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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::J:<6:\"\{\}-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YGF'-%-BOUNDS_WIDTHG6#$"%+!)F*-%.BOUNDS_HEIGHTG6#$"$!pF*-%)CHILDRENG6" FOURIER Series Univ.-Prof. Dr.-Ing. habil. Josef BETTEN RWTH Aachen University Templergraben 55 D-52056 A a c h e n , Germany betten@mmw.rwth-aachen.de Abstract This worksheet is concerned with FOURIER series. Some examples are discussed using MAPLE V, Release 10. Keywords: FOURIER expansion; odd and even functions; HEAVISIDE function; continuous functions with cusps; L-two norm FOURIER Expansion restart: FOURIER_series:= a[0]/2+sum(a[k]*cos(k*x)+b[k]*sin(k*x),k=1..infinity);

NiM+JS9GT1VSSUVSX3Nlcmllc0csJiomIyIiIiIiIyIiIiYlImFHNiMiIiEiIiIiIiItJSRzdW1HNiQsJiomJiUiYUc2IyUia0ciIiItJSRjb3NHNiMqJiUia0ciIiIlInhHIiIiIiIiIiIiKiYmJSJiRzYjJSJrRyIiIi0lJHNpbkc2IyomJSJrRyIiIiUieEciIiIiIiIiIiIvJSJrRzsiIiIlKWluZmluaXR5RyIiIg==

a[k]:=(1/Pi)*Int(f(x)*cos(k*x),x=-Pi..Pi); # k=0,1,2,3,...

NiM+JiUiYUc2IyUia0csJComKiQlI1BpRyEiIiIiIi0lJEludEc2JComLSUiZkc2IyUieEciIiItJSRjb3NHNiMqJiUia0ciIiIlInhHIiIiIiIiLyUieEc7LCQlI1BpRyEiIiUjUGlHIiIiIiIi

a[0]:=simplify(subs(k=0,%));

NiM+JiUiYUc2IyIiISwkKiYqJCUjUGlHISIiIiIiLSUkSW50RzYkLSUiZkc2IyUieEcvJSJ4RzssJCUjUGlHISIiJSNQaUciIiIiIiI=

b[k]:=(1/Pi)*Int(f(x)*sin(k*x),x=-Pi..Pi); # k=1,2,3,...

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ODD and Even Functions odd_function:=f(x)=x;

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A[0]:=value(subs(f(x)=x,a[0]));

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A[k]:=value(subs(f(x)=x,a[k]));

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For odd functions the coefficients A[k], k = 0,1,2.. are identical to zero. B[k]:=value(subs(f(x)=x,b[k]));

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B[k]:=subs({sin(Pi*k)=0,cos(Pi*k)=(-1)^k},%);

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FOURIER_series[f(x)=x][k=4]:=sum(B[k]*sin(k*x),k=1..4);

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for i in [2,4,5] do FOURIER_series[f(x)=x][k=i]:= subs(k=i,sum(B[k]*sin(k*x),k=1..i)) od;

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compact form: y(x,n)[f(x)=x]:=Sum(B[k]*sin(k*x),k=1..n);

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y(x,4)[f(x)=x]:=value(subs(n=4,%));

NiM+Ji0lInlHNiQlInhHIiIlNiMvLSUiZkc2IyUieEclInhHLCoqJiIiIyIiIi0lJHNpbkc2IyUieEciIiIiIiItJSRzaW5HNiMsJComIiIjIiIiJSJ4RyIiIiIiIiEiIiomIyIiIyIiJCIiIi0lJHNpbkc2IywkKiYiIiQiIiIlInhHIiIiIiIiIiIiIiIiKiYjIiIiIiIjIiIiLSUkc2luRzYjLCQqJiIiJSIiIiUieEciIiIiIiIiIiIhIiI=

for i in [1,10,100] do y(x,n=i):=2*subs(n=i,sum((-1)^(n-1)*sin(n*x)/n,n=1..i)) od: plot({y(x,n=1),y(x,n=10),y(x,n=100)},x=-4*Pi..4*Pi,color=black, title="FOURIER-series with n = [1, 10, 100] for f(x) = x");

