q$!3!)*****4e6nk(F^[l7$$!3-MP!z1!p$e&F_q$
\"3(******p-xGW\"F:7$$!383$)QO3LAbF_q$\"3!******Hs!p`=FY7$$!3C#)G([gr4
Y&F_q$\"37+++08K_7F:7$$!3CduNtBh*R&F_q$!3%******4g'Hr>FY7$$!3PJ?%=9`#Q
`F_q$\"3/+++\"\\X(e8F:7$$!3[0mK5R*oF&F_q$\"3%******zU]$)3#FY7$$!3gz6\"
)yY`:_F_q$\"3%*******[\"4Ev\"F:7$$!3iadHZa<a^F_q$\"3_+++Z[`*G'FY7$$!3s
G.y:i\"G4&F_q$\"3!*********zIB>F:7$$!3%G!\\E%)pXJ]F_q$\"33+++])*=%*eFY
7$$!3'pZ\\Fv(4q\\F_q$\"3)******R1e$\\=F:7$$!3(>0M7_Q(3\\F_q$!3$*******
*z8u+$F^[l7$$!33E'=(*Gzt%[F_q$\"35+++Hgk!)=F:7$$!3?+K?e+-'y%F_q$\"3O++
+<03p()F^[l7$$!3@vxoE3mCZF_q$\"3++++*za6g\"F:7$$!3K\\B<&f,Lm%F_q$\"3]*
****z`77(pFY7$$!3WBpljB%>g%F_q$\"3'*******o,t%Q#F:7$$!3c(\\T@8$eSXF_q$
\"3E+++DhZ!3'FY7$$!3csgi+RAzWF_q$\"3%******H'\\m:EF:7$$!3oY16pY'yT%F_q
$\"3++++!zL>R#FY7$$!3!3A&fPa]cVF_q$\"3A+++fa-mDF:7$$!3#\\zzg?Y^H%F_q$!
37+++N(p%4IFY7$$!3#*pVcupyLUF_q$\"3-+++n\")QeBF:7$$!3/W*[IuFC<%F_q$\"3
;+++wQ4'f$FY7$$!3:=N`6&o56%F_q$\"3A+++\\P5zHF:7$$!3G#4=+G4(\\SF_q$\"3S
+++g-*4)zFY7$$!3%om-&[+N))RF_q$\"3()*****p#>T.HF:7$$!3RTs)p\"3*p#RF_q$
!3!)*****>fI/,%F_`l7$$!3'f\"=Z&eJc'QF_q$\"3B+++:?*\\L#F:7$$!32!RcRNsU!
QF_q$!3y******RF1zGFY7$$!3kk4WAJ\"Hu$F_q$\"3C+++3kNVAF:7$$!3uQb#4*Qb\"
o$F_q$\"37+++SibtkFY7$$!3J8,TfY>?OF_q$\"3#)*****pJGq9%F:7$$!3W(o%*yUN)
eNF_q$!3t******pnJCOF^[l7$$!3+i#zj>wu\\$F_q$\"3))*****4(*=%)*GF:7$$!36
OQ'['p6OMF_q$\"3]+++M!)=@nFY7$$!3m5%[LtdZP$F_q$\"3!********Rq(*[%F:7$$
!3z%)H$=])R8LF_q$\"3g+++cOJbvFY7$$!3OfvJq#R?D$F_q$\"3A+++*>5Rf#F:7$$!3
YL@!)Q+o!>$F_q$\"34+++'*y)[8\"F:7$$!3.3nG23KHJF_q$\"3++++y;5#H&F:7$$!3
9#Grddhz1$F_q$\"3A+++JZ(HG'FY7$$!3rceDWBg1IF_q$\"3?+++Q\"Qz[%F:7$$!3#3
VSF6V_%HF_q$\"3++++ZBP(Q$F^[l7$$!3Q0]A\")Q)Q)GF_q$\"3@+++&HT>$RF:7$$!3
&*z&4(\\Y_AGF_q$!3!)*****pvh6*HFY7$$!31aT>=a;hFF_q$\"35+++x'3mT$F:7$$!
