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{SECT 0 {SECT 0 {PARA 257 "" 0 "" {TEXT -1 68 "CorrigŽ du TD 5 : SŽrie
s Enti�res, SŽries de Fourrier et IntŽgration" }}{PARA 256 "" 0 "" 
{TEXT -1 24 "Jean SŽbastien ROY, 1999" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 35 "Ex 1 : Minimisation d'une intŽgrale" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
22 "Le degrŽ du polyn™me :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "N:=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 33 "Un polyn™me unitaire de degrŽ N :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "P:=add(a.i*X^i,i=0..N-1)+X^N
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,,%#a0G\"\"\"*&%#a1GF'%\"XG
F'F'*&%#a2GF')F*\"\"#\"\"\"F'*&%#a3GF')F*\"\"$F/F'*$)F*\"\"%F/F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "On int�gre (pas la peine de prendr
e la racine carrŽe pour minimiser) :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "E:=int(P*P,X=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"EG,@#\"\"\"\"\"*F'%#a3G#F'\"\"%%#a2G#\"\"#\"\"(*$)F)F.\"\"\"#F'F
/%#a1G#F'\"\"$*&F,F'F)F'F5%#a0G#F.\"\"&*&F4F'F)F2F9*$)F,F.F2#F'F:*&F8F
'F)F2#F'F.*&F4F2F,F2F@*&F8F2F,F2#F.F6*$)F4F.F2F5*&F8F2F4F2F'*$)F8F.F2F
'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "On rŽsoud le syst�me formŽ d
e l'ensemble des dŽrivŽes partielles de E :" }}{PARA 0 "" 0 "" {TEXT 
-1 79 "(Si il existe un polyn™me qui minimise E, alors ce polyn™me vŽr
ifie ce syst�me)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "S:=\{se
q(diff(E,a.i),i=0..3)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG<&,,
#\"\"#\"\"&\"\"\"%#a3G#F*F(%#a2G#F(\"\"$%#a1GF*%#a0GF(,,#F*F/F*F+F'F-F
,F0F.F1F*,,#F(\"\"(F*F+F3F-F'F0F,F1F.,,#F*\"\"%F*F+F5F-F3F0F'F1F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sol:=solve(S);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$solG<&/%#a0G#\"\"\"\"#q/%#a1G#!\"#\"\"(/%#a2G
#\"\"*F//%#a3GF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Il n'y a qu'u
ne solution." }}{PARA 0 "" 0 "" {TEXT -1 80 "Calculons la hessienne au
 point pour dŽterminer si c'est bien un minimum local :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "H:=matrix(N,N,(i,j)->diff(E,a.(i-1)
,a.(j-1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG-%'matrixG6#7&7&
\"\"#\"\"\"#F*\"\"$#F+F*7&F+F,F.#F*\"\"&7&F,F.F0#F+F-7&F.F0F3#F*\"\"(
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Assez logiquement, H ne dŽpen
d pas du point. Est-elle dŽfinie positive ?" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warn
ing, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, \+
new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"definite(H,'positive_def');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "On peut vŽrifier le rŽsultat
 ainsi :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(allvalues
(eigenvalues" }{TEXT -1 0 "" }{MPLTEXT 1 0 6 "(H)));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6&,&$\"+g&G/+$!\"*\"\"\"%\"IG$\"+'3c*=6!#?,&$\"*SCGQ$F
&F'F($!+&>a2g\"!#@,&$\"+fAlZ8!#6F'F($\"+^$p)4))!#>,&$\"*6%HM>!#7F'F($!