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7!#:7$$"2MCm'opdW6!#:$!2.f^PiN')>"!#;7$$"2eT=g>&*y9"!#:$!2Em!)fGXs="!#;7$$"2&)eqLU87:"!#:$!2zya-r+N;"!#;7$$"1")*["f6iw6!#9$!1&G@*>!p@8(!#;7$$"2OPF\*)G??"!#:$!1nB%pq[Z'f!#;7$$"2ooBp(HLH7!#:$!2ox7"QbEwC!#<7$$"-#*eqjc7!#5$!0XxauRNQ$!#I-%'CURVESG6#7[u7$$!-#*eqjc7!#5$"1ORQ'oIvx'!#J7$$!2%[.^N[&=?"!#:$"1>*3^]Z7e&!#;7$$!2scSe6*=a6!#:$"1z&=EWUq,"!#:7$$!2=!*p)GSe+6!#:$"2.S_[<xQc"!#;7$$!2AY'e7OiY5!#:$"1f6QR\=6@!#:7$$!1BWmO;x>5!#9$"1"\Z7)yj&Q#!#:7$$!1QQU2m>H**!#:$"2vJS%4ah+E!#;7$$!14Vc70s/)*!#:$"2t)*>jtlWr#!#;7$$!1"y/xTW-o*!#:$"1lSUc&[y"G!#:7$$!1<]Fqj+='*!#:$"1*Rw:E\"fG!#:7$$!1`_%GKodb*!#:$"2X3sr#)Q+)G!#;7$$!17y)4")3-a*!#:$"22<1$)3-6%H!#;7$$!0PI"*H\Y_*!#9$"2;L;l%pu7K!#;7$$!1HHF(y*34&*!#:$"1ld)*GG')pJ!#:7$$!1)[:aFIN\*!#:$"2WeGj#=BCG!#;7$$!1Y!eNwqzZ*!#:$"1Vg8`AGFH!#:7$$!101q^7Ti%*!#:$"2C'R'o!Q'[b$!#;7$$!1lJ%)R<&oW*!#:$"2dw!\;5emL!#;7$$!1Cd)zA#HJ%*!#:$"2$Q;!y(Q2s7!#;7$$!1W%*pa7=:%*!#:$!22Q2()Q)RA=!#;7$$!1jJT"Gq!*R*!#:$!1FFTK+]oN!#:7$$!1#)o73$fHQ*!#:$!1P'ok6wPR$!#:7$$!/1%[L[oO*!#8$!2dB#fM#4L"G!#;7$$!0Ka:OP2N*!#9$!2&)oJ?@zE$H!#;7$$!0/o#)QEYL*!#9$!20ZL0RI^C$!#;7$$!0w")\T:&=$*!#9$!17AGhfb3J!#:7$$!0[&pTWS-$*!#9$!1%4%fa]gnG!#:7$$!1)>4%oMH'G*!#:$!2OWNS'zJiH!#;7$$!1>H7&\#=q#*!#:$!203,,%))e6J!#;7$$!0kO=_rSD*!#9$!2WC]08#p"*H!#;7$$!1e.b[0'zB*!#:$!2Pgb$4'*Q\G!#;7$$!1xSEv&\=A*!#:$!1Ydl5_QJH!#:7$$!1)zx>gQd?*!#:$!2mQI4r!H8I!#;7$$!1=:pGwi*=*!#:$!2E*4C@5+/H!#;7$$!1P_Sbm^t"*!#:$!1GVe/iN6G!#:7$$!1&*\6p)GY/*!#:$!2vU<I5zZr#!#;7$$!1_Z#G3Td"*)!#:$!2xuFTy'>2E!#;7$$!1[-"p@\"\')!#:$!1i;`r`l(Q#!#:7$$!1Vd*4NdDQ)!#:$!2&Rb;Bw54@!#;7$$!1bn:YN3^y!#:$!1iOFRJ<h:!#:7$$!1p_W6uR/t!#:$!1*p>pf)Q65!#:7$$!2l#RUmf(G#o!#;$!1E_n*y5LZ&!#;7$$!2D'RPnoz!G'!#;$"28>3zoxpG#!#@7$$!189YR>\Od!#:$"1Hfy(*4@qb!#;7$$!1Wr;%G`>@&!#:$"1mI!*Hxbi5!#:7$$!17(eM$>iNZ!#:$"2P\yV!*o9c"!#;7$$!1h<r&)y@pT!#:$"1n'4+SrD8#!#:7$$!/.#Qi3$HR!#8$"2&H#*)>Ip]P#!#;7$$!2)Q)G>O*R*o$!#;$"1ycsMrh"f#!#:7$$!28\O%4O')\N!#;$"2N'GX'=$o"y#!#;7$$!2Q9Wp&yK5M!#;$"1j-JW!pC*G!#:7$$!2o0@)=RWvL!#;$"0.\W#pR;H!#97$$!2)pzp!)*f0M$!#;$"2)=X#R[5r*G!#;7$$!2lUO;,=JK$!#;$"1DU=pfttG!#:7$$!2K)[dUgn0L!#;$"2Ms\*3f3oI!#;7$$!2&RL^tSB)G$!#;$"1e%GsT5O1$!#:7$$!2iz^W5#zqK!#;$"2b0im\Y%oG!#;7$$!1`-RN,N`K!#:$"1u_CR5a.I!#:7$$!2$4(Gj;3fB$!#;$"2ngf%*Qy\D$!#;7$$!1mrE(>m%=K!#:$"2e]898m>+$!#;7$$!2Fi0#GU-,K!#;$"2cTUx9WDz#!#;7$$!1zS9fAe$=$!#:$"2_<hSM[kQ$!#;7$$!2d`#3!HSh;$!#;$"2Fk(=5(yi^$!#;7$$!2D*4-@$)p[J!#;$"2Xp;(>SG"Q"!#;7$$!2([%f>Nc78$!#;$!2&*3"))Q*pm%>!#;7$$!1i?"RD<e6$!#:$!2u+%o-I?rN!#;7$$!2Ynke:y.5$!#;$!2L\I%f4M>M!#;7$$!2tG<y0R\3$!#;$!1rpnpU#p$G!#:7$$!2/!*p(f**\pI!#;$!18g%G]'=))G!#:7$$!2O^A<'31aI!#;$!1WrLV"4(>K!#:7$$!2j7vOw@'QI!#;$!2$R'p"*oot;$!#;7$$!1RxilE=BI!#:$!2:j%*Q"45,H!#;7$$!1_.enNu2I!#:$!2j*>&p4di!H!#;7$$!1/3Rvr)f%H!#:$!21mx;V3B(G!#;7$$!2bD,KyIU)G!#;$!2d1T^_aZ#G!#;7$$!1f@#))*zrgF!#:$!2vFqsKLgr#!#;7$$!1iIW9_?PE!#:$!1'G74#[f+E!#:7$$!2v$)oS,zhO#!#;$!2b(Q]&H\vQ#!#;7$$!2Fh%p8G:&4#!#;$!2Z"pLd'Qo4#!#;7$$!2DZbRu!**y:!#;$!2%4?j$*\([c"!#;7$$!1_&RJxH//"!#:$!1vD[*Ql$\5!#:7$$!1[s!R"oiea!#;$!1)>P%R8Chb!#;7$$!21+FY#f_S7!#=$!1n`$zD&*G(H!#=7$$"1vl%)o22<a!#;$"1p2)Hp4f]&!#;7$$"1,)*R%\jS-"!#:$"2w"*Q]'>F;5!#;7$$"2&G?!*R;-X:!#;$"21@"G-5be:!#;7$$"2Y/(**zBA$3#!#;$"2v&QHHC`l?!#;7$$"1&R#H#f%[YB!#:$"0cj6hV>N#!#97$$"1Xxe/ou4E!#:$"2(R*GBXxvk#!#;7$$"2WCu;`mW'G!#;$"2DFwj;R*GH!#;7$$"2Nug(ei=>J!#;$"1k$zQ2*z*Q$!#:7$$"2N0Oboio8$!#;$"1'zZb#GRT$*!#;7$$"2NO6B6RX:$!#;$!1N2O[JjcB!#:7$$"2Nn'3Rb@sJ!#;$!1.oPkg&>n$!#:7$$"1$)>'e'>*)*=$!#:$!2:CY)4%z$4J!#;7$$"2FHPERov?$!#;$!1CbC(4?')y#!#:7$$"2Fg7%>[CDK!#;$!2L%))pJQVdJ!#;7$$"2F"z=Y7#HC$!#;$!2ZI=(>5'H>$!#;7$$"2FAjHn(fgK!#;$!2L8/d8YX*G!#;7$$"2F`Q(*4u#yK!#;$!1p].oj<RH!#:7$$"2F%Q^E0&fH$!#;$!1VD(["fE6J!#:7$$"2F:*G`pi8L!#;$!2K2Fb;%)[(H!#;7$$"2AYk+Q.8L$!#;$!1Rn&yXg'[G!#:7$$"1s(Ro!)z*[L!#:$!122\g4FpH!#:7$$"1#3:OBcmO$!#:$!2Vod-aO-*H!#;7$$"1#R!RgEL%Q$!#:$!0a!>5%[/%G!#97$$"1-d;(34?S$!#:$!1A.f?m%*RG!#:7$$"175%R^&o>M!#:$!1^21y!G0$H!#:7$$"1AjrS>OPM!#:$!1_$eErMh&G!#:7$$"1K;\n$Q]X$!#:$!1&3K'p[ekF!#:7$$"29%pE%z9FZ$!#;$!283\]B_g#G!#;7$$"1^A/@7R!\$!#:$!1sk$))G&*Q%G!#:7$$"1hv"ykn!3N!#:$!2<V,J&G+TF!#;7$$"1rGfuSuDN!#:$!1'=$3n!oVs#!#:7$$"1"=o8]?Ma$!#:$!0nw'*==Ry#!#97$$"21Up%3i79O!#;$!2&ea.4^.4F!#;7$$"0mqb">$[o$!#9$!2ky6uA%z7E!#;7$$"2Y(f%f!3'*QR!#;$!1De_>#*)QM#!#:7$$"1*G@jp*3$>%!#:$!1ZtPS$[>3#!#:7$$"2vqBFOjdt%!#;$!2%[.)yFI8c"!#;7$$"2X9];O1vA&!#;$!2c`(>gWBj5!#;7$$"1(\BeM6^w&!#:$!1;Yqf4b!3&!#;7$$"1/XRRd&4F'!#:$!2Y]\m)ohaG!#>7$$"1HI6$))3(*z'!#:$"1GR0/thi]!#;7$$"2%4MPG<m;t!#;$"2V/2U'**oN5!#;7$$"1@[m[M'y&y!#:$"2dinq,S;c"!#;7$$"1*ob'G]5z$)!#:$"2vUFRVI"*4#!#;7$$"1u2sGuiX')!#:$"2$pT<Py$yQ#!#;7$$"1feyG)\@"*)!#:$"1hGc&3Fgh#!#:7$$"1.+!)evIW!*!#:$"2x(f"=S0`r#!#;7$$"0:9))Glk<*!#9$"2Nf;m/**\"G!#;7$$"1B7#Q:WDC*!#:$"1t#pWB9)oG!#:7$$"1'HG)=Ii3$*!#:$"2O#HO(za.$H!#;7$$"1k+3NF9D$*!#:$"2x(R&etec@$!#;7$$"1K=L^CmT$*!#:$"2(QzVVq$o9$!#;7$$"/Oen@=e$*!#8$"2Ee>LH8\z#!#;7$$"1o`$Q)=qu$*!#:$"2u)R"QqYt.$!#;7$$"1Nr3+;A"R*!#:$"2%)3TTGUUl$!#;7$$"1.*QjJTxS*!#:$"2m2\A14j*G!#;7$$"1r1fK5EC%*!#:$"2Dbv9e+P."!#<7$$"0WU)[2yS%*!#9$!2%o!p4s%osF!#;7$$"1dkS_1'fX*!#:$!1WUDq#)psO!#:7$$"1s/(fbS6Z*!#:$!2x-*\<_'H>$!#;7$$"1'[M&f/K'[*!#:$!1h.!=idix#!#:7$$"1-&)4j.],&*!#:$!0&pf0l@)*H!#97$$"1>Dmm-o;&*!#:$!2C!eL7S)QD$!#;7$$"1LlAq,'=`*!#:$!2uof')=VH4$!#;7$$"1[0zt+/Z&*!#:$!1O6#*e'3#oG!#:7$$"1kXNx*>Ac*!#:$!1v*e*4%)z[H!#:7$$"1E1h"fRHi*!#:$!1jvj(R^1*G!#:7$$"1)ome?fOo*!#:$!2yRq2Gk<$G!#;7$$"16)yVV)40)*!#:$!0sg2*R0:F!#97$$"1N4*GmPl#**!#:$!2&\."e0z,g#!#;7$$"2P0Y$y.\?5!#:$!2ju3'QuL(Q#!#;7$$"19IS!*pK[5!#9$!1,P$>r2`1#!#:7$$"2s#et&y@")4"!#:$!1CW$HpxRd"!#:7$$"2&)eqLU87:"!#:$!2ts(Qd['H1"!#;7$$"2OPF\*)G??"!#:$!1`(R/avOc&!#;7$$"-#*eqjc7!#5$!1ORQ'oIvx'!#J-%%VIEWG6$;$!+iqjc7!")$"+iqjc7!")%(DEFAULTG-%+AXESLABELSG6'Q"x6"Q!6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5%+HORIZONTALG%+HORIZONTALG-%&TITLEG6$-%)_TYPESETG6#QTFOURIER-series~with~~n~=~[1,~10,~100]~~for~f(x)~=~x6"-%-TRANSPARENCYG6#$""!!""-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%%ROOTG6'-%)BOUNDS_XG6#$"#g!""-%)BOUNDS_YG6#$"$?#!""-%-BOUNDS_WIDTHG6#$"%qX!""-%.BOUNDS_HEIGHTG6#$"%SD!""-%)CHILDRENG6"