3iG(ym=1)*p#F_q$\"3#******\\A$y#z*FY7$$!3u-L;bpWQEF_q$\"3N+++zhq6bF:7$
$!3IxykBx3xDF_q$\"3))*****43MWi'FY7$$!3U^C8#\\Gd^#F_q$\"39+++%[_F;&F:7
$$!3)f-<1EpVX#F_q$\"3%******z!4P>BFY7$$!35+;5H+,$R#F_q$\"3Z+++@yxCeF:7
$$!3muhe(z];L#F_q$!3-+++.aDL:FY7$$!3y[22m:HqAF_q$\"3q*****H#*p/2&F:7$$
!3MB`bMB$*3AF_q$\"3#******>bbP\"=F:7$$!3Y(*)RI5tv9#F_q$\"3N+++z7!H5(F:
7$$!3-sW_rQ@'3#F_q$\"3'******p+I1G\"F:7$$!39Y!4+ka[-#F_q$\"3p*****f$\\
^NnF:7$$!3q?O\\3a\\j>F_q$!3%******fnH3,$F]v7$$!3/&>yp<O@!>F_q$\"35+++(
G<&HfF:7$$!3PpFYXpxS=F_q$!34+++T'p2j%FY7$$!3sVt%Rr<%z<F_q$\"3/+++(\\U/
z&F:7$$!30=>V#[e!=<F_q$\"3(******z?YD+#F:7$$!3S#\\;4D*pc;F_q$\"3&*****
*z%4a4))F:7$$!3sm5S>+M&f\"F_q$\"3/+++#f8EU\"F:7$$!32Tc)yy!)R`\"F_q$\"3
1+++M,U!H*F:7$$!3U:-Pc:is9F_q$\"3,+++'fTi7\"F:7$$!3(**ya[Ki7T\"F_q$\"3
C+++lM$[>)F:7$$!3Kk$RL4.*\\8F_qF_an7$$!3mQR#='Qa)G\"F_q$\"3g******)RZT
q)F:7$$!3*H^3.j%=F7F_q$\"3')*****pB,Qr#F:7$$!3K(3$z)RDe;\"F_q$\"3,+++X
xc17F37$$!3nhwFnhY/6F_q$\"3#******>&=B)R#F:7$$!3,OAwNp5V5F_q$\"32+++CM
6?7F37$$!3\\.\"oC/xu\")*F7$!38+++OcS2IF^[l7$$!3)o%QJFZ)Q?*F7$\"3.+++'o
\"oF6F37$$!3E!ff@T#H!f)F7$!3)******>&*GI.\"F:7$$!3mL`+(4+n(zF7$\"3!***
***f<^S7\"F37$$!31x5&=y2JO(F7$\"3w******H\"p-%GF:7$$!3c@opma^\\nF7$\"3
$***********HI<F37$$!3'\\cU::Bf8'F7$\"3#******R\"R%R)=F:7$$!3N3$)QO3LA
bF7$\"3;+++OVfz=F37$$!3u^SB@&Q(3\\F7$\"3/+++)Gl;J\"F:7$$!37&zzg?Y^H%F7
$\"3#)******=U'f'>F37$$!3_Qb#4*Qb\"o$F7$!3)******4zl,0\"F:7$$!3[#Grddh
z1$F7$\"3%*******=Wzb@F37$$!3)e-<1EpVX#F7$\"35+++FQ!=&eF:7$$!3EpFYXpxS
=F7$\"3<+++=&zxS$F37$$!3%H^3.j%=F7F7$\"33+++4/#)[UF:7$$!3okDa^J#f8'F0$
\"3i*****Re:J\"QF3-%&STYLEG6#%&POINTG-%'COLOURG6&%$RGBGF(F($\"*++++\"!
