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:Z:j;<Z:>:1:" }{TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT 326 32 "Module s
ur les s\351ries de Fourier" }}{PARA 18 "" 0 "" {TEXT 327 39 "\251 App
rendre Maple. Alain Le Stang. 2006" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT 303 1 "f" }{TEXT -1 82 " est une fonction p\351r
iodique de classe C1 par morceaux sur IR, \340 valeurs dans IR.." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 304 55 "Param\350tres formels pass\351s \340 la proc\351dure Ser
ie_Fourier:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 305 1 "T" }{TEXT 
-1 14 " : p\351riode de " }{TEXT 325 1 "f" }{TEXT -1 4 ". \n " }{TEXT 
310 11 "subdivision" }{TEXT -1 33 " : subdivision de l'intervalle [-" 
}{TEXT 306 1 "T" }{TEXT -1 5 "/2 , " }{TEXT 307 1 "T" }{TEXT -1 6 "/2]
.\n " }{TEXT 311 14 "liste_fonction" }{TEXT -1 17 " : d\351finition de
 " }{TEXT 308 1 "f" }{TEXT -1 59 " sur chacun des intervalles de la su
bdivision pr\351c\351dente.\n " }{TEXT 309 1 "t" }{TEXT -1 22 " : nom \+
de la variable." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT 263 65 "Variables locales export\351es par le module cr\351
\351 par Serie_Fourier:" }{TEXT -1 2 " \n" }{TEXT 271 7 "periode" }
{TEXT -1 2 ": " }{TEXT 264 1 "T" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT 272 9 "pulsation" }{TEXT -1 3 ":  " }{XPPEDIT 18 0 "omega = 2*Pi
/T;" "6#/%&omegaG*(\"\"#\"\"\"%#PiGF'%\"TG!\"\"" }{TEXT -1 4 " . \n" }
{TEXT 266 1 "a" }{TEXT -1 1 "(" }{TEXT 275 1 "n" }{TEXT -1 3 "), " }
{TEXT 267 1 "b" }{TEXT -1 1 "(" }{TEXT 276 1 "n" }{TEXT -1 33 "): coef
ficients de Fourier r\351els " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG
" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 
-1 4 " de " }{TEXT 265 1 "f" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" 
{TEXT 321 1 "c" }{TEXT -1 1 "(" }{TEXT 322 1 "n" }{TEXT -1 41 "): coef
ficients de Fourier exponentiels  " }{XPPEDIT 18 0 "c[n];" "6#&%\"cG6#
%\"nG" }{TEXT -1 4 " de " }{TEXT 320 1 "f" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT 268 11 "Harmonique " }{TEXT -1 1 "(" }{TEXT 269 5 "n :: \+
" }{TEXT -1 21 "nonnegint) : rend le " }{TEXT 270 1 "n" }{TEXT -1 18 "
-i\350me harmonique  " }{XPPEDIT 18 0 "proc (t) options operator, arro
w; a[0]+Sum(a[k]*cos(k*omega*t)+b[k]*sin(k*omega*t),k = 1 .. n) end pr
oc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\",&&%\"aG6#\"\"!\"\"\"-%$Su
mG6$,&*&&F-6#%\"kGF0-%$cosG6#*(F8F0%&omegaGF0F%F0F0F0*&&%\"bG6#F8F0-%$
sinG6#*(F8F0F=F0F%F0F0F0/F8;F0%\"nGF0F*F*F*" }{TEXT -1 2 ".\n" }{TEXT 
273 6 "graphe" }{TEXT -1 1 "(" }{TEXT 274 7 "Nmin ::" }{TEXT -1 12 " n