+<uv0*)F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Donc le polyn™me min
imisant l'intŽgrale :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "su
bs(sol,P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,#\"\"\"\"#qF%%\"XG#!\"
#\"\"(*$)F'\"\"#\"\"\"#\"\"*F**$)F'\"\"$F.F)*$)F'\"\"%F.F%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "Et la valeur cherchŽe :" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "sqrt(subs(sol,E));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"\"\"\"$5#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "On
 peut aussi utiliser la fonction " }{TEXT 258 8 "minimize" }{TEXT -1 
26 " ˆ titre de vŽrification :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "minimize(sqrt(E));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"
\"$5#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 60 "Ex 2 : Calcul d'intŽgr
ale par intŽgration par parties et DSE" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"f:=t->ln(t)*ln(1-t)/t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%
\"tG6\"6$%)operatorG%&arrowGF(*&*&-%#lnG6#9$\"\"\"-F/6#,&F2F2F1!\"\"F2
\"\"\"F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A:=i
nt(f(t),t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%%ZetaG6#\"
\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Bon. Maple y arrive tout s
eul (parfois). Pour le forcer ˆ ne pas faire le calcul on utilise " }
{TEXT 256 3 "Int" }{TEXT -1 12 " au lieu de " }{TEXT 257 3 "int" }
{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A:=Int(f(
t),t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%$IntG6$*&*&-%#l
nG6#%\"tG\"\"\"-F+6#,&F.F.F-!\"\"F.\"\"\"F-!\"\"/F-;\"\"!F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "VŽrifions l'existence de l'intŽgra
le en 0 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(f(t),t=
0,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"tG,$-%#lnG6#F$!\"\"\"\"!
-%\"OG6#\"\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Ok. IntŽg
ration par parties :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "stu
dent[intparts](A,ln(1-t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$Int
G6$,$*&*$)-%#lnG6#%\"tG\"\"#\"\"\"F0,&\"\"\"F2F.!\"\"!\"\"#F3F//F.;\"
\"!F2F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Il reste donc ˆ calcul
er cette intŽgrale ce que l'on fait via un DSE de 1/(1-t) :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "S:=n->int(t^n*(ln(t))^2/2,t=
0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SGR6#%\"nG6\"6$%)operato
rG%&arrowGF(-%$intG6$,$*&)%\"tG9$\"\"\")-%#lnG6#F2\"\"#\"\"\"#F4F9/F2;
\"\"!F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "S(n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6%,$*&,0**%\"tG\"\"\")-%#lnG6
#F*\"\"#\"\"\"-%$expG6#*&%\"nGF+F-F+F+)F6F0F1F+**F*F1F-F1F2F1F6F1!\"#*
*F*F1F,F1F2F1F6F1F0*(F*F1F-F1F2F1F9F9F+*(F*F1F,F1F2F1F+*&F*F1F2F1F0F1*
&,&F6F+F+F+\"\"\",(F+F+F6F0*$F7F1F+\"\"\"!\"\"#!\"\"F0/F*\"\"!%&rightG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Stupide, Maple n'arrive pas \+
ˆ simplifier cette expression (meme avec des assume adŽquats), donc on
 l'aide :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "op(1,%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,0**%\"tG\"\"\")-%#lnG6#F'\"\"#\"
\"\"-%$expG6#*&%\"nGF(F*F(F()F3F-F.F(**F'F.F*F.F/F.F3F.!\"#**F'F.F)F.F
/F.F3F.F-*(F'F.F*F.F/F.F6F6F(*(F'F.F)F.F/F.F(*&F'F.F/F.F-F.*&,&F3F(F(F
(\"\"\",(F(F(F3F-*$F4F.F(\"\"\"!\"\"#!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 78 "Manifestement, seul le terme '-2' du numŽrateur ne tend
s pas vers zŽro, donc :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
subs(t=0,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&,&%\"nG\"
\"\"F(F(\"\"\",(F(F(F'\"\"#*$)F'F+F$F(\"\"\"!\"\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#*&\"\"\"F$*$),&%\"nG\"\"\"F)F)\"\"$F$!\"\"" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 31 "D'o� la valeur de l'intŽgrale :" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "S:=Sum(1/n^3,n=1..infinity):A=S;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&*&-%#lnG6#%\"tG\"\"\"-F*6#,&F-F-F,!
\"\"F-\"\"\"F,!\"\"/F,;\"\"!F--%$SumG6$*&F2F2*$)%\"nG\"\"$F2F3/F=;F-%)
infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "S=value(S);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*$)%\"nG\"\"$F(!