\")-%+AXESLABELSG6$Q!6\"Ff\\p-%%VIEWG6$%(DEFAULTGF[]p" 1 5 0 1 10 0 2 
6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 118 "The imaginary part is returned as zero, while th
e real part describes a wavefunction of Gaussian shape centered on p0.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 11 "Exer
cise 6:" }}{PARA 0 "" 0 "" {TEXT -1 103 "Change the parameters of the \+
Gaussian wavepacket and observe how they are reflected in the DFT resu
lts." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 215 "As a final check we carry out an error
 analysis: we do have an analytic answer for the FT, i.e., we can figu
re out how accurate the DFT is for the present example. Let us evaluat
e the analytic FT at the grid points:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "FT:=Vector(N): for i from 1 to N do: FT[i]:=evalf(sub
s(p= pv[i],Fp), 16); od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Pick \+
a location on the momentum mesh and compare the results and calculate \+
the relative error:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "i0:=
10: pv[i0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3N3$)QO3LAb!#=" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "FT[i0],yRn[i0],abs((FT[i0]-y
Rn[i0])/FT[i0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"1g3,bw\")p9!#:$
\"+aw\")p9!\"*$\"+J]c.o!#>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "Ph
enomenal! The agreement is with computational accuracy for the analyti
c result controlled by the Digits variable; (increase Digits and repea
t the calculation to confirm this)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 36 "However, if we go to larger momenta:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "i0:=30: pv[i0];" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 44 "FT[i0],yRn[i0],abs((FT[i0]-yRn[i0])/FT[i0])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3sVt%Rr<%z<!#<" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%$\"1C_s5R*p3$!#@$\"+VR!p3$!#:$\"+R5U9H!#9" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "In fact, if one calculates the ave
rage relative error over the entire mesh one obtains:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "add(abs((FT[i]-yRn[i])/FT[i]),i=1..
N)/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+rXB'z$\"$?#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 221 "A ridiculously large number! This is the
 result of applying periodic boundary conditions in the DFT. Instead o
f falling to zero the 'Gaussian' as computed per DFT connects periodic