even_function:=f(x)=x^2;

NiM+JS5ldmVuX2Z1bmN0aW9uRy8tJSJmRzYjJSJ4RyokKSUieEciIiMiIiI=

A[0]:=value(subs(f(x)=x^2,a[0]));

NiM+JiUiQUc2IyIiISwkKigiIiMiIiIiIiQhIiIpJSNQaUciIiMiIiIiIiI=

A[k]:=value(subs(f(x)=x^2,a[k]));

NiM+JiUiQUc2IyUia0csJCoqIiIjIiIiJSNQaUchIiIsKComIiIjIiIiLSUkc2luRzYjKiYlI1BpRyIiIiUia0ciIiIiIiIhIiIqKCklImtHIiIjIiIiLSUkc2luRzYjKiYlI1BpRyIiIiUia0ciIiIiIiIpJSNQaUciIiMiIiIiIiIqKiIiIyIiIi0lJGNvc0c2IyomJSNQaUciIiIlImtHIiIiIiIiJSJrRyIiIiUjUGlHIiIiIiIiIiIiKSUia0ciIiQhIiIiIiI=

A[k]:=subs({sin(Pi*k)=0,cos(Pi*k)=(-1)^k},%);

NiM+JiUiQUc2IyUia0csJCooIiIlIiIiKSEiIiUia0ciIiIpJSJrRyIiIyEiIiIiIg==

B[k]:=value(subs(f(x)=x^2,b[k]));

NiM+JiUiQkc2IyUia0ciIiE=

For even functions the coefficients B[k] are identical to zero. y(x,n)[f(x)=x^2]:=A[0]/2+Sum(A[k]*cos(k*x),k=1..n);

NiM+Ji0lInlHNiQlInhHJSJuRzYjLy0lImZHNiMlInhHKiQpJSJ4RyIiIyIiIiwmKiYiIiQhIiIpJSNQaUciIiMiIiIiIiItJSRTdW1HNiQsJCoqIiIlIiIiKSEiIiUia0ciIiIpJSJrRyIiIyEiIi0lJGNvc0c2IyomJSJrRyIiIiUieEciIiIiIiIiIiIvJSJrRzsiIiIlIm5HIiIi

y(x,4)[f(x)=x^2]:=value(subs(n=4,%));

NiM+Ji0lInlHNiQlInhHIiIlNiMvLSUiZkc2IyUieEcqJCklInhHIiIjIiIiLCwqJiIiJCEiIiklI1BpRyIiIyIiIiIiIiomIiIlIiIiLSUkY29zRzYjJSJ4RyIiIiEiIi0lJGNvc0c2IywkKiYiIiMiIiIlInhHIiIiIiIiIiIiKiYjIiIlIiIqIiIiLSUkY29zRzYjLCQqJiIiJCIiIiUieEciIiIiIiIiIiIhIiIqJiMiIiIiIiUiIiItJSRjb3NHNiMsJComIiIlIiIiJSJ4RyIiIiIiIiIiIiIiIg==

for i in [1,10,100] do y(x,n=i):=Pi^2/3+subs(n=i,4*sum((-1)^n*cos(n*x)/n^2, n=1..i)) od: plot({y(x,n=1),y(x,n=10),y(x,n=100)}, x=-4*Pi..4*Pi,color=black, title="FOURIER-series # n = [1, 10, 100] for f(x) = x^2");

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Function f(x) = (x - X)^2 in several ranges: f(x)[-2*Pi]:=(x+2*Pi)^2; x = [-3*Pi, -Pi];

NiM+Ji0lImZHNiMlInhHNiMsJComIiIjIiIiJSNQaUciIiIhIiIqJCksJiUieEciIiIqJiIiIyIiIiUjUGlHIiIiIiIiIiIjIiIi NiMvJSJ4RzckLCQqJiIiJCIiIiUjUGlHIiIiISIiLCQlI1BpRyEiIg==

f(x)[0]:=x^2; x = [-Pi, Pi];

NiM+Ji0lImZHNiMlInhHNiMiIiEqJCklInhHIiIjIiIi NiMvJSJ4RzckLCQlI1BpRyEiIiUjUGlH

f(x)[2*Pi]:=(x-2*Pi)^2; x = [Pi, 3*Pi];

NiM+Ji0lImZHNiMlInhHNiMsJComIiIjIiIiJSNQaUciIiIiIiIqJCksJiUieEciIiIqJiIiIyIiIiUjUGlHIiIiISIiIiIjIiIi NiMvJSJ4RzckJSNQaUcsJComIiIkIiIiJSNQaUciIiIiIiI=

alias(H=Heaviside,th=thickness,co=color): p[1]:=plot(f(x)[-2*Pi],x=-3*Pi..-Pi,0..Pi^2,th=2,co=black): p[2]:=plot(f(x)[0],x=-Pi..Pi,th=3,co=black): p[3]:=plot(f(x)[2*Pi],x=Pi..3*Pi,th=2,co=black): p[4]:=plot({Pi^2,Pi^2*H(x+3*Pi),Pi^2*H(x-3*Pi), Pi^2*H(x+2*Pi),Pi^2*H(x-2*Pi)}, x=-3.001*Pi..3.001*Pi,co=black, title="Quadratic functions with the period 2*Pi"): plots[display]({seq(p[k],k=1..4)});