\"\"/F+;\"\"\"%)infinityG-%%ZetaG6#\"\"$" }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 48 "Ex 3 : DSE utilisant une Žquation diffŽrentielle" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart:with(powseries):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=x->(arcsin(x/2))^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowG
F(*$)-%'arcsinG6#,$9$#\"\"\"\"\"#F5\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 49 "Le package Powerseries ne traite pas la fonction " }
{TEXT 259 6 "arcsin" }{TEXT -1 16 ". On va dŽriver " }{TEXT 262 11 "ar
csin(x/2)" }{TEXT -1 44 ", calculer le DSE puis intŽgrer le rŽsultat.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g:=unapply(diff(arcsin(
x/2),x)):g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$-%%sqrtG
6#,&\"\"%\"\"\"*$)%\"xG\"\"#F$!\"\"F$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "On calcule le DSE de " }{TEXT 260 1 "g" }{TEXT -1 40 ", o
n l'int�gre, et on l'Žl�ve au carrŽ :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "evalpow(g(x)):powint(%):S:=evalpow(%^2):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "tpsform(%,x,10);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#+-%\"xG#\"\"\"\"\"%\"\"##F&\"#[\"\"%#F&\"$g$\"\"'#F&
\"%SA\"\")-%\"OG6#F&\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "VŽrif
ions ˆ partir d'un DL de " }{TEXT 261 1 "f" }{TEXT -1 2 " :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(f(x),x=0,9);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"xG#\"\"\"\"\"%\"\"##F&\"#[\"\"%#F
&\"$g$\"\"'#F&\"%SA\"\")-%\"OG6#F&\"#5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 72 "DŽterminons une Žquation diffŽrentielle. ˆ coefficents po
lyn™miaux dont " }{TEXT 263 1 "f" }{TEXT -1 15 " est solution :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eq1:=dy=diff(f(x),x):eq1;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#dyG,$*&-%'arcsinG6#,$%\"xG#\"\"\"
\"\"#\"\"\"*$-%%sqrtG6#,&\"\"%F-*$)F+F.F/!\"\"F/!\"\"F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eq2:=dy2=diff(f(x),x$2):eq2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$dy2G,&*&\"\"\"F',&\"\"%\"\"\"*$)%\"
xG\"\"#F'!\"\"!\"\"F.*&*&-%'arcsinG6#,$F-#F*F.F*F-F*F'*$)F(#\"\"$F.F'F
0F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "On isole " }{TEXT 264 11 "a
rcsin(x/2)" }{TEXT -1 29 " dans la premi�re Žquation..." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "student[isolate](eq1,arcsin(1/2*x))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arcsinG6#,$%\"xG#\"\"\"\"\"#,
$*&%#dyGF*-%%sqrtG6#,&\"\"%F**$)F(F+\"\"\"!\"\"F6F)" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 35 "Et l'on substitue dans la seconde :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(%,eq2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$dy2G,&*&\"\"\"F',&\"\"%\"\"\"*$)%\"xG\"\"#F'!\"\"!\"
\"F.*&*&%#dyGF*F-F*F'F(F0F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "La
 voilˆ. " }{TEXT 267 3 "dy2" }{TEXT -1 5 " est " }{TEXT 265 3 "f''" }
{TEXT -1 4 " et " }{TEXT 268 2 "dy" }{TEXT -1 5 " est " }{TEXT 266 2 "
f'" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eqy:=
collect(numer(rhs(%)-lhs(%)),dy2):eqy=0;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,(*&,&\"\"%\"\"\"*$)%\"xG\"\"#\"\"\"!\"\"F(%$dy2GF(F(!\"#F(*&%#
dyGF(F+F(F.\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 " On cherche a
lors une solution DSE :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "
dy:=sum(a(k)*x^k,k=0..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dyG-%$
sumG6$*&-%\"aG6#%\"kG\"\"\")%\"xGF,F-/F,;\"\"!%\"NG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "dy2:=diff(dy,x):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "ex:=simplify(combine(eqy));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#exG,&-%$sumG6$,(*(-%\"aG6#%\"kG\"\"\"F.F/)%\"xG,&F.F
/!\"\"F/F/\"\"%*(F+\"\"\"F.F6)F1,&F.F/F/F/F/F3*&F+F6F7F6F3/F.;\"\"!%\"
NGF/!\"#F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Les conditions init
iales :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "collect(expand(s
ubs(N=3,ex)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&-%\"aG6#\"\"$
\"\"\")%\"xG\"\"%\"\"\"!\"%*&-F&6#\"\"#F))F+F(F-!\"$*&,&-F&6#F)!\"#F%