ally to the neighbours in adjacent cells." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "P1:=plot([seq([pv[i],log(yRn[i])],i=1..N/2)]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P2:=plot([seq([pv[i],log(FT[
i])],i=1..N/2)],color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "display([P1,P2],view=[0..4,-25..1]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 800 234 234 {PLOTDATA 2 "6&-%'CURVESG6$7]q7$$\"\"!F)$!+\"HNI
f\"!\"*7$$\"+_J#f8'!#6$!+#3\"GK6F,7$$\"+IY=F7!#5$!+k:m<tF67$$\"+XpxS=F
6$!+g^!\\\"RF67$$\"+h#pVX#F6$!+/;a96F67$$\"+w:'z1$F6$\"+4\"HM3\"F67$$
\"+\"*Qb\"o$F6$\"+qp+zEF67$$\"+1i9&H%F6$\"+*)>>sOF67$$\"+@&Q(3\\F6$\"+
hT)H1%F67$$\"+O3LAbF6$\"+#[$Q^QF67$$F/F6$\"+j**QPIF67$$\"+na^\\nF6$\"+
&f.5i\"F67$$\"+#y2JO(F6$!+jivxRF07$$\"+(4+n(zF6$!+!pZ*=IF67$$\"+7CH!f)
F6$!+(f7DC'F67$$\"+FZ)Q?*F6$!+Oq%o+\"F,7$$\"+UqZ<)*F6$!+&4#o\\9F,7$$\"
+Op5V5F,$!+Tkv_>F,7$$\"+nhY/6F,$!+r+2;DF,7$$\"+*RDe;\"F,$!+')HiRJF,7$$
F5F,$!+'=:M#QF,7$$\"+iQa)G\"F,$!+smWnXF,7$$\"+$4.*\\8F,$!+Zurr`F,7$$\"
+DBE69F,$!+<vAOiF,7$$\"+c:is9F,$!+3p(4;(F,7$$\"+)y!)R`\"F,$!+)plf9)F,7
$$\"+>+M&f\"F,$!+\\T>\">*F,7$$\"+^#*pc;F,$!+:jmH5!\")7$$\"+#[e!=<F,$!+
pvBY6F^t7$$\"+9xTz<F,$!+$>M)o7F^t7$$F;F,$!+t&fuR\"F^t7$$\"+xh8->F,$!+`
X7K:F^t7$$\"+3a\\j>F,$!+j1'Gn\"F^t7$$\"+SY&[-#F,$!+Ngt>=F^t7$$\"+sQ@'3
#F,$!+\"z5F(>F^t7$$\"+.JdZ@F,$!+i!H)H@F^t7$$\"+NB$*3AF,$!+?JsuAF^t7$$
\"+m:HqAF,$!+Wsr`BF^t7$$\"+)z];L#F,$!+r]\\iBF^t7$$\"+H+,$R#F,$!+;P-dBF
^t7$$F@F,$!+y>)oN#F^t7$$\"+#\\Gd^#F,$!+(34ZO#F^t7$$\"+Cx3xDF,$!+lF'3Q#
F^t7$$\"+bpWQEF,$!+7vj0CF^t7$$\"+(=1)*p#F,$!+IX(*QCF^t7$$\"+=a;hFF,$!+
'oa&yCF^t7$$\"+]Y_AGF,$!+`>#[^#F^t7$$\"+\")Q)Q)GF,$!+BfHGDF^t7$$\"+8JC
XHF,$!+;&o6^#F^t7$$\"+WBg1IF,$!+k$\\3[#F^t7$$FEF,$!+WHR`CF^t7$$\"+23KH
JF,$!+mcwMCF^t7$$\"+R+o!>$F,$!+Q1zECF^t7$$\"+q#R?D$F,$!+Uh)4V#F^t7$$\"
+-&)R8LF,$!+Wq*3X#F^t7$$\"+LxvuLF,$!+3HB(\\#F^t7$$\"+lp6OMF,$!+^c(Gj#F
^t7$%*undefinedGF^]l7$$\"+aBF/QF,$!+32bFDF^t7$$\"+&eJc'QF,$!+@@J3CF^t7
$$\"+<3*p#RF,$!+$pR'[BF^t7$$\"+\\+N))RF,$!+(f?-J#F^t7$$\"+!G4(\\SF,$!+
\"3JUG#F^t7$$\"+7&o56%F,$!+H!*>nAF^t7$$\"+VxUsTF,$!+d[]dAF^t7$$\"+vpyL
UF,$!+\"4JWD#F^t7$$FOF,$!+0#zyD#F^t7$$\"+Qa]cVF,$!+UuQoAF^t7$$\"+pY'yT
%F,$!+XV](G#F^t7$$\"+,RAzWF,$!+\"*)\\!>BF^t7$$\"+KJeSXF,$!+O>PuBF^t7$$
\"+kB%>g%F,$!+ShfDDF^tF]]l7$$\"+tBh*R&F,$!+!)HNqCF^t7$$\"+0;(4Y&F,$!+<
@S4CF^t7$$FYF,$!+?*zRQ#F^t7$$\"+o+p$e&F,$!+/)zdP#F^t7$$\"+*H\\]k&F,$!+
x8_!Q#F^t7$$\"+J&3kq&F,$!+[3C)R#F^t7$$\"+ixwndF,$!+XW%HV#F^t7$$\"+%*p7
HeF,$!+g**y*\\#F^t7$$\"+Di[!*eF,$!+6hF*z#F^tF]]l7$$\"+4$>FW'F,$!+H'))y
e#F^t7$$\"+T&yS]'F,$!+Q,rNDF^t7$$\"+sxVllF,$!+oNLADF^t7$$\"+/qzEmF,$!+
iPdKDF^t7$$\"+Ni:)o'F,$!+VNimDF^t7$$F\\oF,$!+eUuPEF^t7$$\"+)pu3\"oF,$!