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7$$"1;T[q.^*Q*!#:$""!!""7$$"1WF=DMJ-%*!#:$""!!""7$$"1t8))zk6:%*!#:$""!!""7$$"-eM&>zU*!#6$"1e$*3,Wgp)*!#:-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%'CURVESG6$7hn7$$!-eM&>zU*!#6$""!!""7$$!1kwW.f"p,*!#:$""!!""7$$!1#*QiTQIf')!#:$""!!""7$$!1z]iw;8d#)!#:$""!!""7$$!1prX`OH_y!#:$""!!""7$$!1tbua(z$\u!#:$""!!""7$$!1K32%*p#e2(!#:$""!!""7$$!1sCTZZ.*o'!#:$""!!""7$$!1@%4+?C!*['!#:$""!!""7$$!0P1El8!*G'!#9$""!!""7$$!18j;$3`lF'!#:$"1e$*3,Wgp)*!#:7$$!1bis8D4ki!#:$"1e$*3,Wgp)*!#:7$$!1(>'GW>j^i!#:$"1e$*3,Wgp)*!#:7$$!1Th%[Pr"Ri!#:$"1e$*3,Wgp)*!#:7$$!1Eg'fB]U@'!#:$"1e$*3,Wgp)*!#:7$$!17f3(4H$*='!#:$"1e$*3,Wgp)*!#:7$$!1$oD$>o[Rh!#:$"1e$*3,Wgp)*!#:7$$!1`acTXk*3'!#:$"1e$*3,Wgp)*!#:7$$!1%*\/')*f**)f!#:$"1e$*3,Wgp)*!#:7$$!1OX_IaF!*e!#:$"1e$*3,Wgp)*!#:7$$!1y,C%z*>&o&!#:$"1e$*3,Wgp)*!#:7$$!1@e&z:C,[&!#:$"1e$*3,Wgp)*!#:7$$!11t#Qpi)=^!#:$"1e$*3,Wgp)*!#:7$$!0mj(\y;7Z!#9$"1e$*3,Wgp)*!#:7$$!1aefxI!QI%!#:$"1e$*3,Wgp)*!#:7$$!2M'*f9&zE5R!#;$"1e$*3,Wgp)*!#:7$$!2Bs*eb.!Hb$!#;$"1e$*3,Wgp)*!#:7$$!1*4m(=d&z7$!#:$"1e$*3,Wgp)*!#:7$$!2Br'Hr=(zw#!#;$"1e$*3,Wgp)*!#:7$$!1Dd1y]A\B!#:$"1e$*3,Wgp)*!#:7$$!27xU@@j&y>!#;$"1e$*3,Wgp)*!#:7$$!1P)z%)R))=d"!#:$"1e$*3,Wgp)*!#:7$$!2Wqcc!yj%="!#;$"1e$*3,Wgp)*!#:7$$!1pl+tV#e!y!#;$"1e$*3,Wgp)*!#:7$$!1J"[rwM`4%!#;$"1e$*3,Wgp)*!#:7$$!1#)ya\d/2$*!#=$"1e$*3,Wgp)*!#:7$$"2FBTVMdT1%!#<$"1e$*3,Wgp)*!#:7$$"09hnOOIo(!#:$"1e$*3,Wgp)*!#:7$$"1:0M&[_"f6!#:$"1e$*3,Wgp)*!#:7$$"1.#)f!fPHc"!#:$"1e$*3,Wgp)*!#:7$$"2bu[,aiz&>!#;$"1e$*3,Wgp)*!#:7$$"2POV1\p,M#!#;$"1e$*3,Wgp)*!#:7$$"1wkmW^akF!#:$"1e$*3,Wgp)*!#:7$$"2#efnWb'e9$!#;$"1e$*3,Wgp)*!#:7$$"1%y&*GY1Ib$!#:$"1e$*3,Wgp)*!#:7$$"2FiTykO>#R!#;$"1e$*3,Wgp)*!#:7$$"1y!>syu_K%!#:$"1e$*3,Wgp)*!#:7$$"1Vq\VXy/Z!#:$"1e$*3,Wgp)*!#:7$$"1VKd*e"[,^!#:$"1e$*3,Wgp)*!#:7$$"2v=J<YD$*[&!#;$"1e$*3,Wgp)*!#:7$$"1H7Q_?O&*e!#:$"1e$*3,Wgp)*!#:7$$"1L2xZNU'G'!#:$"1e$*3,Wgp)*!#:7$$"0yp!4/M'o'!#9$"1e$*3,Wgp)*!#:7$$"10O%ovXH3(!#:$"1e$*3,Wgp)*!#:7$$"1_(4;)[QZu!#:$"1e$*3,Wgp)*!#:7$$"0*e_XK2ly!#9$"1e$*3,Wgp)*!#:7$$"0i6vpe'Q#)!#9$"1e$*3,Wgp)*!#:7$$"1'zn)3(ypj)!#:$"1e$*3,Wgp)*!#:7$$"1**QA%y@#=!*!#:$"1e$*3,Wgp)*!#:7$$"-eM&>zU*!#6$"1e$*3,Wgp)*!#:-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%'CURVESG6$7hn7$$!-eM&>zU*!#6$""!!""7$$!1kwW.f"p,*!#:$""!!""7$$!1#*QiTQIf')!#:$""!!""7$$!1z]iw;8d#)!#:$""!!""7$$!1prX`OH_y!#:$""!!""7$$!1tbua(z$\u!#:$""!!""7$$!1K32%*p#e2(!#:$""!!""7$$!1sCTZZ.*o'!#:$""!!""7$$!0P1El8!*G'!#9$""!!""7$$!1OX_IaF!*e!#:$""!!""7$$!1@e&z:C,[&!#:$""!!""7$$!11t#Qpi)=^!#:$""!!""7$$!0mj(\y;7Z!#9$""!!""7$$!1aefxI!QI%!#:$""!!""7$$!2M'*f9&zE5R!#;$""!!""7$$!2Bs*eb.!Hb$!#;$""!!""7$$!1*4m(=d&z7$!#:$""!!""7$$!2Br'Hr=(zw#!#;$""!!""7$$!1Dd1y]A\B!#:$""!!""7$$!27xU@@j&y>!#;$""!!""7$$!1P)z%)R))=d"!#:$""!!""7$$!2Wqcc!yj%="!#;$""!!""7$$!1pl+tV#e!y!#;$""!!""7$$!1J"[rwM`4%!#;$""!!""7$$!1#)ya\d/2$*!#=$""!!""7$$"2FBTVMdT1%!#<$""!!""7$$"09hnOOIo(!#:$""!!""7$$"1:0M&[_"f6!#:$""!!""7$$"1.#)f!fPHc"!#:$""!!""7$$"2bu[,aiz&>!#;$""!!""7$$"2POV1\p,M#!#;$""!!""7$$"1wkmW^akF!#:$""!!""7$$"2#efnWb'e9$!#;$""!!""7$$"1%y&*GY1Ib$!#:$""!!""7$$"2FiTykO>#R!#;$""!!""7$$"1y!>syu_K%!#:$""!!""7$$"1Vq\VXy/Z!#:$""!!""7$$"1VKd*e"[,^!#:$""!!""7$$"2v=J<YD$*[&!#;$""!!""7$$"13i0dPM#p&!#:$""!!""7$$"1H7Q_?O&*e!#:$""!!""7$$"10'GiUFJ*f!#:$""!!""7$$"1")f2+G*34'!#:$""!!""7$$"1p'**p[v(Rh!#:$""!!""7$$"1dL#R<e')='!#:$""!!""7$$"1,_Q<&*48i!#:$""!!""7$$"1Wq%3'3aPi!#:$""!!""7$$"1nzdK:w\i!#:$""!!""7$$"1*))3V?#)>E'!#:$""!!""7$$"0")Rg(G?ui!#9$""!!""7$$"1L2xZNU'G'!#:$"1e$*3,Wgp)*!#:7$$"2lD?%y>Q'['!#;$"1e$*3,Wgp)*!#:7$$"0yp!4/M'o'!#9$"1e$*3,Wgp)*!#:7$$"10O%ovXH3(!#:$"1e$*3,Wgp)*!#:7$$"1_(4;)[QZu!#:$"1e$*3,Wgp)*!#:7$$"0*e_XK2ly!#9$"1e$*3,Wgp)*!#:7$$"0i6vpe'Q#)!#9$"1e$*3,Wgp)*!#:7$$"1'zn)3(ypj)!#:$"1e$*3,Wgp)*!#:7$$"1**QA%y@#=!*!#:$"1e$*3,Wgp)*!#:7$$"-eM&>zU*!#6$"1e$*3,Wgp)*!#:-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%%VIEWG6$;$!+b&>zU*!"*$"+b&>zU*!"*;$""!!""$"+/Wgp)*!"*-%+AXESLABELSG6'Q"x6"Q!6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5%+HORIZONTALG%+HORIZONTALG-%&TITLEG6$-%)_TYPESETG6#QJQuadratic~functions~with~the~period~~2*Pi6"-%-TRANSPARENCYG6#$""!!""-%%ROOTG6'-%)BOUNDS_XG6#$"#q!""-%)BOUNDS_YG6#$"$S#!""-%-BOUNDS_WIDTHG6#$"%5N!""-%.BOUNDS_HEIGHTG6#$"%S6!""-%)CHILDRENG6"

constant_load:=q;

NiM+JS5jb25zdGFudF9sb2FkRyUicUc=

A[0]:=value(subs(f(x)=q,a[0]));

NiM+JiUiQUc2IyIiISwkKiYiIiMiIiIlInFHIiIiIiIi

A[k]:=value(subs(f(x)=q,a[k]));

NiM+JiUiQUc2IyUia0csJCosIiIjIiIiJSNQaUchIiItJSRzaW5HNiMqJiUjUGlHIiIiJSJrRyIiIiIiIiUicUciIiIlImtHISIiIiIi

limit_value:=Limit(A[k],k=0)=limit(%,k=0);

NiM+JSxsaW1pdF92YWx1ZUcvLSUmTGltaXRHNiQsJCosIiIjIiIiJSNQaUchIiItJSRzaW5HNiMqJiUjUGlHIiIiJSJrRyIiIiIiIiUicUciIiIlImtHISIiIiIiLyUia0ciIiEsJComIiIjIiIiJSJxRyIiIiIiIg==

B[k]:=value(subs(f(x)=q,b[k]));

NiM+JiUiQkc2IyUia0ciIiE=

y(x,n)[f=q]:=q+Sum((2*q*sin(Pi*k)/Pi/k)*cos(k*x),k=1..n);

NiM+Ji0lInlHNiQlInhHJSJuRzYjLyUiZkclInFHLCYlInFHIiIiLSUkU3VtRzYkLCQqLiIiIyIiIiUicUciIiItJSRzaW5HNiMqJiUjUGlHIiIiJSJrRyIiIiIiIiUjUGlHISIiJSJrRyEiIi0lJGNvc0c2IyomJSJrRyIiIiUieEciIiIiIiIiIiIvJSJrRzsiIiIlIm5HIiIi

y(x,n)[f=q]:= simplify(subs({sin(Pi*k)=0,cos(Pi*k)=(-1)^k},%));

NiM+Ji0lInlHNiQlInhHJSJuRzYjLyUiZkclInFHJSJxRw==

The solution is trivial. For all k yields: y(x, k...n) = q. another_example:=f=1+x;

NiM+JTBhbm90aGVyX2V4YW1wbGVHLyUiZkcsJiIiIiIiIiUieEciIiI=

A[0]:=value(subs(f(x)=1+x,a[0]));

NiM+JiUiQUc2IyIiISIiIw==

A[k]:=value(subs(f(x)=1+x,a[k]));

NiM+JiUiQUc2IyUia0csJCoqIiIjIiIiJSNQaUchIiIlImtHISIiLSUkc2luRzYjKiYlI1BpRyIiIiUia0ciIiIiIiIiIiI=

limit_value:=Limit(A[k],k=0)=limit(2*sin(Pi*k)/Pi/k,k=0);

NiM+JSxsaW1pdF92YWx1ZUcvLSUmTGltaXRHNiQsJCoqIiIjIiIiJSNQaUchIiIlImtHISIiLSUkc2luRzYjKiYlI1BpRyIiIiUia0ciIiIiIiIiIiIvJSJrRyIiISIiIw==

B[k]:=value(subs(f(x)=1+x,b[k]));

NiM+JiUiQkc2IyUia0csJCoqIiIjIiIiJSNQaUchIiIsJi0lJHNpbkc2IyomJSNQaUciIiIlImtHIiIiISIiKigtJSRjb3NHNiMqJiUjUGlHIiIiJSJrRyIiIiIiIiUia0ciIiIlI1BpRyIiIiIiIiIiIiklImtHIiIjISIiISIi

B[k]:=subs({sin(Pi*k)=0,cos(Pi*k)=(-1)^k},%);

NiM+JiUiQkc2IyUia0csJCooIiIjIiIiKSEiIiUia0ciIiIlImtHISIiISIi

y(x,n)[f=1+x]:=1+Sum(B[k]*sin(k*x),k=1..n);

NiM+Ji0lInlHNiQlInhHJSJuRzYjLyUiZkcsJiIiIiIiIiUieEciIiIsJiIiIiIiIi0lJFN1bUc2JCwkKioiIiMiIiIpISIiJSJrRyIiIiUia0chIiItJSRzaW5HNiMqJiUia0ciIiIlInhHIiIiIiIiISIiLyUia0c7IiIiJSJuRyIiIg==

y(x,4)[f=1+x]:=value(subs(n=4,%));

NiM+Ji0lInlHNiQlInhHIiIlNiMvJSJmRywmIiIiIiIiJSJ4RyIiIiwsIiIiIiIiKiYiIiMiIiItJSRzaW5HNiMlInhHIiIiIiIiLSUkc2luRzYjLCQqJiIiIyIiIiUieEciIiIiIiIhIiIqJiMiIiMiIiQiIiItJSRzaW5HNiMsJComIiIkIiIiJSJ4RyIiIiIiIiIiIiIiIiomIyIiIiIiIyIiIi0lJHNpbkc2IywkKiYiIiUiIiIlInhHIiIiIiIiIiIiISIi

for i in [1,4,100] do y(x,n=i):=1+subs(n=i,sum(2*sin(Pi*n)*cos(n*x)/Pi/n- (2*(-sin(Pi*n)+cos(Pi*n)*Pi*n)*sin(n*x))/Pi/n^2, n=1..i)) od: plot({1,y(x,n=1),y(x,n=4),y(x,n=100)},x=-4*Pi..4*Pi,co=black, title="FOURIER series # n = [1, 4, 100] for f(x) = 1+x");

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FOURIER Representation of the HEAVISIDE Function alias(H=Heaviside): F(x)[H_odd]:=H(x)+2*sum((-1)^n*H(x-n*Pi),n=1..N)- H(-x)+2*sum((-1)^n*H(x+n*Pi),n=1..N);

NiM+Ji0lIkZHNiMlInhHNiMlJkhfb2RkRywqLSUiSEc2IyUieEciIiIqJiIiIyIiIi0lJHN1bUc2JComKSEiIiUibkciIiItJSJIRzYjLCYlInhHIiIiKiYlI1BpRyIiIiUibkciIiIhIiIiIiIvJSJuRzsiIiIlIk5HIiIiIiIiLSUiSEc2IywkJSJ4RyEiIiEiIiomIiIjIiIiLSUkc3VtRzYkKiYpISIiJSJuRyIiIi0lIkhHNiMsJiUieEciIiIqJiUjUGlHIiIiJSJuRyIiIiIiIiIiIi8lIm5HOyIiIiUiTkciIiIiIiI=

f(x)[H_odd]:=subs(N=4,%);

NiM+Ji0lImZHNiMlInhHNiMlJkhfb2RkRywqLSUiSEc2IyUieEciIiIqJiIiIyIiIi0lJHN1bUc2JComKSEiIiUibkciIiItJSJIRzYjLCYlInhHIiIiKiYlI1BpRyIiIiUibkciIiIhIiIiIiIvJSJuRzsiIiIiIiUiIiIiIiItJSJIRzYjLCQlInhHISIiISIiKiYiIiMiIiItJSRzdW1HNiQqJikhIiIlIm5HIiIiLSUiSEc2IywmJSJ4RyIiIiomJSNQaUciIiIlIm5HIiIiIiIiIiIiLyUibkc7IiIiIiIlIiIiIiIi

plot(%,x=-4.5*Pi..4.5*Pi,co=black);

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

A[0]:=value(subs(f(x)=f(x)[H_odd],a[0]));

NiM+JiUiQUc2IyIiISIiIQ==

A[k]:=value(simplify(subs(f(x)=f(x)[H_odd],a[k])));

NiM+JiUiQUc2IyUia0ciIiE=

B[k]:=simplify(value(subs(f(x)=f(x)[H_odd],b[k])));

NiM+JiUiQkc2IyUia0csJCoqIiIjIiIiLCYtJSRjb3NHNiMqJiUjUGlHIiIiJSJrRyIiIiIiIiIiIiEiIiIiIiUjUGlHISIiJSJrRyEiIiEiIg==

B[k]:=subs(cos(Pi*k)=(-1)^k,%);

NiM+JiUiQkc2IyUia0csJCoqIiIjIiIiLCYpISIiJSJrRyIiIiIiIiEiIiIiIiUjUGlHISIiJSJrRyEiIiEiIg==

y(x,n)[f=H_odd]:=sum(B[k]*sin(k*x),k=1..n);

NiM+Ji0lInlHNiQlInhHJSJuRzYjLyUiZkclJkhfb2RkRy0lJHN1bUc2JCwkKiwiIiMiIiIsJikhIiIlImtHIiIiIiIiISIiIiIiJSNQaUchIiIlImtHISIiLSUkc2luRzYjKiYlImtHIiIiJSJ4RyIiIiIiIiEiIi8lImtHOyIiIiUibkc=

y(x,4)[H_odd]:=value(subs(n=4,%));

NiM+Ji0lInlHNiQlInhHIiIlNiMlJkhfb2RkRy0lJHN1bUc2JCwkKiwiIiMiIiIsJikhIiIlImtHIiIiIiIiISIiIiIiJSNQaUchIiIlImtHISIiLSUkc2luRzYjKiYlImtHIiIiJSJ4RyIiIiIiIiEiIi8lImtHOyIiIiIiJQ==

for i in [1,2,3,4,5,6] do y(x,n=i):=subs(n=i,2*sum(((1-(-1)^n)/Pi/n)*sin(n*x), n=1..i)) od;

NiM+LSUieUc2JCUieEcvJSJuRyIiIiwkKigiIiUiIiItJSRzaW5HNiMlInhHIiIiJSNQaUchIiIiIiI= NiM+LSUieUc2JCUieEcvJSJuRyIiIywkKigiIiUiIiItJSRzaW5HNiMlInhHIiIiJSNQaUchIiIiIiI= NiM+LSUieUc2JCUieEcvJSJuRyIiJCwmKigiIiUiIiItJSRzaW5HNiMlInhHIiIiJSNQaUchIiIiIiIqJiMiIiUiIiQiIiIqJiUjUGlHISIiLSUkc2luRzYjLCQqJiIiJCIiIiUieEciIiIiIiIiIiIiIiIiIiI= NiM+LSUieUc2JCUieEcvJSJuRyIiJSwmKigiIiUiIiItJSRzaW5HNiMlInhHIiIiJSNQaUchIiIiIiIqJiMiIiUiIiQiIiIqJiUjUGlHISIiLSUkc2luRzYjLCQqJiIiJCIiIiUieEciIiIiIiIiIiIiIiIiIiI= NiM+LSUieUc2JCUieEcvJSJuRyIiJiwoKigiIiUiIiItJSRzaW5HNiMlInhHIiIiJSNQaUchIiIiIiIqJiMiIiUiIiQiIiIqJiUjUGlHISIiLSUkc2luRzYjLCQqJiIiJCIiIiUieEciIiIiIiIiIiIiIiIiIiIqJiMiIiUiIiYiIiIqJiUjUGlHISIiLSUkc2luRzYjLCQqJiIiJiIiIiUieEciIiIiIiIiIiIiIiIiIiI= NiM+LSUieUc2JCUieEcvJSJuRyIiJywoKigiIiUiIiItJSRzaW5HNiMlInhHIiIiJSNQaUchIiIiIiIqJiMiIiUiIiQiIiIqJiUjUGlHISIiLSUkc2luRzYjLCQqJiIiJCIiIiUieEciIiIiIiIiIiIiIiIiIiIqJiMiIiUiIiYiIiIqJiUjUGlHISIiLSUkc2luRzYjLCQqJiIiJiIiIiUieEciIiIiIiIiIiIiIiIiIiI=

for i in [1,5,99] do y(x,n=i):=subs(n=i,2*sum(((1-(-1)^n)/Pi/n)*sin(n*x), n=1..i)) od: plot({f(x)[H_odd],y(x,n=1),y(x,n=5),y(x,n=99)}, x=-4.5*Pi..4.5*Pi,co=black);

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Interval ( 0, 2*Pi ): y(x):=alpha[0]/2+ Sum(alpha[k]*cos(k*x)+beta[k]*sin(k*x),k=1..infinity);

NiM+LSUieUc2IyUieEcsJiomIyIiIiIiIyIiIiYlJmFscGhhRzYjIiIhIiIiIiIiLSUkU3VtRzYkLCYqJiYlJmFscGhhRzYjJSJrRyIiIi0lJGNvc0c2IyomJSJrRyIiIiUieEciIiIiIiIiIiIqJiYlJWJldGFHNiMlImtHIiIiLSUkc2luRzYjKiYlImtHIiIiJSJ4RyIiIiIiIiIiIi8lImtHOyIiIiUpaW5maW5pdHlHIiIi

alpha[k]:=(1/Pi)*Int(phi(x)*cos(k*x),x=0..2*Pi); # k = 0,1,2,3,...

NiM+JiUmYWxwaGFHNiMlImtHLCQqJiokJSNQaUchIiIiIiItJSRJbnRHNiQqJi0lJHBoaUc2IyUieEciIiItJSRjb3NHNiMqJiUia0ciIiIlInhHIiIiIiIiLyUieEc7IiIhLCQqJiIiIyIiIiUjUGlHIiIiIiIiIiIiIiIi

alpha[0]:=simplify(subs(k=0,%));

NiM+JiUmYWxwaGFHNiMiIiEsJComKiQlI1BpRyEiIiIiIi0lJEludEc2JC0lJHBoaUc2IyUieEcvJSJ4RzsiIiEsJComIiIjIiIiJSNQaUciIiIiIiIiIiIiIiI=

beta[k]:=(1/Pi)*Int(phi(x)*sin(k*x),x=0..2*Pi); # k = 1,2,3,...

NiM+JiUlYmV0YUc2IyUia0csJComKiQlI1BpRyEiIiIiIi0lJEludEc2JComLSUkcGhpRzYjJSJ4RyIiIi0lJHNpbkc2IyomJSJrRyIiIiUieEciIiIiIiIvJSJ4RzsiIiEsJComIiIjIiIiJSNQaUciIiIiIiIiIiIiIiI=

alias(H=Heaviside,th=thickness,co=color): phi(x,N):=H(x)+2*Sum((-1)^n*H(x-n*Pi),n=1..N);

NiM+LSUkcGhpRzYkJSJ4RyUiTkcsJi0lIkhHNiMlInhHIiIiKiYiIiMiIiItJSRTdW1HNiQqJikhIiIlIm5HIiIiLSUiSEc2IywmJSJ4RyIiIiomJSNQaUciIiIlIm5HIiIiISIiIiIiLyUibkc7IiIiJSJORyIiIiIiIg==

phi(x,4):=subs(N=4,%);

NiM+LSUkcGhpRzYkJSJ4RyIiJSwmLSUiSEc2IyUieEciIiIqJiIiIyIiIi0lJFN1bUc2JComKSEiIiUibkciIiItJSJIRzYjLCYlInhHIiIiKiYlI1BpRyIiIiUibkciIiIhIiIiIiIvJSJuRzsiIiIiIiUiIiIiIiI=

plot(%,x=0..4.5*Pi,co=black);

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

Alpha[0]:=value(subs({phi(x)=phi(x,4),k=0}, alpha[k]));

NiM+JiUmQWxwaGFHNiMiIiEiIiE=

Alpha[k]:=simplify(value(subs(phi(x)=phi(x,4), alpha[k])));

NiM+JiUmQWxwaGFHNiMlImtHLCQqLCIiIyIiIi0lJHNpbkc2IyomJSNQaUciIiIlImtHIiIiIiIiLCYtJSRjb3NHNiMqJiUjUGlHIiIiJSJrRyIiIiIiIiIiIiEiIiIiIiUjUGlHISIiJSJrRyEiIiEiIg==

A[k]:=subs({sin(Pi*k)=0,cos(Pi*k)=(-1)^k},%);

NiM+JiUiQUc2IyUia0ciIiE=

BETA[k]:=simplify(value(subs(phi(x)=phi(x,4), beta[k])));

NiM+JiUlQkVUQUc2IyUia0csJCosIiIjIiIiLSUkY29zRzYjKiYlI1BpRyIiIiUia0ciIiIiIiIsJi0lJGNvc0c2IyomJSNQaUciIiIlImtHIiIiIiIiIiIiISIiIiIiJSNQaUchIiIlImtHISIiIiIi

BETA[k]:=subs(cos(Pi*k)=(-1)^k,%);

NiM+JiUlQkVUQUc2IyUia0csJCosIiIjIiIiKSEiIiUia0ciIiIsJikhIiIlImtHIiIiIiIiISIiIiIiJSNQaUchIiIlImtHISIiIiIi

y(x,n):=Sum(BETA[k]*sin(k*x),k=1..n);