\"#7F))F+F2F-F)*&,&-F&6#\"\"!!\"\"F0\"\")F)F+F)F)F7F,F9F)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 5 "Donc " }{TEXT 269 8 "a(1)=1/2" }{TEXT -1 
5 ", ..." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "A partir de cette ex
pression, on cherche ˆ obtenir une relation de rŽcurrence vŽrifiŽe par
 les coefficients du DSE." }}{PARA 0 "" 0 "" {TEXT -1 36 "Il faut un p
eu manipuler ˆ la main :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"rel[A]:=op([1,1],ex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$relG6#%
\"AG,(*(-%\"aG6#%\"kG\"\"\"F-F.)%\"xG,&F-F.!\"\"F.F.\"\"%*(F*\"\"\"F-F
5)F0,&F-F.F.F.F.F2*&F*F5F6F5F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 
"On s'arrange pour avoir les puissances de " }{TEXT 270 1 "x" }{TEXT 
-1 16 " toutes Žgales :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "
subs(k=k+2,op(1,rel[A]))+op(2,rel[A])+op(3,rel[A]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,(*(-%\"aG6#,&%\"kG\"\"\"\"\"#F*F*F(F*)%\"xG,&F)F*F*F
*F*\"\"%*(-F&6#F)F*F)F*F,\"\"\"!\"\"*&F1F3F,F3F4" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 28 "rel[B]:=simplify(%/x^(k+1));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%$relG6#%\"BG,**&-%\"aG6#,&%\"kG\"\"\"\"\"#F/F/F
.F/\"\"%F*\"\")*&-F+6#F.F/F.\"\"\"!\"\"F4F7" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 37 "On exprime la relation de rŽcurence :" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 38 "rel[C]:=student[isolate](rel[B],a(k));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$relG6#%\"CG/-%\"aG6#%\"kG*&,&*&-F
*6#,&F,\"\"\"\"\"#F3F3F,F3!\"%F0!\")\"\"\",&F,!\"\"F9F3!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Apr�s simplification et en remarq
uant que seuls les coefficients impairs peuvent �tre non nuls (par par
itŽ de " }{TEXT 274 1 "f" }{TEXT -1 7 " (donc " }{TEXT 275 2 "f'" }
{TEXT -1 26 " est impaire)), on change " }{TEXT 271 4 "a(k)" }{TEXT 
-1 4 " en " }{TEXT 272 4 "b(p)" }{TEXT -1 6 " avec " }{TEXT 273 7 "k=2
*p-1" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:
=k->b((k+1)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6#%\"kG6\"6$
%)operatorG%&arrowGF(-%\"bG6#,&9$#\"\"\"\"\"#F1F2F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "rel[D]:=eval(subs(k=2*p+1,rel[C]));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$relG6#%\"DG/-%\"bG6#,&%\"pG\"
\"\"F.F.*&,&*&-F*6#,&F-F.\"\"#F.F.,&F-F5F.F.F.!\"%F2!\")\"\"\",&F-!\"#
F;F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "On rŽsoud :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rb:=unapply(rsolve(\{rel[D],
b(1)=1/2\},b(p)),p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rbGR6#%\"pG
6\"6$%)operatorG%&arrowGF(*&*()\"\"%,$9$!\"\"\"\"\"-%&GAMMAG6#F1F3-%%s
qrtG6#%#PiG\"\"\"F;-F56#,&F1F3#F3\"\"#F3!\"\"F(F(F(" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 10 "Donc pour " }{TEXT 276 1 "k" }{TEXT -1 37 " imp
air le terme du developpement de " }{TEXT 277 2 "f'" }{TEXT -1 6 " est
 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(p=(k+1)/2,rb(p))
*x^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&**)\"\"%,&%\"kG#!\"\"\"\"#F
)\"\"\"F,-%&GAMMAG6#,&F(#F,F+F1F,F,-%%sqrtG6#%#PiG\"\"\")%\"xGF(F,F6-F
.6#,&F(F1F,F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "On int�gre \+
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "c:=unapply(subs(k=n-1,
int(%,x)),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGR6#%\"nG6\"6$%)
operatorG%&arrowGF(*&**)\"\"%,$9$#!\"\"\"\"#\"\"\"-%&GAMMAG6#,$F1#F5F4
F5-%%sqrtG6#%#PiG\"\"\")%\"xGF1F5F?*&-F76#,&F1F:F:F5\"\"\"F1\"\"\"!\"
\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Voilˆ le terme gŽnŽra
l du developpment :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c(2*n
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&**)\"\"%,$%\"nG!\"\"\"\"\"-%
&GAMMAG6#F)F+-%%sqrtG6#%#PiG\"\"\")%\"xG,$F)\"\"#F+F3*&-F-6#,&F)F+#F+F
7F+\"\"\"F)\"\"\"!\"\"F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Pas d
e chance, Maple ne sait pas simplifier plus. Comparons avec la solutio
n connue (notŽe " }{TEXT 278 1 "v" }{TEXT -1 17 " dans la suite) :" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "v:=n->1/2*(n-1)!^2/(2*n)!;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vGR6#%\"nG6\"6$%)operatorG%&ar
rowGF(,$*&*$)-%*factorialG6#,&9$\"\"\"!\"\"F5\"\"#\"\"\"F8-F16#,$F4F7!
\"\"#F5F7F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "C'est bon :" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(v(n)*x^(2*n)-c(2*
n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "sum(c(2*n),n=1..4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,**$)%\"xG\"\"#\"\"\"#\"\"\"\"\"%*$)F&F+F(#F*\"#[*$)F&
\"\"'F(#F*\"$g$*$)F&\"\")F(#F*\"%SA" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "series(f(x),x=0,9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#+-%\"xG#\"\"\"\"\"%\"\"##F&\"#[\"\"%#F&\"$g$\"\"'#F&\"%SA\"\")-%\"OG
6#F&\"#5" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Ex 4 : SŽrie de Fou
rier des polygones" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restar
t:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=4:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "On dŽfinit un polygone par ses som
mets :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "z:=[1,1+I,-1+I,-I
,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG7'\"\"\",&F&F&%\"IGF&,&!
\"\"F&F(F&,$F(F*F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Puis en fo
rmant une fonction affine par morceaux, continue, qui reprŽsente le di
t polygone sur 0..1 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "g:
=t->piecewise(seq(op([t<=j/n,n*(z[j]*(j/n-t)+z[j+1]*(t-(j-1)/n))]),j=1
..n)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(t);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6&7$,(\"\"\"F(%\"tG!\"%*&,&\"\"%F(
%\"IGF-F(F)F(F(1F)#F(F-7$,&*&F,\"\"\",&#F(\"\"#F(F)!\"\"F(F(*&,&F*F(F.
F-F(,&F)F(#F8F-F(F(F(1F)F67$,&*&F:F4,&#\"\"$F-F(F)F8F(F(*&F.F(,&F)F(#F
8F7F(F(F*1F)FB7$,(*&F.F4,&F(F(F)F8F(F*F)F-!\"$F(1F)F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "pf:=complexplot(g(t),t=0..1,color=b
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$%(DEFAULTGFjz" 1 2 0 1 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 }}}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Calculons le dŽveloppement en sŽr
ie de Fourier de g. Tout d'abord les coefficients :" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "c:=m->int(g(t)*exp(2*I*Pi*t*m),t=0..1);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGR6#%\"mG6\"6$%)operatorG%&arro
wGF(-%$intG6$*&-%\"gG6#%\"tG\"\"\"-%$expG6#,$**%\"IGF4%#PiGF4F3F49$F4
\"\"#F4/F3;\"\"!F4F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Puis \+
la somme partielle :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "s:=
(t,nn)->sum('c(m)*exp(2*I*Pi*t*m)',m=-nn..nn);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"sGR6$%\"tG%#nnG6\"6$%)operatorG%&arrowGF)-%$sumG6$.
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%!\"\"FCF)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Puis le graphe \+
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "p:=k->complexplot(s(t,
k),t=0..1,numpoints=400,color=COLOR(RGB,(6-k)/5,0,(k-1)/5));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"pGR6#%\"kG6\"6$%)operatorG%&arrowGF(-%,c
omplexplotG6&-%\"sG6$%\"tG9$/F2;\"\"!\"\"\"/%*numpointsG\"$+%/%&colorG
-%&COLORG6&%$RGBG,&#\"\"'\"\"&F7F3#!\"\"FDF6,&F3#F7FDFEF7F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Et on trace les 5 premi�res sommes
 partielles ainsi que le polygone :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "display(p(1),p(2),p(3),p(4),p(5),pf);" }}{PARA 13 "" 
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