+LY0qGF^tF]]l7$$\"+$R_\\*pF,$!+5p())z#F^t7$$\"+C;JcqF,$!+^T$>k#F^t7$$
\"+c3n<rF,$!+7m'=f#F^t7$$\"+(3I!zrF,$!+g&3]e#F^t7$$\"+>$*QSsF,$!+1X%pj
#F^tF]]l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7\\s7$F($!31+++\"HNIf\"!#
<7$$\"3okDa^J#f8'!#>$!3\"******43\"GK6Fhhl7$$\"3%H^3.j%=F7!#=$!3X*****
*f:m<tFbil7$$\"3EpFYXpxS=Fbil$!3x*****f:0\\\"RFbil7$$\"3)e-<1EpVX#Fbil
$!3%******zfTX6\"Fbil7$$\"3[#Grddhz1$Fbil$\"31+++4\"HM3\"Fbil7$$\"3_Qb
#4*Qb\"o$Fbil$\"3w*****z(p+zEFbil7$$\"37&zzg?Y^H%Fbil$\"3!******f*>>sO
Fbil7$$\"3u^SB@&Q(3\\Fbil$\"31+++oT)H1%Fbil7$$\"3N3$)QO3LAbFbil$\"3#)*
*****)[$Q^QFbil7$$\"3'\\cU::Bf8'Fbil$\"33+++j**QPIFbil7$$\"3c@opma^\\n
Fbil$\"38+++&f.5i\"Fbil7$$\"31x5&=y2JO(Fbil$!3C++++ivxRF\\il7$$\"3mL`+
(4+n(zFbil$!30+++$oZ*=IFbil7$$\"3E!ff@T#H!f)Fbil$!3G+++$f7DC'Fbil7$$\"
3)o%QJFZ)Q?*Fbil$!31+++Nq%o+\"Fhhl7$$\"3\\.\"oC/xu\")*Fbil$!3++++&4#o
\\9Fhhl7$$\"3,OAwNp5V5Fhhl$!3#*******Rkv_>Fhhl7$$\"3nhwFnhY/6Fhhl$!30+
++q+2;DFhhl7$$\"3K(3$z)RDe;\"Fhhl$!3%)*****R)HiRJFhhl7$$\"3*H^3.j%=F7F
hhl$!3$)*****H=:M#QFhhl7$$\"3mQR#='Qa)G\"Fhhl$!3e*****pmYuc%Fhhl7$$\"3
Kk$RL4.*\\8Fhhl$!3))*****\\V<<P&Fhhl7$$\"3(**ya[Ki7T\"Fhhl$!3$******z[
FiB'Fhhl7$$\"3U:-Pc:is9Fhhl$!3u*****f#o(4;(Fhhl7$$\"32Tc)yy!)R`\"Fhhl$
!3?+++\\a'f9)Fhhl7$$\"3sm5S>+M&f\"Fhhl$!3U*****fN$>\">*Fhhl7$$\"3S#\\;
4D*pc;Fhhl$!3/+++bgmH5!#;7$$\"30=>V#[e!=<Fhhl$!3/+++-nBY6Fbam7$$\"3sVt
%Rr<%z<Fhhl$!3/+++y7$)o7Fbam7$$\"3PpFYXpxS=Fhhl$!31+++$y\\uR\"Fbam7$$
\"3/&>yp<O@!>Fhhl$!3.+++;A4K:Fbam7$$\"3q?O\\3a\\j>Fhhl$!3%)*****zdeFn
\"Fbam7$$\"39Y!4+ka[-#Fhhl$!3#******z')[%>=Fbam7$$\"3-sW_rQ@'3#Fhhl$!3
7+++'3j@(>Fbam7$$\"3Y(*)RI5tv9#Fhhl$!3;+++L7!48#Fbam7$$\"3MB`bMB$*3AFh
hl$!3'******zIjcH#Fbam7$$\"3y[22m:HqAFhhl$!3'******>J\\kY#Fbam7$$\"3mu
he(z];L#Fhhl$!39+++X#fKk#Fbam7$$\"35+;5H+,$R#Fhhl$!3++++0J4EGFbam7$$\"
3)f-<1EpVX#Fhhl$!38+++&*3&\\,$Fbam7$$\"3U^C8#\\Gd^#Fhhl$!3I+++7E$)4KFb
am7$$\"3IxykBx3xDFhhl$!3&******zDQ2T$Fbam7$$\"3u-L;bpWQEFhhl$!3:+++Lym
<OFbam7$$\"3iG(ym=1)*p#Fhhl$!35+++O8iIQFbam7$$\"31aT>=a;hFFhhl$!3!****
**zw)f\\SFbam7$$\"3&*z&4(\\Y_AGFhhl$!3;+++G,guUFbam7$$\"3Q0]A\")Q)Q)GF
hhl$!3<+++;ai0XFbam7$$\"3#3VSF6V_%HFhhl$!3.+++LYnUZFbam7$$\"3rceDWBg1I
Fhhl$!3u******yxu&)\\Fbam7$$\"39#Grddhz1$Fhhl$!3*)*****H&[%[B&Fbam7$$
\"3.3nG23KHJFhhl$!3#)*****\\&e'**[&Fbam7$$\"3YL@!)Q+o!>$Fhhl$!3G+++'y5
6v&Fbam7$$\"3OfvJq#R?D$Fhhl$!3z*****\\kz#=gFbam7$$\"3z%)H$=])R8LFhhl$!