NiM+LSUieUc2JCUieEclIm5HLSUkU3VtRzYkLCQqLiIiIyIiIikhIiIlImtHIiIiLCYpISIiJSJrRyIiIiIiIiEiIiIiIiUjUGlHISIiJSJrRyEiIi0lJHNpbkc2IyomJSJrRyIiIiUieEciIiIiIiIiIiIvJSJrRzsiIiIlIm5H

y(x,4):=value(subs(n=4,%));

NiM+LSUieUc2JCUieEciIiUsJiooIiIlIiIiLSUkc2luRzYjJSJ4RyIiIiUjUGlHISIiIiIiKiYjIiIlIiIkIiIiKiYlI1BpRyEiIi0lJHNpbkc2IywkKiYiIiQiIiIlInhHIiIiIiIiIiIiIiIiIiIi

for i in [1,3,99] do y(x,n=i):=subs(n=i,y(x,n)) od: plot({y(x,n=1),y(x,n=3),y(x,n=99)}, x=-4*Pi..8*Pi,co=black);

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Continuous Functions with Cusps h(x):=piecewise(x>=0 and x<=1,x,x>=1 and x<=2*Pi,1);

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

plot(h(x),x=0..2*Pi,co=black);

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

Alpha[k]:=simplify(value(subs(phi(x)=h(x),alpha[k])));

NiM+JiUmQWxwaGFHNiMlImtHKigsKCIiIiEiIi0lJGNvc0c2IyUia0ciIiIqKiIiIyIiIiUia0ciIiItJSRzaW5HNiMqJiUjUGlHIiIiJSJrRyIiIiIiIi0lJGNvc0c2IyomJSNQaUciIiIlImtHIiIiIiIiIiIiIiIiKSUia0ciIiMhIiIlI1BpRyEiIg==

Alpha[k]:=subs(sin(k*Pi)=0,%);

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Alpha[0]:=simplify(value(subs(phi(x)=h(x),alpha[0])));

NiM+JiUmQWxwaGFHNiMiIiEsJCooIiIjISIiLCYiIiIhIiIqJiIiJSIiIiUjUGlHIiIiIiIiIiIiJSNQaUchIiIiIiI=

BETA[k]:=simplify(value(subs(phi(x)=h(x),beta[k])));

NiM+JiUlQkVUQUc2IyUia0csJCooLCgtJSRzaW5HNiMlImtHISIiKigiIiMiIiIlImtHIiIiKS0lJGNvc0c2IyomJSNQaUciIiIlImtHIiIiIiIjIiIiIiIiJSJrRyEiIiIiIiklImtHIiIjISIiJSNQaUchIiIhIiI=

BETA[k]:=subs((cos(k*Pi))^2=1,%);

NiM+JiUlQkVUQUc2IyUia0csJCooLCYtJSRzaW5HNiMlImtHISIiJSJrRyIiIiIiIiklImtHIiIjISIiJSNQaUchIiIhIiI=

y(x,n):=Alpha[0]/2+sum(Alpha[k]*cos(k*x)+ BETA[k]*sin(k*x),k=1..n);

NiM+LSUieUc2JCUieEclIm5HLCYqKCIiJSEiIiwmIiIiISIiKiYiIiUiIiIlI1BpRyIiIiIiIiIiIiUjUGlHISIiIiIiLSUkc3VtRzYkLCYqKiwmIiIiISIiLSUkY29zRzYjJSJrRyIiIiIiIiklImtHIiIjISIiJSNQaUchIiItJSRjb3NHNiMqJiUia0ciIiIlInhHIiIiIiIiIiIiKiosJi0lJHNpbkc2IyUia0chIiIlImtHIiIiIiIiKSUia0ciIiMhIiIlI1BpRyEiIi0lJHNpbkc2IyomJSJrRyIiIiUieEciIiIiIiIhIiIvJSJrRzsiIiIlIm5HIiIi

y(x,1):=evalf(subs(n=1,%),4);

NiM+LSUieUc2JCUieEciIiIsKCQiJTAjKiEiJSIiIiomJCIlajkhIiUiIiItJSRjb3NHNiMlInhHIiIiISIiKiYkIiVZXSEiJiIiIi0lJHNpbkc2IyUieEciIiIhIiI=

y(x,3):=evalf(subs(n=3,%%),4);

NiM+LSUieUc2JCUieEciIiQsMCQiJTAjKiEiJSIiIiomJCIlajkhIiUiIiItJSRjb3NHNiMlInhHIiIiISIiKiYkIiVZXSEiJiIiIi0lJHNpbkc2IyUieEciIiIhIiIqJiQiJUY2ISIlIiIiLSUkY29zRzYjLCQqJiQiIiMiIiEiIiIlInhHIiIiIiIiIiIiISIiKiYkIiUhbykhIiYiIiItJSRzaW5HNiMsJComJCIiIyIiISIiIiUieEciIiIiIiIiIiIhIiIqJiQiJVFxISImIiIiLSUkY29zRzYjLCQqJiQiIiQiIiEiIiIlInhHIiIiIiIiIiIiISIiKiYkIiU2NSEiJSIiIi0lJHNpbkc2IywkKiYkIiIkIiIhIiIiJSJ4RyIiIiIiIiIiIiEiIg==

y(x,99):=evalf(subs(n=99,%%%),4): for i in [1,3,99] do y(x,n=i):=subs(n=i,y(x,n)) od: plot({y(x,n=1),y(x,n=3),y(x,n=99)}, x=0..2*Pi,color=black);

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L-two Norm L_two[n]:=sqrt((1/2/Pi)*Int((H(x)-Y(x,n))^2, x=0..2*Pi));

NiM+JiUmTF90d29HNiMlIm5HLCQqJiMiIiIiIiMiIiIqJikiIiMjIiIiIiIjIiIiKSwkKiYqJCUjUGlHISIiIiIiLSUkSW50RzYkKiQpLCYtJSJIRzYjJSJ4RyIiIi0lIllHNiQlInhHJSJuRyEiIiIiIyIiIi8lInhHOyIiISwkKiYiIiMiIiIlI1BpRyIiIiIiIiIiIiIiIiMiIiIiIiMiIiIiIiIiIiI=

for i in [1,3,99] do L_two[n=i]:=evalf(sqrt((1/2/Pi)*value(int((h(x)-y(x,i))^2, x=0..2*Pi))),4) od;

NiM+JiUmTF90d29HNiMvJSJuRyIiIiQiJWw9ISIl NiM+JiUmTF90d29HNiMvJSJuRyIiJCQiJTA4ISIl NiM+JiUmTF90d29HNiMvJSJuRyIjKiokIiVkQSEiJg==

For n = 99 the FOURIER series y(x, n = 99) represents a good approximation to the above given function h(x). g(x):=piecewise(x<-1,-1,x>-1 and x<1,x,x>1,1);

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

plot(g(x),x=-Pi..Pi,co=black);

NictJSdDVVJWRVNHNiM3XW83JCQhLUpaRWZUSiEjNiQhIzUhIiI3JCQhMj0qcHIpM1BZKyQhIzskISM1ISIiNyQkITEzRVsqeXNhKUchIzokISM1ISIiNyQkITJjYSkpPjJnOXYjISM7JCEjNSEiIjckJCExLEBASiFmbGgjISM6JCEjNSEiIjckJCEyPFFNOjoqSCNbIyEjOyQhIzUhIiI3JCQhMEQ3bTBCeU4jISM5JCEjNSEiIjckJCExJz1kLUZOKkdBISM6JCEjNSEiIjckJCEyWipHQlAkUmM0IyEjOyQhIzUhIiI3JCQhMlpLY2YpM3hpPiEjOyQhIzUhIiI3JCQhMnByNEFOKjRFPSEjOyQhIzUhIiI3JCQhMiVRUSozKio9ZHEiISM7JCEjNSEiIjckJCEyMT5qZ0AqPnE6ISM7JCEjNSEiIjckJCEybGg7UylIN005ISM7JCEjNSEiIjckJCEyeTRGLEspKUhJIiEjOyQhIzUhIiI3JCQhMiUqNCFSI1swUj0iISM7JCEjNSEiIjckJCExQmhRd1w1ODYhIzokISM1ISIiNyQkITJuOSNRcVdJVTUhIzskISM1ISIiNyQkITIjKlExOjk1dC0iISM7JCEjNSEiIjckJCEyPWpJRSJlSjc1ISM7JCEjNSEiIjckJCExYEY+Wyc9WysiISM6JCEjNSEiIjckJCExVihbdiRbQHQqKiEjOyQhMVYoW3YkW0B0KiohIzs3JCQhMWIqcEo+ViMpKikqISM7JCExYipwSj5WIykqKSohIzs3JCQhMW82els6RkIpKiEjOyQhMW82els6RkIpKiEjOzckJCExPmdGclxRQiYqISM7JCExPmdGclxRQiYqISM7NyQkITFwM3ckUilcQiMqISM7JCExcDN3JFIpXEIjKiEjOzckJCExSmM4bzM5R3khIzskITFKYzhvMzlHeSEjOzckJCExI1tCUC04SWYnISM7JCExI1tCUC04SWYnISM7NyQkITFCMzxAPyl5QiYhIzskITFCMzxAPyl5QiYhIzs3JCQhMiYqKVI8WW9aWlIhIzwkITImKilSPFlvWlpSISM8NyQkITFUcVg9VzIsRSEjOyQhMVRxWD1XMixFISM7NyQkITI8OWolKW9jWU8iISM8JCEyPDlqJSlvY1lPIiEjPDckJCExLHo3VktKLEohIz0kITEsejdWS0osSiEjPTckJCIyLHUjZjN4RWE4ISM8JCIyLHUjZjN4RWE4ISM8NyQkIjF1cSlIdmUsYyMhIzskIjF1cSlIdmUsYyMhIzs3JCQiMmpsKFI8VGJpUSEjPCQiMmpsKFI8VGJpUSEjPDckJCIxIio9Sm9mMDNfISM7JCIxIio9Sm9mMDNfISM7NyQkIjFeMVhJcU9DbCEjOyQiMV4xWElxT0NsISM7NyQkIjEkb0JsbWx6eighIzskIjEkb0JsbWx6eighIzs3JCQiMXYnZiM0KXo/QCohIzskIjF2J2YjNCl6P0AqISM7NyQkIjF6JilRQTR1SCYqISM7JCIxeiYpUUE0dUgmKiEjOzckJCIxJFs8Yi4tdSUpKiEjOyQiMSRbPGIuLXUlKSohIzs3JCQiMHIqejh0Im8jKiohIzokIjByKno4dCJvIyoqISM6NyQkIjJPPjMjZktpKzUhIzskIiM1ISIiNyQkIjJqVE9xeWsmMzUhIzskIiM1ISIiNyQkIjIoUVknW0oxbCwiISM7JCIjNSEiIjckJCIyUTNAMFAqUUs1ISM7JCIjNSEiIjckJCIyIkh2PEVDRls1ISM7JCIjNSEiIjckJCIycyU+Xk1tNTs2ISM7JCIjNSEiIjckJCIyYU9ZRyUzJVI9IiEjOyQiIzUhIiI3JCQiMkQhKVJFZndvSSIhIzskIiM1ISIiNyQkIjJZdF0oUXlGVDkhIzskIiM1ISIiNyQkIjJ5MzJzJCpReGMiISM7JCIjNSEiIjckJCIvSz9CcyMqKnAiISM4JCIjNSEiIjckJCIyQUgkZU1hO0g9ISM7JCIjNSEiIjckJCIxIz5DWidlWWs+ISM6JCIjNSEiIjckJCIybVcoeWZpeCU0IyEjOyQiIzUhIiI3JCQiMiU0cSkpZnUuR0EhIzskIiM1ISIiNyQkIjFAdycqKj0mPmdCISM6JCIjNSEiIjckJCIwIzNhPVdqIlsjISM5JCIjNSEiIjckJCIyLWkmKil5dSIzaSMhIzskIiM1ISIiNyQkIjFpLUhuV0lYRiEjOiQiIzUhIiI3JCQiMik9UldoTi55RyEjOyQiIzUhIiI3JCQiMkZ4KlJTQTIwSSEjOyQiIzUhIiI3JCQiLVhdRWZUSiEjNiQiIzUhIiItJSVWSUVXRzYkOyQhK2FFZlRKISIqJCIrYUVmVEohIiolKERFRkFVTFRHLSUrQVhFU0xBQkVMU0c2J1EieDYiUSE2Ii0lJUZPTlRHNiUlKERFRkFVTFRHJShERUZBVUxURyIjNSUrSE9SSVpPTlRBTEclK0hPUklaT05UQUxHLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiMhKiEiIi0lKUJPVU5EU19ZRzYjJCIkPyIhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVJPiEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiQ1KSEiIi0lKUNISUxEUkVORzYi

A[k]:=simplify(value(subs(f(x)=g(x),a[k])));

NiM+JiUiQUc2IyUia0ciIiE=

A[0]:=value(subs(f(x)=g(x),a[0]));

NiM+JiUiQUc2IyIiISIiIQ==

B[k]:=simplify(value(subs(f(x)=g(x),b[k])));

NiM+JiUiQkc2IyUia0csJCoqIiIjIiIiLCYqJi0lJGNvc0c2IyomJSNQaUciIiIlImtHIiIiIiIiJSJrRyIiIiIiIi0lJHNpbkc2IyUia0chIiIiIiIpJSJrRyIiIyEiIiUjUGlHISIiISIi

B[k]:=subs(cos(Pi*k)=(-1)^k,%);

NiM+JiUiQkc2IyUia0csJCoqIiIjIiIiLCYqJikhIiIlImtHIiIiJSJrRyIiIiIiIi0lJHNpbkc2IyUia0chIiIiIiIpJSJrRyIiIyEiIiUjUGlHISIiISIi

y(x,n):=sum(B[k]*sin(k*x),k=1..n);

NiM+LSUieUc2JCUieEclIm5HLSUkc3VtRzYkLCQqLCIiIyIiIiwmKiYpISIiJSJrRyIiIiUia0ciIiIiIiItJSRzaW5HNiMlImtHISIiIiIiKSUia0ciIiMhIiIlI1BpRyEiIi0lJHNpbkc2IyomJSJrRyIiIiUieEciIiIiIiIhIiIvJSJrRzsiIiIlIm5H

y(x,1):=evalf(value(subs(n=1,%)),4);

NiM+LSUieUc2JCUieEciIiIsJComJCIlczYhIiQiIiItJSRzaW5HNiMlInhHIiIiIiIi

y(x,3):=evalf(value(subs(n=3,%%)),4);

NiM+LSUieUc2JCUieEciIiQsKComJCIlczYhIiQiIiItJSRzaW5HNiMlInhHIiIiIiIiKiYkIiVPPCEiJSIiIi0lJHNpbkc2IywkKiYkIiIjIiIhIiIiJSJ4RyIiIiIiIiIiIiEiIiomJCIlQUEhIiUiIiItJSRzaW5HNiMsJComJCIiJCIiISIiIiUieEciIiIiIiIiIiIiIiI=

for i in[1,3,99] do y(x,n=i):=subs(n=i,y(x,n)) od: plot({y(x,n=1),y(x,n=3),y(x,n=99)}, x=-2*Pi..2*Pi,co=black);

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L_two[n]:=sqrt((1/Pi)*Int((G(x)-Y(x,n))^2,x=0..Pi));

NiM+JiUmTF90d29HNiMlIm5HKiQpLCQqJiokJSNQaUchIiIiIiItJSRJbnRHNiQqJCksJi0lIkdHNiMlInhHIiIiLSUiWUc2JCUieEclIm5HISIiIiIjIiIiLyUieEc7IiIhJSNQaUciIiIiIiIjIiIiIiIjIiIi

for i in [1,3,99] do L_two[n=i]:=evalf(sqrt((1/Pi)*int((g(x)-y(x,i))^2, x=0..Pi)),4) od;

NiM+JiUmTF90d29HNiMvJSJuRyIiIiQiJXNKISIl NiM+JiUmTF90d29HNiMvJSJuRyIiJCQiJW9DISIl NiM+JiUmTF90d29HNiMvJSJuRyIjKiokIiVrQSEiJg==

For n = 99 the FOURIER series y(x, n = 99) represents a good approximation to the above given function g(x). Legal Notice: The copyright for this application is owned by the author. Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.