onnegint, " }{TEXT 277 7 "Nmax ::" }{TEXT -1 12 " nonnegint, " }{TEXT 
278 1 "a" }{TEXT -1 2 ", " }{TEXT 279 1 "b" }{TEXT -1 4 ") : " }}
{PARA 0 "" 0 "" {TEXT -1 3 "   " }{TEXT 280 4 "Nmin" }{TEXT -1 2 ", " 
}{TEXT 281 4 "Nmax" }{TEXT -1 59 "  : ordre minimum et maximum des har
moniques souhait\351s.\n   " }{TEXT 282 1 "a" }{TEXT -1 1 "," }{TEXT 
283 1 "b" }{TEXT -1 64 " : entier minimum et maximum( ... la valeur -1
 correspond \340 [-3*" }{TEXT 284 1 "T" }{TEXT -1 6 "/2 , -" }{TEXT 
285 1 "T" }{TEXT -1 33 "/2] , la valeur 0 correspond \340 [-" }{TEXT 
286 1 "T" }{TEXT -1 5 "/2 , " }{TEXT 287 1 "T" }{TEXT -1 4 "/2]," }}
{PARA 0 "" 0 "" {TEXT -1 37 "           la valeur 1 correspond \340 [
" }{TEXT 288 1 "T" }{TEXT -1 7 "/2 , 3*" }{TEXT 289 1 "T" }{TEXT -1 
30 "/2] etc ...) \n   La proc\351dure " }{TEXT 299 6 "graphe" }{TEXT 
-1 11 " repr\351sente" }{TEXT 290 2 " f" }{TEXT -1 49 " et les harmoni
ques correspondant aux entiers de " }{TEXT 291 4 "Nmin" }{TEXT -1 3 " \+
\340 " }{TEXT 292 4 "Nmax" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 
22 "   sur l'intervalle [-" }{TEXT 293 1 "T" }{TEXT -1 3 "/2+" }{TEXT 
294 1 "a" }{TEXT -1 1 "*" }{TEXT 295 1 "T" }{TEXT -1 3 " , " }{TEXT 
298 1 "T" }{TEXT -1 3 "/2+" }{TEXT 296 1 "b" }{TEXT -1 1 "*" }{TEXT 
297 1 "T" }{TEXT -1 2 "]\n" }{TEXT 300 6 "serieR" }{TEXT -1 54 "() : r
end la s\351rie de Fourier (coefficients r\351els) de " }{TEXT 301 1 "
f" }{TEXT -1 16 " , d\351finie par  " }{XPPEDIT 18 0 "proc (t) options
 operator, arrow; a[0]+Sum(a[k]*cos(k*omega*t)+b[k]*sin(k*omega*t),k =
 1 .. infinity) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\",&&%
\"aG6#\"\"!\"\"\"-%$SumG6$,&*&&F-6#%\"kGF0-%$cosG6#*(F8F0%&omegaGF0F%F
0F0F0*&&%\"bG6#F8F0-%$sinG6#*(F8F0F=F0F%F0F0F0/F8;F0%)infinityGF0F*F*F
*" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT 323 6 "serieC" }{TEXT -1 
58 "() : rend la s\351rie de Fourier (coefficients complexes) de " }
{TEXT 324 1 "f" }{TEXT -1 16 " , d\351finie par  " }{XPPEDIT 18 0 "pro
c (t) options operator, arrow; Sum(c[k]*exp(I*k*omega*t),k = -infinity
 .. infinity) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"-%$Sum
G6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#**%\"IGF3F2F3%&omegaGF3F%F3F3/F2;,$%)
infinityG!\"\"F=F*F*F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT 302 
8 "Parseval" }{TEXT -1 48 "() : rend l'identit\351 de Parseval , d\351
finie par  " }{XPPEDIT 18 0 "1/T;" "6#*&\"\"\"F$%\"TG!\"\"" }{XPPEDIT 
18 0 "Int(f(t)^2,t = 0 .. T) = a[0]^2;" "6#/-%$IntG6$*$-%\"fG6#%\"tG\"
\"#/F+;\"\"!%\"TG*$&%\"aG6#F/F," }{TEXT -1 3 " + " }{XPPEDIT 18 0 "1/2
;" "6#*&\"\"\"F$\"\"#!\"\"" }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k = 1 ..
 infinity)" "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/F
+;F-%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT 312 49 "Proc\351dures locales de la proc\351dure Serie_Four
ier:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 9 "i
ntegrale" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 
40 ") : calcule l'int\351grale de la fonction  " }{XPPEDIT 18 0 "proc \+
(t) options operator, arrow; f(t)*phi(t) end proc;" "6#f*6#%\"tG7\"6$%
)operatorG%&arrowG6\"*&-%\"fG6#F%\"\"\"-%$phiG6#F%F/F*F*F*" }{TEXT -1 
8 " sur [0," }{TEXT 314 1 "T" }{TEXT -1 5 "] .  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 3 "def" }{TEXT -1 16 " : d
\351finition de" }{TEXT 319 3 "  f" }{TEXT -1 55 "  en une seule formu
le \340 partir des  intervalles de la " }{TEXT 318 11 "subdivision" }
{TEXT -1 6 " de [-" }{TEXT 316 1 "T" }{TEXT -1 5 "/2 , " }{TEXT 317 1 
"T" }{TEXT -1 3 "/2]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3766 "Serie_Fourier:=proc(T,subdivision::list,list
e_fonction::list,t::name)\n module()\n        description \"Ce module \+
comporte des outils sur les s\351ries de Fourier\";\n        option `C
opyright (c) Octobre 2001  A.Le Stang`;\n        export periode, pulsa
tion, a, b, c,\n               harmonique, graphe, serieR, serieC, Par
seval;\n        local p,q,integrale,def;\n        periode := T;\n     \+
   pulsation := 2*Pi/T;\n        p:=nops(subdivision);q:=nops(liste_fo
nction);\n        if q<>p-1 or subdivision[1]<>-T/2 or subdivision[p]<
>T/2\n        then error \"Param\350tres non conformes\" end if;\n\n\n
        integrale:=proc(phi)\n          local i,s; \n          s:=0;\n
          for i to nops(liste_fonction) do \ns:=s+int(liste_fonction[i
](t)*phi(t),t=subdivision[i]..subdivision[i+1]);\n          end do;\n \+
         s;\n        end proc;\n\n        a:=proc(n)\n         local i
,cond,T,w;\n           T:=periode;\n           w:=pulsation;        \n
           i:=1/T*integrale(unapply(cos(n*w*t),t));\n           if n=0
 then return(i) end if;\n           if type(n,name) then \n           \+
cond:=cos(n*Pi)=(-1)^n,sin(n*Pi)=0,((-1)^n)^2=1,cos(n*Pi/2)=0:\n      \+
     factor(subs(cond,2*i));\n           else 2*i end if;   \n        \+
end proc;             \n\n        b:=proc(n)\n         local i,cond,T,
w;\n           T:=periode;\n           w:=pulsation;        \n        \+
   i:=2/T*integrale(unapply(sin(n*w*t),t)); \n           if type(n,nam
e) then\n           cond:=cos(n*Pi)=(-1)^n,sin(n*Pi)=0,((-1)^n)^2=1,co
s(n*Pi/2)=0:\n           factor(subs(cond,i));\n           else i end \+
if;  \n        end proc;\n\n       c:=proc(n)\n          local i,cond,
T,w;\n           T:=periode;\n           w:=pulsation;        \n      \+
     i:=1/T*integrale(unapply(exp(-I*n*w*t),t));\n           if n=0 th
en return(i) end if;\n           if type(n,name) then         cond:=ex
p(I*n*Pi)=(-1)^n,exp(-I*Pi*n)=(-1)^n,exp(2*I*n*Pi)=1,((-1)^n)^2=1:    \+
       factor(subs(cond,i));\n           else i end if;  \n       end \+
proc;\n  \n       harmonique:=proc(n::nonnegint)\n         local sf,k,
w;\n          w:=pulsation;\n          sf:=array(0..n);\n          sf[
0]:=unapply(a(0),t);\n          if n=0 then return sf[0] end if;\n    \+
      for k from 1 to n do    \n            sf[k]:=unapply(sf[k-1](t)+
a(k)*cos(k*w*t)+b(k)*sin(k*w*t),t);\n          end do;\n       end pro
c;\n\n       def:=proc()                                              \+
              local i,s;\n         s:=0;\n         for i to nops(liste
_fonction) do \ns:=s+(Heaviside(t-subdivision[i])-Heaviside(t-subdivis
ion[i+1]))*liste_fonction[i](t)\n         end do;\n         s:=unapply
(s,t); \n       end proc;\n\n      graphe:=proc(Nmin::nonnegint,Nmax::
nonnegint,a, b)\n       local i,s,F,CF,CH,T;\n        if a>b or Nmin>N
max then error \"Param\350tres non conformes\" end if;\n        T:=per
iode;\n        F:=def();\n        s:=0;\n        for i from a to b do
\n          s:=s+F(t-i*T)   \n        end do;\n        F:=unapply(s,t)
;        \n        CF:=plot(F(t),t=-T/2+a*T..T/2+b*T,thickness=2,color
=black):                                CH:=plot(\{seq(harmonique(i)(t
),i=Nmin..Nmax)\},t=-T/2+a*T..T/2+b*T):\n        plots[display](\{CF,C
H\});\n      end proc;\n\n      serieR:=proc()\n       local n,w;\n   \+
     w:=pulsation;        unapply(a(0)+Sum(a(n)*cos(n*w*t)+b(n)*sin(n*
w*t),n=1..infinity),t);\n      end proc; \n      \n      serieC:=proc(
)\n       local n,w;\n        w:=pulsation;                           \+
unapply(Sum(c(n)*exp(I*n*w*t),n=-infinity..infinity),t);\n      end pr
oc; \n \n      Parseval:=proc()\n       local i,k,L,T,n;\n        T:=p
eriode;\n        L:=map(t->t^2,liste_fonction);\n        i:=0;\n      \+
  for k to nops(L) do\n          i:=i+int(L[k](t),t=subdivision[k]..su
bdivision[k+1]);\n        end do;\n        1/T*i=a(0)^2+1/2*Sum(a(n)^2
+b(n)^2,n=1..infinity);\n      end proc;\n  \n     end module;\nend pr
oc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%.Serie_FourierGf*6&%\"TG'%,su
bdivisionG%%listG'%/liste_fonctionGF*'%\"tG%%nameG6\"F0F0b6#%+thismodu
leG6&%\"pG%\"qG%*integraleG%$defG6#%GCopyright~(c)~Octobre~2001~~A.Le~
StangG6,%(periodeG%*pulsationG%\"aG%\"bG%\"cG%+harmoniqueG%'grapheG%'s
erieRG%'serieCG%)ParsevalGC1>8(T$>8),$*&%#PiG\"\"\"FI!\"\"\"\"#>8$-%%n
opsG6#T&>8%-FU6#T(@$550FY,&FSFOFOFP0&FW6#FO,$FI#FPFQ0&FW6#FS,$FI#FOFQY
Q9Param|cytres~non~conformesF0>8&f*6#%$phiG6$%\"iG%\"sGF0F0C%>FY\"\"!?
(FSFOFO-FU6#FI%%trueG>FY,&FYFO-%$intG6$*&-&FIFcoFVFO-9$FVFO/FW;&FfnFco
&Ffn6#,&FSFOFOFOFOFYF0F06(F,FfnF.T*F)FW>8*f*6#%\"nG6&F^p%%condGF'%\"wG
F0F0C'>FioT#>8'T%>FS*&FioFP-T)6#-%(unapplyG6$-%$cosG6#*(F`qFOFerFOFfnF
OFfnFO@$/F`qFbpOFS@%-%%typeG6$F`qF/C$>FY6&/-F`s6#*&F`qFOFNFO)FPF`q/-%$
sinGF_tFbp/*$)FatFQFOFO/-F`s6#,$F`tFeoFbp-%'factorG6#-%%subsG6$FY,$FSF
QFbuF0F06*F<FHF=FKF.FhqF7Fio>8+f*F\\rF^rF0F0C&>FioFcr>FerFfr>FS,$*&Fio
FP-Fjr6#-F]s6$-FdtFasFfnFOFQ@%FgsC$>FYF\\t-F]u6#-F`u6$FYFSFSF0F0Fcu>8,
f*F\\rF^rF0F0C'>FioFcr>FerFfr>FS*&FioFP-Fjr6#-F]s6$-%$expG6#**^#FPFOF`
qFOFerFOFfnFOFfnFO@$FdsOFS@%FgsC$>FY6&/-Ffw6#*(^#FOFOF`qFOFNFOFat/-Ffw
6#*(FiwFOF`qFOFNFOFat/-Ffw6#*(^#FQFOF`qFOFNFOFbpFetFevFSF0F0Fcu>8-f*6#
'F]r%*nonnegintG6%%#sfG%\"kGF`rF0F0C'>FioFcr>FS-%&arrayG6#;FbpF`q>&FS6
#Fbp-F]s6$-FfrF`zFfn@$FdsOF_z?(FYFOFOF`qFfp>&FS6#FY-F]s6$,(-&FS6#,&FYF
OFOFPFenFO*&-FfrFizFO-F`s6#*(FYFOFioFOFfnFOFOFO*&-FjrFizFO-FdtFd[lFOFO
FfnF0F06*F=FKF>FjqF.FhqF?Feu>Ferf*F0F]pF0F0C%>FYFbp?(FSFOFOFdpFfp>FY,&
FYFO*&,&-%*HeavisideG6#,&FWFOFcqFPFO-Fd\\l6#,&FWFOFdqFPFPFOF]qFOFO>FY-
F]s6$FYFWF0F0Fgq>8.f*6&'%%NminGFcy'%%NmaxGFcyF>F?6(F^pF_p%\"FG%#CFG%#C
HGF'F0F0C+@$529'9&29%F`qYFgo>FKFcr>Fio-FfrF0>FYFbp?(FSF^^lFOF]^lFfp>FY
,&FYFO-Fio6#,&FfnFO*&FSFOFKFOFPFO>Fio-F]s6$FYFfn>Fer-%%plotG6&-FioFen/
Ffn;,&FKF`o*&F^^lFOFKFOFO,&FKFeo*&F]^lFOFKFOFO/%*thicknessGFQ/%&colorG
%&blackG>FH-Fb_l6$<#-%$seqG6$--FjrFcoFen/FS;F`qF`^lFe_l-&%&plotsG6#%(d
isplayG6#<$FerFHF0F06*F<FHF8FerF.FhqFAF_y>8/f*F06$F]rF`rF0F0C$>FYFcr-F
]s6$,&FczFO-%$SumG6$,&*&-FfrFcoFO-F`s6#*(FSFOFYFOFfnFOFOFO*&Fh`lFO-Fdt
FcblFOFO/FS;FO%)infinityGFOFfnF0F0Fi[l>80f*F0FfalF0F0C$>FYFcr-F]s6$-F]
bl6$*&FablFO-Ffw6#**FdxFOFSFOFYFOFfnFOFO/FS;,$FiblFPFiblFfnF0F06(F=FKF
@FjvF.Fhq>81f*F06'F^pFfy%\"LGF'F]rF0F0C'>FerFcr>Fio-%$mapG6$f*6#F.F06$
%)operatorG%&arrowGF0*$)F`qFQFOF0F0F0FW>FSFbp?(FYFOFO-FU6#FioFfp>FS,&F
SFO-Fjp6$-&FioFizFen/Ffn;&FhqFiz&Fhq6#,&FYFOFOFOFO/*&FerFPFSFO,&*$)-T+
F`zFQFOFO*&FeoFO-F]bl6$,&*$)-Fcfl6#FHFQFOFO*$)-T-F[glFQFOFO/FHFhblFOFO
F0F06.F<FHF,FfnF.FhqF)FWF>FjqF?Feu6#QXCe~module~comporte~des~outils~su
r~les~s|dyries~de~FourierF0F06*F'F`qF)F`^lF,F^^lF.F]^lF0F0F0F0" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 16 "P
remier exemple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Sf:=Seri
e_Fourier(2*Pi,[-Pi,0,Pi],[t->-1,t->1],t):\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "Sf:-periode;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,$%#PiG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Sf:-pulsati
on;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "Sf:-a(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Sf:-a(n);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "Sf:-b(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%#PiG!\"\",&F&\"\"
\")F&%\"nGF(F(F*F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Sf
:-c(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%#PiG!\"\",&F%\"\"\")F%%
\"nGF'F'F)F%^#F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Sf:-h
armonique(9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"tG6\"6$%)opera
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#,$F1F6F2F2*(#F3\"\"&F2F,F--F/6#,$F1F<F2F2*(#F3\"\"(F2F,F--F/6#,$F1FBF
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{MPLTEXT 1 0 21 "Sf:-graphe(1,9,-1,1);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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*7$$\"39;Mm#*QzbYF*$!3')R\")H$*4?r7F*7$F[[l$!3K*=Zq)f=t7F*7$$\"3]=Yy6u
y_ZF*$!3/g+R,2?s7F*7$Fbhz$!3%>xuo%**3o7F*7$$\"3#o:KiiG>&[F*$!31tWcAQ'3
E\"F*7$Fh]p$!3_T?=\"3S0D\"F*7$F_iz$!32l=,Y(32A\"F*7$F^[l$!3q3u)G\"*)))
y6F*7$F__p$!3c^]I&GWT1\"F*7$Fa[l$!3)*3;.&)\\L&4*F^t7$Fd[l$!343@g+Nv7rF
^t7$Fg[l$!3Gh#e*p#4#Q[F^t7$F]\\l$!3yaiVDUkeCF^t7$Fe]l$\"3]a,Y)o.fX\"Fh
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*7$Fei[l$\"3(*[jK6Pf97F*7$Fbjp$\"39$y,%>EyZ7F*7$$\"3/InA&[req(F*$\"3g4
Gur(*Hf7F*7$Fbj[l$\"3w\"o8v9(Qn7F*7$$\"3Yi1W!ye-\"yF*$\"3Y^NF?F-s7F*7$
Fd^l$\"3gD*oZ'Q>t7F*7$$\"3A!Hs'[]84zF*$\"3yHSV-NIr7F*7$Fjj[l$\"3I#)eML
Jkm7F*7$$\"3b6;#**G+D+)F*$\"3)p2\\=\"HAf7F*7$Fjjp$\"3U&\\^c+f!\\7F*7$F
g[\\l$\"3HbAT_Yf?7F*7$Fg^l$\"3C6q9>!)\\\"=\"F*7$Fa\\q$\"3)>@\\YF,V1\"F
*7$Fj^l$\"3;Q%QKGm00*F^t7$Fc]q$\"3#3V/]_/0>(F^t7$F]_l$\"31!H=IuF,2&F^t
7$Fi_q$\"37r;#*zz!)\\QF^t7$Fc`q$\"3?WqZWH;*e#F^t7$F]aq$\"3Bj,I8uR,8F^t
7$F`_l$\"35dK1(Q/()R#F``l-Fc_l6&Fe_lF[bqF]bqF[bq-%+AXESLABELSG6%Q\"t6
\"Q!Fad^l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+izxC%*!\"*$\"+izxC%*F]e^lFf
d^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "Sf:-serieR();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#f*6#%\"tG6\"6$%)operatorG%&arrowGF&-%$SumG6$,$**%#PiG!
\"\",&F0\"\"\")F0%\"nGF2F2F4F0-%$sinG6#*&F4F29$F2F2!\"#/F4;F2%)infinit
yGF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Sf:-serieC();" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"tG6\"6$%)operatorG%&arrowGF&-
%$SumG6$*,%#PiG!\"\",&F/\"\"\")F/%\"nGF1F1F3F/-%$expG6#*(F3F19$F1^#F1F
1F1F9F1/F3;,$%)infinityGF/F=F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "Sf:-Parseval();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
\"\"\",$-%$SumG6$,$*(%#PiG!\"#,&!\"\"F$)F.%\"nGF$\"\"#F0F,\"\"%/F0;F$%
)infinityG#F$F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT 257 17 "Deuxi\350me exemple:" }{TEXT -1 2 "  " }{XPPEDIT 
18 0 "alpha;" "6#%&alphaG" }{TEXT -1 24 " est un r\351el non entier." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Sf:=Serie_Fourier(2*Pi,[-
Pi,Pi],[t->cos(alpha*t)],t):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "Sf:-periode;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"\"#" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Sf:-pulsation;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "Sf:-a(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%#PiG!
\"\"-%$sinG6#*&F$\"\"\"%&alphaGF*F*F+F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "Sf:-a(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.%#PiG
!\"\"%&alphaG\"\"\"-%$sinG6#*&F%F(F'F(F()F&%\"nGF(,&F'F&F.F(F&,&F'F(F.
F(F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Sf:-b(n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "Sf:-c(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.%#PiG
!\"\"%&alphaG\"\"\"-%$sinG6#*&F%F(F'F(F()F&%\"nGF(,&F'F&F.F(F&,&F'F(F.
F(F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Sf:-harmonique(3)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"tG6\"6$%)operatorG%&arrowG
F&,**(%#PiG!\"\"-%$sinG6#*&F,\"\"\"%&alphaGF2F2F3F-F2*0\"\"#F2F,F-,&F3
F2F2F-F-,&F3F2F2F2F-F3F2F.F2-%$cosG6#9$F2F-*0F5F2F,F-,&F3F2F5F-F-,&F3F
2F5F2F-F3F2F.F2-F96#,$F;F5F2F2*0F5F2F,F-,&F3F2\"\"$F-F-,&F3F2FDF2F-F3F
2F.F2-F96#,$F;FDF2F-F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "La \+
s\351rie de Fourier de " }{TEXT 258 2 "f " }{TEXT -1 14 "co\357ncide a
vec " }{TEXT 259 1 "f" }{TEXT -1 6 " sur [" }{XPPEDIT 18 0 "-Pi,Pi;" "
6$,$%#PiG!\"\"F$" }{TEXT -1 4 "] : " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "cos(alpha*t) = Sf:-serieR()(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$cosG6#*&%&alphaG\"\"\"%\"tGF),&*(%#PiG!\"\"-%$sinG6
#*&F-F)F(F)F)F(F.F)-%$SumG6$,$*0F-F.F(F)F/F))F.%\"nGF),&F(F)F9F.F.,&F(
F)F9F)F.-F%6#*&F9F)F*F)F)\"\"#/F9;F)%)infinityGF)" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "Sf:-serieC()(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$SumG6$*0)!\"\"%\"nG\"\"\"%&alphaGF*-%$sinG6#*&%#PiGF
*F+F*F*F0F(,&F+F*F)F(F(,&F+F*F)F*F(-%$expG6#*(F)F*%\"tGF*^#F*F*F*/F);,
$%)infinityGF(F<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
" 0 "" {TEXT 260 18 "Troisi\350me exemple:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 69 "Sf:=Serie_Fourier(2*Pi,[-Pi,0,Pi],[t->-t*(2*Pi+t),t
->t*(2*Pi-t)],t):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Sf:-
periode;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Sf:-pulsation;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Sf:
-a(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#PiG\"\"#\"\"\"#F'\"\"
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Sf:-a(n);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$)%\"nG\"\"#F%!\"\"!\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Sf:-b(n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Sf:-
c(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$)%\"nG\"\"#F%!
\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Sf:-harmonique(
5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"tG6\"6$%)operatorG%&arro
wGF&,.*$)%#PiG\"\"#\"\"\"#F.\"\"$*&\"\"%F/-%$cosG6#9$F/!\"\"-F56#,$F7F
.F8*&#F3\"\"*F/-F56#,$F7F1F/F8*&#F/F3F/-F56#,$F7F3F/F8*&#F3\"#DF/-F56#
,$F7\"\"&F/F8F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "La s\351ri
e de Fourier de " }{TEXT 261 2 "f " }{TEXT -1 14 "co\357ncide avec " }
{TEXT 262 1 "f" }{TEXT -1 6 " sur [" }{XPPEDIT 18 0 "0,Pi;" "6$\"\"!%#
PiG" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "t*(
2*Pi-t) = expand(Sf:-serieR()(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
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