3')*****HVs9H'Fbam7$$\"3m5%[LtdZP$Fhhl$!3S+++\\\"*oqlFbam7$$\"36OQ'['p
6OMFhhl$!3M*****RzHf&oFbam7$$\"3+i#zj>wu\\$Fhhl$!3=+++nV>ZrFbam7$$\"3W
(o%*yUN)eNFhhl$!3V******oG[WuFbam7$$\"3J8,TfY>?OFhhl$!3e+++*H&zZxFbam7
$$\"3uQb#4*Qb\"o$Fhhl$!39+++e;8d!)Fbam7$$\"3kk4WAJ\"Hu$Fhhl$!3;+++X>\\
s$)Fbam7$$\"32!RcRNsU!QFhhl$!3m+++gh(Qp)Fbam7$$\"3'f\"=Z&eJc'QFhhl$!3d
*****RI%G@!*Fbam7$$\"3RTs)p\"3*p#RFhhl$!3u*****pP;ZN*Fbam7$$\"3%om-&[+
N))RFhhl$!3Q+++yB<%p*Fbam7$$\"3G#4=+G4(\\SFhhl$!3'******4BlR+\"!#:7$$
\"3:=N`6&o56%Fhhl$!3$******ph:\"R5Ff]n7$$\"3/W*[IuFC<%Fhhl$!3-+++&Ro[2
\"Ff]n7$$\"3#*pVcupyLUFhhl$!3(******pcB76\"Ff]n7$$\"3#\\zzg?Y^H%Fhhl$!
31+++J6=[6Ff]n7$$\"3!3A&fPa]cVFhhl$!3/+++)3Td=\"Ff]n7$$\"3oY16pY'yT%Fh
hl$!3(******zV.RA\"Ff]n7$$\"3csgi+RAzWFhhl$!3%******4=oEE\"Ff]n7$$\"3c
(\\T@8$eSXFhhl$!38+++<`.-8Ff]n7$$\"3WBpljB%>g%Fhhl$!3()*****\\%[+U8Ff]
n7$$\"3K\\B<&f,Lm%Fhhl$!3!******pwwDQ\"Ff]n7$$\"3@vxoE3mCZFhhl$!31+++
\"3^PU\"Ff]n7$$\"3?+K?e+-'y%Fhhl$!3)******zyFbY\"Ff]n7$$\"33E'=(*Gzt%[
Fhhl$!3)******z)o!z]\"Ff]n7$$\"3(>0M7_Q(3\\Fhhl$!3/+++\"Q))3b\"Ff]n7$$
\"3'pZ\\Fv(4q\\Fhhl$!3#******fEsWf\"Ff]n7$$\"3%G!\\E%)pXJ]Fhhl$!37+++X
&e'Q;Ff]n7$$\"3sG.y:i\"G4&Fhhl$!39+++;sW$o\"Ff]n7$$\"3iadHZa<a^Fhhl$!3
%*******z#Q)G<Ff]n7$$\"3gz6\")yY`:_Fhhl$!33+++P<$[x\"Ff]n7$$\"3[0mK5R*
oF&Fhhl$!3-+++(eF9#=Ff]n7$$\"3PJ?%=9`#Q`Fhhl$!3++++Ieio=Ff]n7$$\"3CduN
tBh*R&Fhhl$!31+++mkU;>Ff]n7$$\"3C#)G([gr4Y&Fhhl$!3'******R\\H['>Ff]n7$
$\"383$)QO3LAbFhhl$!3\"******\\\"\\$Q,#Ff]n7$$\"3-MP!z1!p$e&Fhhl$!3#**
*****GFWj?Ff]n7$$\"3!*f\">%*H\\]k&Fhhl$!3,+++OHl8@Ff]n7$$\"3*[eM4`3kq&
Fhhl$!3))*****f`lW;#Ff]n7$$\"3y5+XixwndFhhl$!33+++H0)e@#Ff]n7$$\"3lOa'
R*p7HeFhhl$!37+++9z*yE#Ff]n7$$\"3kh3[Di[!*eFhhl$!3*)*****Hp<0K#Ff]n7$$
\"3a(G'*pXX=&fFhhl$!31+++k)RPP#Ff]n7$$\"3T8<^)o/K,'Fhhl$!3-+++GWcFCFf]
n7$$\"3IRr-?RcugFhhl$!3/+++&Q\"*>[#Ff]n7$$\"3HkDa^J#f8'Fhhl$!37+++N2-P
DFf]n7$$\"3;!*z0$Q#G(>'Fhhl$!3u*****pZ_Ef#Ff]n7$$\"31;Md9;keiFhhl$!3C+
