{VERSION 4 0 "IBM INTEL LINUX" "4.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 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 "Text Outp ut" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Tim es" 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 "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{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 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 } {PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 30 "Fourier-sarjojen perusty \366arkki" }}{PARA 19 "" 0 "" {TEXT -1 13 "17.11.00 HA" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Apufunktioita: JJ, trigsiev, jana" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(plots):\n" }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "#read(\"/p/ed u/mat-1.414/maple/ohjelmat.mpl\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "#read(\"c:\\\\opetus\\\\maple\\\\ohjelmat.mpl\"):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "read(\"/home/apiola/opetus/ peruskurssi/v2-3/maple/ohjelmat.mpl\"):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "M\344\344ritell\344\344n uudet koodit varmuuden vuoksi my \366s t\344ss\344, vaikka ovatkin ohjelmat.mpl:ss\344." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "trigsiev:=proc(lauseke,n)\nlocal t ulos;\ntulos:=subs(sin(n*Pi)=0,lauseke);\ntulos:=subs(cos(n*Pi)=(-1)^n ,tulos);\ntulos:=subs((-1)^(2*n)=1,tulos);\nsimplify(tulos);\nend:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "jana:=(x,y)->plot([[x,0],[x ,y]],thickness=3,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "#display([seq(jana(n,abs(an)),n=1..6)],axes=box);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Fourier-kertoimet" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Reaalinen muoto, cos ja sin" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 95 "N\344m\344 on syyt\344 ottaa pohjaksi ty\366skentelyyn \+ ja olla valmis modifioimaan esim. integrointirajoja," }}{PARA 0 "" 0 " " {TEXT -1 93 "integrointia paloittain ym. (Kokemus opettaa v\344ltt \344m\344\344n liian pitk\344lle viety\344 automatiikkaa.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "f:='f': T:='T': omega:='omega': #om ega:=2*Pi/T:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "a0:=(2/T)* int(f(t),t=0..T):\nan:=(2/T)*int(f(t)*cos(n*omega*t),t=0..T):\nbn:=(2/ T)*int(f(t)*sin(n*omega*t),t=0..T):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "sarja:=a0/2+sum(an*cos(n*omega*t)+bn*sin(n*omega*t),n =1..infinity):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "osasumma: =N->a0/2+sum(an*cos(n*omega*t)+bn*sin(n*omega*t),n=1..N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 "Sinisarja ja kosinisarja" }}{PARA 0 "" 0 "" {TEXT -1 296 "J\344 ljemp\344n\344 esit\344mme parillisen ja parittoman laajennusoperaatto rin. K\344yt\344nn\366ss\344 parillista/paritonta laajennusta vastaa k osini/sinisarja, jossa l\344hdet\344\344n liikkeelle v\344lill\344 [0, d] m\344\344ritellyst\344 funktiosta f, joka laajennetaan parilliseks i/parittomaksi. T\344ll\366in yll\344 olevat kaavat tulevat muotoon:" }}{PARA 0 "" 0 "" {TEXT 261 15 "1) kosinisarja:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "f:='f': T:='T': omega:='omega': #omega:=2*Pi/T :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "a0:=(1/d)*int(f(t),t=- d..d);an:=(1/d)*int(f(t)*cos(n*omega*t),t=-d..d);bn:=0;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G*&-%$ intG6$-%\"fG6#%\"tG/F,;,$%\"dG!\"\"F0\"\"\"F0F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#anG*&-%$intG6$*&-%\"fG6#%\"tG\"\"\"-%$cosG6#*(%\"nGF .%&omegaGF.F-F.F./F-;,$%\"dG!\"\"F8F.F8F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bnG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Ko ska yll\344 olevat integroitavat ovat parilllisia, voidaan integroida \+ [0..d], kunhan tulos kerrotaan 2:lla." }}{PARA 0 "" 0 "" {TEXT -1 151 "T\344ll\366in tullaan toimeen alkuper\344isell\344, v\344lill\344 [0, d] m\344\344ritellyll\344 funktiolla, eik\344 tarvitse kohdistaa integ rointia parilliseen laajennukseen. Siisp\344 " }{TEXT 262 18 "kosinisa rjakaavat " }{TEXT -1 5 "ovat:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "f:='f':T:='T':omega:='omega':#omega:=2*Pi/T:\na0:=(2/d)*int(f (t),t=0..d);\nan:=(2/d)*int(f(t)*cos(n*omega*t),t=0..d);bn:=0;\nsarja: ='a0'/2+sum('an'*cos(n*omega*t),n=1..infinity);" }}{PAGEBK }{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G,$*&-%$in tG6$-%\"fG6#%\"tG/F-;\"\"!%\"dG\"\"\"F1!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#anG,$*&-%$intG6$*&-%\"fG6#%\"tG\"\"\"-%$cosG6#*(%\"n GF/%&omegaGF/F.F/F//F.;\"\"!%\"dGF/F9!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bnG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sarja G,&%#a0G#\"\"\"\"\"#-%$sumG6$*&%#anGF(-%$cosG6#*(%\"nGF(%&omegaGF(%\"t GF(F(/F3;F(%)infinityGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 10 "Sini sarja " }{TEXT -1 26 "saadaan aivan vastaavasti:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "f:='f':T:='T':omega:='omega':#omega:=2*Pi/T: \nbn:=(2/d)*int(f(t)*sin(n*omega*t),t=0..d);\nsarja:=sum('bn'*cos(n*om ega*t),n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bnG,$*&-% $intG6$*&-%\"fG6#%\"tG\"\"\"-%$sinG6#*(%\"nGF/%&omegaGF/F.F/F//F.;\"\" !%\"dGF/F9!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sarjaG-%$su mG6$*&%#bnG\"\"\"-%$cosG6#*(%\"nGF*%&omegaGF*%\"tGF*F*/F/;F*%)infinity G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Kompleksinen muoto" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 308 "Kokeillaan vaihteeksi sellaista t yyli\344, ett\344 c on n:n funktio, t\344ss\344 se on mukavaa, kun kaa vatkin ovat yhten\344iset. c-funktio on syyt\344 m\344\344ritell\344 \+ unapply-tavalla. Laskentatehossa nimitt\344in on dramaattinen ero. (un apply evaluoi (esim. integroi) m\344\344rittelyaikana, kun taas -> m \344\344rittely tekee sen kutsuaikana.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "T:='T': f:='f': omega: ='omega': omega:=2*Pi/T:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "c:=unapply(1/T*int(f(t)*exp(-I*n*omega*t),t=0..T),n);c(0):=1/T*int(f( t),t=0..T);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Vaikka seuraava to imii, niin ei kannata tehd\344, kts. yll\344." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "#c:=n->1/T*int(f(t)*exp(-I*n*omega*t),t=0..T);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "sarja:=sum('c(n)'*exp(I*n *'omega'*t),n=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&sarjaG-%$sumG6$*&-%\"cG6#%\"nG\"\"\"-%$expG6#**^#F-F-F,F-%&omegaGF -%\"tGF-F-/F,;,$%)infinityG!\"\"F8" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 6 "Esim. " }}{PARA 0 "" 0 "" {TEXT -1 213 "Muodosta alla m \344\344ritelt\344v\344n funktion Fourier-sarja, piirr\344 jaksollinen laajennus ja Fourier-osasummien kuvaajia. Piirr\344 my\366s taajuussp ektrit. K\344sittele sek\344 reaalimuotoista sin/cos-sarjaa ett\344 k ompleksimuotoista." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f:=t-> piecewise(-Pi%\"fGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6& 32,$%#PiG!\"\"9$2F4\"\"!,&F4\"\"\"F2F832F6F42F4F2F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(f,-Pi..Pi);" }}{PARA 13 "" 1 " " {GLPLOT2D 370 277 277 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$!3*****4tk#fTJ !#<$\"3*)H&[*HJzzi!#E7$$!3w\"*pr)3PY+$F*$\"3h8!G([cbp8!#=7$$!3@3E[*ysa )GF*$\"3-\\=2T')>hDF37$$!3fX&))>2g9v#F*$\"3IvC,;eK,RF37$$!3)4577.flh#F *$\"3L@pxBiL]_F37$$!3v\"QM::*H#[#F*$\"3u8Tb?]$Hf'F37$$!3y\\AhcI#yN#F*$ \"3PLaxpfpPyF37$$!3&e=d-FN*GAF*$\"3ksgKLQdE\"*F37$$!3w%*GBP$Rc4#F*$\"3 P)*oN;L&f/\"F*7$$!3mCj&f)3xi>F*$\"3YoMjn<#)y6F*7$$!3'or4AN*4E=F*$\"3Dw +Q,L\\:8F*7$$!3QQQ*3**=dq\"F*$\"3tafpiO(eV\"F*7$$!3d!>jg@*>q:F*$\"3a-m _PMRr:F*7$$!3`;m,%)H7M9F*$\"3fwJdp'puq\"F*7$$!3v(4F,K))HI\"F*$\"3O&piM L/'Q=F*7$$!3S*4!R#[0R=\"F*$\"3r$p*>rrod>F*7$$!3pY@QqWIU5F*$\"3WYw?$=)G *4#F*7$$!3Wp3w$R)\\B#*F3$\"3<1P@9GC>AF*7$$!3EJc8o39GyF3$\"3**Hixm&y(eB F*7$$!3'=[BP-8If'F3$\"3%\\W<7N\"H#[#F*7$$!3CB3<@?)yB&F3$\"3z5FZ^W!yh#F *7$$!3`*)R$\"33s]ai\"4=2$ F*7$$!38,z7VKJ,J!#?$\"3AlmMS8\\QJF*7$$\"3p0t!3)GF;mF_s$\"37$z*e`EfTJF* 7$$\"3:SFf3xEa8F3F[t7$$\"3'R2()Hve,c#F3F[t7$$\"3IcwR>CZ'eYk>F*F[t7$$\"3kYuyfix%4#F*F[t7$$\"3T4q ))fu.GAF*F[t7$$\"3z?w'**=&>gBF*F[t7$$\"3!*>3a=Wj\"[#F*F[t7$$\"3A?c*)yu \"3i#F*F[t7$$\"3,i-HnWIXFF*F[t7$$\"3v=RWhN.yGF*F[t7$$\"3ss(*RSA20IF*F[ t7$$\"3!)***\\/l#fTJF*F[t-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F]yF\\y-%+A XESLABELSG6$Q!6\"Fay-%%VIEWG6$;$!+aEfTJ!\"*$\"+aEfTJFiy%(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 21 "jaksf:=JJ(f,-Pi..Pi):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(jaksf,-2*Pi..2*Pi);" }} {PARA 13 "" 1 "" {GLPLOT2D 370 277 277 {PLOTDATA 2 "6%-%'CURVESG6$7eo7 $$!3(****>YH&=$G'!#<$\"37$z*e`EfTJF*7$$!3dp3dxTF4gF*F+7$$!3W07Azb%4x&F *F+7$$!3[DqOW,#H]&F*F+7$$!3()o*[H1=JB&F*F+7$$!3gAxs.$)fk\\F*F+7$$!3/sy +9hk:ZF*F+7$$!3ellUT0(yX%F*F+7$$!3#>@6bny7>%F*F+7$$!3/t34t_OF*F+7$$!3:T&yWM==`$F*F+7$$!35AGA$)zV6MF*F+7$$!3Ah8%e4yOM$F* F+7$$!3!***)f%3#=fF$F*F+7$$!3Bp\"pZEQ?C$F*F+7$$!3dQ%y5Ke\"3KF*F+7$$!3C tIB\\$=7>$F*F+7$$!3!zq(Qx$yU<$F*F+7$$!3YD]Y\"R3e;$F*F+7$$!3-VBa0%Qt:$F *F+7$$!38g'>'>%o)[JF*F+7$$!3pxppL%)RSJF*$\"31Ha\"G*)>U>\"!#?7$$!3.2%=< ?AV+$F*$\"3!4'Qr=Xqs8!#=7$$!3OO)R(pfCoGF*$\"3^n&*\\QoYLFFap7$$!3In=4Um (fg#F*$\"3:e#z\\6ghN&Fap7$$!3<$*otm4\"yO#F*$\"3T***G&oo\"yt(Fap7$$!3I; D'G%*3Y3#F*$\"3#oFF2r$)p0\"F*7$$!38\"Hq4o*pW=F*$\"3+-&>E(H*oH\"F*7$$!3 %zq%)fP&\\*yFap$\"3MV&4;G(4_BF*7$$!3 WCVrl)[@?&Fap$\"3ng$=qwx8i#F*7$$!3S5B'pS8$HFFap$\"32iN*GJh'oGF*7$$!3#z #pf^)pcR\"Fap$\"3K+,Voc--IF*7$$!3.VX:B'HE?'F[p$\"3dQm'R-!RNJF*7$$\"31$ =+TaaKK\"FapF+7$$\"3b6>V%QN&3FFapF+7$$\"3H.`,suJ?^FapF+7$$\"30FZ`*>3^s (FapF+7$$\"3)y$=+!>6;/\"F*F+7$$\"3tD!)H-M([I\"F*F+7$$\"3nO`QHJff:F*F+7 $$\"3g&>\"zdfTU=F*F+7$$\"3M=t;[[a'4#F*F+7$$\"3\"3Rp8o\")yO#F*F+7$$\"3q #RL3=`Ph#F*F+7$$\"3SB4#HnbD)GF*F+7$$\"3IySJrn,4IF*F+7$$\"3>LsqpyZNJF*F +7$$\"3ohydP'RP9$F*$\"3hi&o!))R)p9#F[p7$$\"3t*[[aS,?:$F*$\"3xg'pe=v3/ \"!#>7$$\"3x<\">L0n=Fgw7$$\"3DY(*=T\\_oJF*$\"3?9`**f (GKp#Fgw7$$\"3z-5$pZ[]=$F*$\"3in47MBeXVFgw7$$\"3LfAn7?d,KF*$\"3/@mC3f$ z*fFgw7$$\"3SsZ:%3>YB$F*$\"3*y#z\\cIk-$*Fgw7$$\"3Z&GPc:mwE$F*$\"3ZB\\Z ?]tg7Fap7$$\"3g6Bg)HgPL$F*$\"3%[=D,Xw;#>Fap7$$\"3sPtcTW&)*R$F*$\"3@Yax zyh#e#Fap7$$\"3CVG)Gl#4HNF*$\"3B,0$H****\\(QFap7$$\"3u[$)>k3LeOF*$\"3E cb31@Qn^Fap7$$\"3+;\\MC<$*GRF*$\"3$)G7b22RtyFap7$$\"3Oz6M9Db*=%F*$\"3D 'Q^2')fz/\"F*7$$\"3n8rS9\\2cWF*$\"3c?t\"3E#[98F*7$$\"3wWiVu.R?ZF*$\"3l ^k%3s(zy:F*7$$\"3Im7YJ)oK'\\F*$\"3=t9(yW)HU+5#F*7$$\"3w8mpG*31\\&F*$\"3l?o5vi,\\BF*7$$\"3Ql7(o6ngv&F*$\"3Es9 GjWZ9EF*7$$\"3b(*fluW95gF*$\"3V/i1@=boGF*7$$\"3(****>YH&=$G'F*$\"3'o?I 5k#fTJF*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fd^lFc^l-%+AXESLABELSG6$Q!6 \"Fh^l-%%VIEWG6$;$!+3`=$G'!\"*$\"+3`=$G'F`_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "T:=2*Pi:omega:=2*Pi/T:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "a0:=(2/T)*int(f(t),t=-Pi..Pi);\nan:=(2/T )*int(f(t)*cos(n*omega*t),t=-Pi..Pi):an:=trigsiev(an,n);\nbn:=(2/T)*in t(f(t)*sin(n*omega*t),t=-Pi..Pi):bn:=trigsiev(bn,n);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G,$%#PiG#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#anG,$*&,&)!\"\"%\"nG\"\"\"F+F)F+*&%#PiGF+)F*\"\"#F+F )F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bnG,$*&)!\"\"%\"nG\"\"\"F)F( F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Koska halusimme v\344ltty \344 " }{TEXT 0 6 "assume" }{TEXT -1 62 "-komennon aiheuttamilta sivuv aikutuksilta, sievensimme omalla " }{TEXT 0 9 "trigsiev-" }{TEXT -1 28 "komennolla. Huomaamme, ett\344 " }{TEXT 0 3 "int" }{TEXT 260 1 " \+ " }{TEXT -1 16 "osasi k\344sitell\344 " }{TEXT 0 9 "piecewise" }{TEXT -1 111 "-m\344\344ritelty\344 funktiota ihan n\344tisti. Luultavaa on, ett\344 se osaa my\366s ep\344jatkuvalle, mutta kannattaa tarkkailla. 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}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Piirt\344minen hoituu o ikein n\344pp\344r\344sti yll\344 m\344\344rittelem\344ll\344mme " } {TEXT 0 4 "jana" }{TEXT -1 12 "-funktiolla." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A0:=a0/2; An:=abs(an-I*bn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A0G,$%#PiG#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AnG-%$absG6#,&*&,&)!\"\"%\"nG\"\"\"F.F,F.*&%#PiGF.)F-\"\"#F.F ,F,*&*&^#F.F.F+F.F.F-F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "phi0:=0: phin:=argument(an-I*bn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%phinG-%)argumentG6#,&*&,&)!\"\"%\"nG\"\"\"F.F,F.*&%#PiGF.)F-\"\" #F.F,F,*&*&^#F.F.F+F.F.F-F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "display([jana(0,A0),seq(jana(n,An),n=1..6)],axes=box,title=\"Am plitudispektri\");display([jana(0,phi0),seq(jana(n,phin),n=1..6)],titl e=\"Vaihespektri\");" }}{PARA 13 "" 1 "" {GLPLOT2D 370 277 277 {PLOTDATA 2 "6--%'CURVESG6%7$7$$\"\"!F)F(7$F($\"3$[M#>!\\%>cB!#<-%'COL 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45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 18 "Kompleksinen muoto" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "c:=unapply(1/T*int(f(t)*exp(-I*n*om ega*t),t=0..T),n);c(0):=1/T*int(f(t),t=0..T);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "sarja:=sum('c(n)'*exp(I*n*'omega'*t),n=-infinity..inf inity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGR6#%\"nG6\"6$%)operat orG%&arrowGF(*&*&^##\"\"\"\"\"#F0,&!\"\"F0-%$expG6#*(^#F3F0%#PiGF09$F0 F0F0F0F:F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"cG6#\"\"!,$%# PiG#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sarjaG-%$sumG6$* &-%\"cG6#%\"nG\"\"\"-%$expG6#**^#F-F-F,F-%&omegaGF-%\"tGF-F-/F,;,$%)in finityG!\"\"F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c(0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^ #!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 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