++8m))[EFf]n7$$\"3%>%))3Y3+?jFhhl$!3G+++TJs0FFf]n7$$\"3%pE/w2g8Q'Fhhl$
!35+++i?;jFFf]n7$$\"3\"Gp>\"4$>FW'Fhhl$!3G+++wL?@GFf]n7$$\"3q=^jS&yS]'
Fhhl$!3A+++$3Z)zGFf]n7$$\"3eW0:sxVllFhhl$!3t*****>=$4RHFf]n7$$\"3epfm.
qzEmFhhl$!3!)*****\\nT*)*HFf]n7$$\"3Y&R\"=Ni:)o'Fhhl$!3++++gDRfIFf]n7$
$\"3M@opma^\\nFhhl$!3)******z$eW?JFf]n7$$\"3BZA@)pu3\"oFhhl$!3u******3
:5#=$Ff]n7$$\"3AswsHRBsoFhhl$!3%)*****HdfVC$Ff]n7$$\"35)4V7;$fLpFhhl$!
3s******H+A2LFf]n7$$\"3)R_eFR_\\*pFhhl$!3'*******zGoqLFf]n7$$\"3)*[RFC
;JcqFhhl$!3u*****>7[ZV$Ff]n7$$\"3'[P*yb3n<rFhhl$!3')*****pv:%*\\$Ff]n7
$$\"3u+[I(3I!zrFhhl$!3y*****\\y&okNFf]n7$$\"3jE-#)=$*QSsFhhl$!3/+++1#e
0j$Ff]n7$$\"3i^cL]&[<I(Fhhl$!33+++?I.(p$Ff]n7$$\"3]x5&=y2JO(Fhhl$!3!**
****pA5Tw$Ff]n7$$\"3R.lO8qYCuFhhl$!3$)*****f#)*yJQFf]n7$$\"3GH>)[CEe[(
Fhhl$!3y******==2+RFf]n7$$\"3EatRwa=ZvFhhl$!3%)*****R?c*oRFf]n7$$\"3:!
y7zqW&3wFhhl$!3E+++#)HWQSFf]n7$$\"3-1#G%RR!*pwFhhl$!3))*****H:K&3TFf]n
7$$\"3#>jV4<j7t(Fhhl$!3')*****prB#zTFf]n7$$\"3!p0fCSAEz(Fhhl$!3%******
Hn<0D%Ff]n-F\\hl6&F^hlF(F($\"*++++\"F^t-%+AXESLABELSG6%Q!6\"F_ao-%%FON
TG6#%(DEFAULTG-%%VIEWG6$;F($\"\"%F);$!#DF)$\"\"\"F)" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 40.000000 44.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 264 11 "Exercise 7:" }}{PARA 0 "" 0 "" {TEXT -1 
112 "Compare the analytic and numerical FT for different choices of pa
rameteres, particularly for smaller grid sizes." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 11 "Exercise 8:" }}{PARA 0 "
" 0 "" {TEXT -1 117 "Carry out the comparison between analytic and dis
crete FT for a wavepacket of a different shape, namely a Lorentzian." 
}}{PARA 0 "" 0 "" {TEXT 19 30 "exp(I*p0*x)*a^2/(a^2+(x-x0)^2)" }{TEXT 
-1 56 ". Why are the results less accurate than for a Gaussian?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "74 0 0" 66 }
{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }