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1 2 2 2 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Hel vetica" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 262 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu t" -1 263 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 256 19 "Fourier Polynomials" }} {PARA 258 "" 0 "" {TEXT 257 14 "Brian E. Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 83 "Click on a [+] \+ sign to expand a section. Click on a [-] sign to collapse a section." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 2 " I" }{TEXT 271 11 "ntroduction " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 7 "In th is" }{TEXT -1 2 " " }{TEXT 258 5 "MAPLE" }{TEXT -1 4 " " } {HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT 268 14 ", we will us e" }{TEXT -1 2 " " }{TEXT 259 5 "MAPLE" }{TEXT -1 3 " " }{TEXT 267 33 "to calculate Fourier polynomials." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 262 "" 0 "" {TEXT 278 9 " Code for" }{TEXT 277 2 " " }{TEXT 276 14 "fourierPolynom" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 379 74 "This fun ction is not a builtin. To use it you must execute the next group." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6964 "fourierPolynom := proc()\n\noption `Copyright (c) 2000 by Brian \+ E. Blank.\nAll rights to the code are reserved.\nHowever, all use of t he code including distribution is permitted provided this copyright no tice is not removed.`:\n\n############################################ ###############\n# Description of Local Variables \+ #\n# # \n # degreeFlag: Inititialized to 0, set to 1 if the second # \+ # argument, the degree of the Fourier polyn omial, #\n# is not numeric # \n# # \n# func tionFlag: Inititialized to 0, set to 1 if the #\n# \+ function does not evaluate to numeric #\n# \+ # \n# rangeFlag: Inititialized to 0, set to 1 if the third # # arg ument a..b does not have numeric endpoints #\n# \+ #\n# \+ # \n# ff: the name given to the first \+ argument #\n# \+ #\n# NN: the name given to the second argument, the \+ #\n# degree of the Fourier polynomial # \n# \+ #\n# aa: th e lower limit of integration # \n# \+ #\n# bb: the upper limit of \+ integration # \n# \+ #\n# LL: the length of the interval of integr ation #\n# \+ #\n# jj: index of summation #\n# \+ #\n# AA: th e coefficients of the cosine polynomials #\n# \+ # \n# BB: the coefficients of the sine polynomials # \n# \+ # ######################################### ################## \n\nlocal ff, NN, xx, aa , bb, LL, jj, AA, BB, \n \+ degreeFlag, functionFlag, rangeFlag, t:\ndegreeFlag := 0:\nfuncti onFlag := 0:\nrangeFlag := 0:\n\n##################################### ######################\n# \+ #\n# Begin general argument checking \+ #\n# #\n### ########################################################\nif nargs < 3 or nargs > 4 then \n ERROR(`fourierPolynom expects 3 or 4 argument s.`);\nelif not type(args[1],\{name,procedure,function\}) \n and not type(args[1],realcons) then \n ERROR(`fourierPolynom expects its f irst argument to be the name of a function.`);\nelif type(args[2],alge braic) and not type(args[2],numeric)\n then degreeFlag := 1:\nelif n ot type(args[2],nonnegint) \n then ERROR(`fourierPolynom expects it s second argument to be a nonnegative integer`);\nfi:\n############### ############################################\n# \+ #\n# End general argument checking #\n# \+ #\n################################################### ########\n\n\n######################################################## ###\n# #\n# \+ Set local variables in three argument case #\n# \+ #\n###################### #####################################\nif nargs = 3 then\n if not ty pe(args[3],algebraic) then\n ERROR(`when called with three arguments fourierPolynom expects its third argument to be an algebraic expressi on`);\n fi:\nff := args[1]:\nNN := args[2]:\nxx := args[3]:\naa := - Pi:\nbb := Pi:\nLL := bb-aa:\nfi:\n\n################################# ##########################\n# \+ #\n# Set local variables in four argument case \+ #\n# # \n###########################################################\nif narg s = 4 then\n\n if not type(args[4],algebraic) then\n ERROR(`when c alled with four arguments fourierPolynom expects its fourth argument t o be an algebraic expression`);\n fi:\n\n\n if not type(args[3],ra nge) then\n ERROR(`when called with four arguments fourierPolynom ex pects its third argument to be a range a..b`);\n\n\n\nelif not type(op (1,args[3]),realcons) or not type(op(2,args[3]),realcons) then\n ran geFlag := 1;\n ff := args[1]:\n NN := args[2]:\n xx := args[4]: \n aa := op(1,args[3]):\n bb := op(2,args[3]):\n LL := bb-aa: \+ \n elif evalf(op(2,args[3])) <= evalf(op(1,args[3])) then\n ERROR(`whe n called with four arguments fourierPolynom expects its third argument to be a range a..b with b > a`);\nelse\n ff := args[1]:\n NN := a rgs[2]:\n xx := args[4]:\n aa := op(1,args[3]):\n bb := op(2,arg s[3]):\n LL := bb-aa:\nfi:\nfi:\n\nif rangeFlag = 0 and (not type(ev alf(ff( (aa+bb)/3)), realcons) or not type(evalf(ff( (sqrt(2)*aa+sq rt(3)*bb)/(sqrt(2)+sqrt(3)))), realcons)) \n then functionFlag := 1: \nfi:\n\n########################################################### \n# #\n# \+ Case of symbolic degree #\n# \+ #\n######################### ##################################\n\nif degreeFlag = 1 then\nRETURN( \+ 1/LL*Int(args[1](tau),tau=aa..bb)+2/LL*Sum(Int(args[1](tau)*cos(2*Pi/L L*j*tau),tau=aa..bb)*cos(2*Pi/LL*j*xx)+ Int(args[1](tau)*sin(2*Pi/LL*j *tau),tau=aa..bb)*sin(2*Pi/LL*j*xx),j=1..NN));\nfi;\n\n############### ############################################\n# \+ #\n# Calculate Coeffic ients #\n# \+ #\n################################################### ########\n \n\nfor jj from 0 to NN do\n AA[jj] := 2/LL*int(ff(tau)*co s(2*Pi/LL*jj*tau),tau=aa..bb):\n BB[jj] := 2/LL*int(ff(tau)*sin(2*Pi/ LL*jj*tau),tau=aa..bb):\n if functionFlag = 0 and rangeFlag = 0 the n\n if has(AA[jj],int) then \n AA[jj] := evalf(AA[jj ]); \n fi:\n if has(BB[jj],int) then \n BB[jj] := evalf(BB[jj]): \n fi:\nfi:\nod:\n\njj := 'jj':\n\n\n####### ####################################################\n# \+ #\n# Gener ate Return #\n# \+ #\n########################################### ################\n\n RETURN(AA[0]/2 + sum(AA[jj]*cos(2*Pi/LL*jj*xx )+\n BB[jj]*sin(2*Pi/LL*jj*xx),jj=1..NN));\nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT 273 1 " " }{TEXT 275 14 "Description of " }{TEXT 274 2 " " }{TEXT 272 14 "fourierPolynom" }}{PARA 3 "" 0 "" {TEXT -1 14 "fourierPolynom" }{TEXT 30 33 " - Calculates Fourier polyn omials" }}{PARA 4 "" 0 "usage" {TEXT -1 18 "Calling Sequences:" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 279 50 "fourierPolynom(f,N,t )\n fourierPolynom(f,N,a..b,t)" }}{PARA 4 "" 0 "" {TEXT -1 11 "Parame ters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 48 "f - a \+ name (of a function) or a procedure " }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }{TEXT 23 71 "N - a nonnegative integer (the degree of the \+ Fourier polynomial) " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 74 "t - an algebraic expression (the argument of the Fourier pol ynomial)" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 28 "a..b - a range (optional)" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT 321 1 "\n" }{TEXT 280 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 9 "The call " }{TEXT 281 30 " fourierPolynom(f, N, a..b, t)" }{TEXT -1 22 " returns the degree " }{TEXT 351 1 "N" }{TEXT -1 25 " \+ Fourier polynomial of " }{TEXT 282 1 "f" }{TEXT -1 14 " evaluated at " }{TEXT 352 1 "t" }{TEXT -1 58 ". To be specific, the returned Fouri er polynomial is\n\n " }{TEXT 355 6 " " }{TEXT 357 4 " " } {XPPEDIT 356 1 "a[0]/2 + Sum(a[k]*cos(2*Pi/(b-a)*k*t)+b[k]*sin(2*Pi*k* t/(b-a)),k=1..N)" ",&*&&%\"aG6#\"\"!\"\"\"\"\"#!\"\"F(-%$SumG6$,&*&&F% 6#%\"kGF(-%$cosG6#*,F)F(%#PiGF(,&%\"bGF(F%F*F*F2F(%\"tGF(F(F(*&&F96#F2 F(-%$sinG6#*,F)F(F7F(F2F(F:F(,&F9F(F%F*F*F(F(/F2;F(%\"NGF(" }{TEXT -1 17 "\n\nwhere\n " }{TEXT 358 10 " " }{XPPEDIT 359 1 " a[k] = 2/(b-a)*Int(f(t)*cos(2*Pi/(b-a)*k*t),t=a..b)" "/&%\"aG6#%\"kG*( \"\"#\"\"\",&%\"bGF)F$!\"\"F,-%$IntG6$*&-%\"fG6#%\"tGF)-%$cosG6#*,F(F) %#PiGF),&F+F)F$F,F,F&F)F4F)F)/F4;F$F+F)" }{TEXT -1 20 " and \+ " }{TEXT 360 1 " " }{XPPEDIT 361 1 "b[k] = 2/(b-a)*Int(f(t)*sin(2 *Pi/(b-a)*k*t),t=a..b)" "/&%\"bG6#%\"kG*(\"\"#\"\"\",&F$F)%\"aG!\"\"F, -%$IntG6$*&-%\"fG6#%\"tGF)-%$sinG6#*,F(F)%#PiGF),&F$F)F+F,F,F&F)F4F)F) /F4;F+F$F)" }{TEXT -1 3 " ." }}{PARA 15 "" 0 "" {TEXT -1 3 "If " } {TEXT 363 2 " N" }{TEXT -1 98 " is not numeric, then the return is g iven as an inert sum - i.e., the return is expressed using " }{TEXT 362 4 " Sum" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 167 "If the F ourier coefficients cannot be calculated exactly in closed form and if the Fourier coefficients are real constants, as would be the case whe n (i) the interval " }{TEXT 365 5 " a..b" }{TEXT -1 51 " is specified in terms of real constants and (ii) " }{TEXT 366 1 "f" }{TEXT -1 97 " does not depend on symbolic parameters, then a numerical evaluation o f the coefficients is used." }}{PARA 15 "" 0 "" {TEXT -1 25 "The three argument call " }{TEXT 353 29 "fourierPolynom(f, N, a..b, t)" } {TEXT -1 42 " is equivalent to the four argument call " }{TEXT 354 32 "fourierPolynom(f, N, -Pi..Pi, t)" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 364 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "fourier Polynom(g, N, a..b, t); # Undetermined degree" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%\"bG\"\"\"%\"aG!\"\"F)-%$IntG6$-%\"gG6#%$tauG/F0 ;F(F&F'F'*&F%F)-%$SumG6$,&*&-F+6$*&F-F'-%$cosG6#,$**%#PiGF'F%F)%\"jGF' F0F'\"\"#F'F1F'-F=6#,$**FAF'F%F)FBF'%\"tGF'FCF'F'*&-F+6$*&F-F'-%$sinGF >F'F1F'-FNFEF'F'/FB;F'%\"NGF'FC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "fourierPolynom(t -> t, 2, -1..1, t); # Standard call" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#PiG!\"\"-%$sinG6#*&F%\"\"\"%\"tG F+F+\"\"#*&F%F&-F(6#,$F*F-F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "fourierPolynom(t -> t, 1, 0..b, t); # symbolic range" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"bG#\"\"\"\"\"#*(%#PiG!\"\"F$F&-%$sinG6# ,$*(F)F&F$F*%\"tGF&F'F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "fourierPolynom(g, 2, -1..1, t); # symbolic function" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#,,-%$intG6$-%\"gG6#%$tauG/F*;!\"\"\"\"\"#F.\"\"# *&-F%6$*&F'F.-%$cosG6#*&%#PiGF.F*F.F.F+F.-F66#*&F9F.%\"tGF.F.F.*&-F%6$ *&F'F.-%$sinGF7F.F+F.-FCF;F.F.*&-F%6$*&F'F.-F66#,$F8F0F.F+F.-F66#,$F " 0 "" {MPLTEXT 1 0 102 "fourierPolynom(t -> exp(-t)*sin(t^2), 1, 0..1, t); \+ \n# Numerical evaluation of the Fourier coefficients" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,($\"+sc74:!#5\"\"\"-%$cosG6#,$*&%#PiGF'%\"tGF'\"\"# $\"+j>4%R\"!#7-%$sinGF*$!+Aw4O7F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 377 8 "Examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 322 13 "t -> sin(t)^2 " }{TEXT -1 43 " (This is a disguised Fourier polynomial )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f \+ := t -> sin(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"tG6\" 6$%)operatorG%&arrowGF(*$-%$sinG6#9$\"\"#F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The next set of execution groups illustrates that " }{TEXT 323 1 " " }{TEXT 324 12 "t-> sin(t)^ 2" }{TEXT -1 35 " is a degree 2 Fourier polynomial:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fourierPol ynom(f,0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fourierPolynom(f,1,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fourierPolynom(f,2,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"#F%-%$cosG6#,$%\"tGF&#!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fourierPolynom(f,3,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"#F%-%$cosG6#,$%\"tGF&#!\"\"F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Here we have used the default interval of lengt h " }{TEXT 327 1 " " }{XPPEDIT 328 1 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" } {TEXT -1 38 " - which happens to be the period of " }{TEXT 325 1 " " }{TEXT 326 1 "f" }{TEXT -1 16 ". The function " }{TEXT 342 14 "fourie rPolynom" }{TEXT -1 77 " does not need this default period to be spec ified: the three argument call " }{TEXT 343 21 "fourierPolynom(f,N,t) " }{TEXT -1 28 " and the four argument call " }{TEXT 344 29 "fourierPo lynom(f,N,0..2*Pi,t)" }{TEXT -1 23 " have the same result." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "If we specify \+ a different interval on which to base our Fourier expansion, say " } {TEXT 331 5 "[0,5]" }{TEXT -1 52 ", then our expansion will approximat e but not equal " }{TEXT 329 1 " " }{TEXT 330 1 "f" }{TEXT -1 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fourierPolynom(f,2,0..5,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.* &-%$cosG6#\"\"&\"\"\"-%$sinGF'F)#!\"\"\"#5#F)\"\"#F)**-F+6#F.F),&%#PiG F)!\"&F)F-,&F5F)F(F)F--F&6#,$*&F5F)%\"tGF)#F0F(F)#F(F0*&,&**,(*$F5F0F- \"#DF)*&-F&F3F)F5F0F)F)F5F-F7F-F4F-F/*(F5F-F7F-F4F-#!#DF0F)-F+F9F)F)** F2F),&F5F0F6F)F-,&F5F0F(F)F--F&6#,$F;#\"\"%F(F)F>*&,&**,(FC!\"%FDF)FEF RF)F5F-FMF-FLF-#F)FR*(F5F-FMF-FLF-#FIFRF)-F+FOF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([fourierPolynom(f,2,0..5,t),f(t)], t=0 ..5,color=[NAVY,plum]);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7c p7$\"\"!$\"1+1e;F`J_!#;7$$\"1LLL3x&)*3\"F+$\"1Q/2*Hy$RWF+7$$\"1nm\"H2P \"Q?F+$\"1e[(H)pe]QF+7$$\"1LL$eRwX5$F+$\"1C#p^*3laLF+7$$\"1MLL$eI8k$F+ $\"1wPR\")f3'=$F+7$$\"1ML$3x%3yTF+$\"1-pOVB$)F+$\"1K\\#ztS:Q%F+7$$\"1++DJ bw!Q*F+$\"1&*R%>LTuA&F+7$$\"1nm;/j$o/\"!#:$\"1$yhnpod@'F+7$$\"1LL3_>jU 6Fdp$\"1Tn-?*>k7(F+7$$\"1++]i^Z]7Fdp$\"1`=m`x([7)F+7$$\"1++](=h(e8Fdp$ \"1h,>GZvA!*F+7$$\"1++]P[6j9Fdp$\"1)*)yB$)4)>(*F+7$$\"1n\"HKR'\\5:Fdp$ \"1'p:f;1f'**F+7$$\"1L$e*[z(yb\"Fdp$\"1*)=QQ[;;5Fdp7$$\"1Uh\"Fdp$\"11[`QdPK5Fdp7$$\"1$eky#)*QU;Fdp $\"1]%3ZZau.\"Fdp7$$\"1nm;a/cq;Fdp$\"1***RUg[//\"Fdp7$$\"1n;zpYU%p\"Fd p$\"1dLh!oD8/\"Fdp7$$\"1nmT&)))G=5Fdp7$$\"1+]iSj0x=Fdp$\"1U6,[0#p'**F+7$$\"1nmm\"pW`(>Fdp$\" 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{TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 341 33 "t -> \+ ((1+2+cos(2*t))^(1/2))^(1/2)" }{TEXT -1 65 " (This is a periodic func tion that is not a Fourier polynomial )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "In the next two execution groups we \+ define and plot this function with period " }{TEXT 345 1 " " } {XPPEDIT 346 1 "Pi" "I#PiG%*protectedG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f := t -> (1+(2+cos(2*t))^(1/2))^(1/2); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"tG6\"6$%)operatorG%&arrowG F(*$,&\"\"\"F.*$,&\"\"#F.-%$cosG6#,$9$F1F.#F.F1F.F7F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot(f(t), t = 0 .. 2*Pi, color = P LUM, thickness = 2);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6#7ar7$ \"\"!$\"1q5G];*Gl\"!#:7$$\"1v_Sq*))QU$!#<$\"1\\y$o$po_;F+7$$\"1]0\"3%z xZoF/$\"1b(pZ7t?l\"F+7$$\"1#e@6pmr-\"!#;$\"1J?@,70^;F+7$$\"15@;)eb&p8F 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Because the period is not " }{XPPEDIT 19 1 "2*Pi" "*&\"\"# \"\"\"%#PiGF$" }{TEXT -1 60 " we must use the four argument call, spec ifying the period. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fourierPolynom(f,2,0..Pi,t);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#,,$\"+i/\"Ra\"!\"*\"\"\"-%$cosG6#,$%\"tG\"\"#$\" +H;by6!#5-%$sinGF*$!+7.kHs!#E-F)6#,$F,\"\"%$!+fD655!#6-F2F7$!+g]Ws^!#D " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 406 "Notice the difference between this retur n and that of the preceding example. When the Fourier coefficients can be calculated exactly, they are. When an exact calculation of the Fou rier coefficients is not possible, a numerical evaluation is used inst ead. If you look at the coefficients of the sine functions, then you w ill see that minute inaccuracies have been introduced. These coefficie nts should be 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot([fourie rPolynom(f,2,0..Pi,t), f(t)], t = 0..2*Pi, color = [NAVY,PLUM]);" }} {PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7ar7$\"\"!$\"1V1&**\\k;l\"!#: 7$$\"1v_Sq*))QU$!#<$\"1.$p)oG[^;F+7$$\"1]0\"3%zxZoF/$\"1ko,*\\P4l\"F+7 $$\"1#e@6pmr-\"!#;$\"1>>n()p-];F+7$$\"15@;)eb&p8F:$\"12)=n3\\([;F+7$$ \"1n]I&3x`'>F:$\"13OhDF:$\"1Wd'Q\"3RT;F+7$$\"1>Fd fdK,RF:$\"1Zfz;'pvi\"F+7$$\"19Ubk4$ed%F:$\"11%*QzTN=;F+7$$\"15d`phL]_F :$\"1tGP#fXwg\"F+7$$\"1a$=!pbj@fF:$\"1L:Y*[jbf\"F+7$$\"1**4]o\\$Hf'F:$ \"1JQf)fj@e\"F+7$$\"1zf7WaJ:sF:$\"1-I.!)opo:F+7$$\"1g4v>fpPyF:$\"1Z&eU C&Ra:F+7$$\"1`ld_[8#[)F:$\"1GX+1r#*Q:F+7$$\"1Y@S&ytl7*F:$\"1Dg`(p 9F+$\"1pF:rj;>9F+7$$\"1F?GYNj.:F+$\"1#R1p:ytT\"F+7$$\"1mM;=Nd?:F+$\"1! e!3c8v;9F+7$$\"10\\/!\\8v`\"F+$\"1iu')oXI;9F+7$$\"1Wj#>Y`Wb\"F+$\"1n$4 '*pQgT\"F+7$$\"1$y2QV$Rr:F+$\"1m\\%oGafT\"F+7$$\"1E8%G'HS)e\"F+$\"1^#p ]F_gT\"F+7$$\"1p[(=\\7ag\"F+$\"19l,&=LjT\"F+7$$\"17%34-ACi\"F+$\"10c-V kz;9F+7$$\"1b>%*\\:VR;F+$\"1OGx.6W<9F+7$$\"1T!4!31Xt;F+$\"1A:'H,p#>9F+ 7$$\"1Fh2m'puq\"F+$\"1t[;f?!=U\"F+7$$\"1Wx7)*p.tF+$\"1yu*4>x$f9F+7$$\"1)zHUn([G?F+$\"1)=IxuvYZ\"F+7$$\"1h\"\\.=) G*4#F+$\"1)3mhJ07\\\"F+7$$\"1R*\\f\\l#f@F+$\"1hZH)34e]\"F+7$$\"1<2b6GC >AF+$\"1sgUyYl?:F+7$$\"1$oXzo5!*G#F+$\"1i\"Gksgy`\"F+7$$\"1\\1Mk&y(eBF +$\"1Gp\")Q,ia:F+7$$\"1.>mc\\`?CF+$\"19/_)z-)o:F+7$$\"1eJ)*[8H#[#F+$\" 1([0*o>;#e\"F+7$$\"1<7A**y/]DF+$\"1]L-w5o&f\"F+7$$\"1w#f%\\W!yh#F+$\"1 [%p7ccyg\"F+7$$\"1pUJ3ZK#o#F+$\"1@1n7Z5=;F+7$$\"1j#pr'\\%ou#F+$\"1V$*4 vs)pi\"F+7$$\"1(Qg,@&[\")GF+$\"1BWD(>l5k\"F+7$$\"1M3w'41L%HF+$\"18U!e( =`X;F+7$$\"1\"GhL)p70IF+$\"1Grlu*p([;F+7$$\"1;kJs!o%QIF+$\"1\"G$yIR,]; 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Since the function is not periodic, we will have to make a somewhat arbitra ry choice, say " }{TEXT 348 1 " " }{TEXT 349 5 "[0,2]" }{TEXT -1 109 " , for the interval. We will see that the approximation over this inter val is reasonable. 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\"19\"f3e(Gp6F+7$$\"1+vV)e#3'[&Fez$\"1s^#3pz;=\"F+7$$\"1++]Ujp-bFez$\" 1_WZQ?t.7F+7$$\"1+](=U:C`&Fez$\"1)e*4pbLb7F+7$$\"1++D,X8ibFez$\"1(oURM *>)H\"F+7$$\"1]P4rUcpbFez$\"1*owIJ`\\I\"F+7$$\"1+v$4/%*pd&Fez$\"1[S>r* f)48F+7$$\"1\\7y5QU%e&Fez$\"1?Rz%=xHJ\"F+7$$\"1+]i!e`=f&Fez$\"1%Q**3Sn WJ\"F+7$$\"1+DJ?Jr1cFez$\"1xb$[dFez$\"1#HeMV(GG9F+7$$\"1+++5h (*3eFez$\"1x5BQu%=X\"F+7$$\"1++D6EjpeFez$\"1%G>*3bfu9F+7$$\"1,]i0j\"[$ fFez$\"1zlifbYo:F+7$$\"\"'F(F)-%'COLOURG6&%$RGBG$\")!\\DP\"!\")Ffdp$\" )viobFhdp-F$6$7S7$F(F(7$F-$\"1ib:ZtT5i!*R_h#3$Fez7$Fgfl$\"1)*[aLl&[\\$Fez7$Feh l$\"1xnn7?r=SFez7$Fjhl$\"14QQ9n4\"\\%Fez7$F_il$\"1VzYG.ht]Fez7$Fdil$\" 1w(*Gx2')=cFez7$F\\\\m$\"1X;Zwb4\\iFez7$F^_m$\"1&[>)*Qe.)oFez7$Fh_m$\" 1*4'p\"*))QrvFez7$F`bm$\"1_K-K84N#)Fez7$Fffm$\"1jVqX&RA)*)Fez7$Fbjm$\" 1L:&3oiEz*Fez7$Fj\\n$\"1J$)\\)>jE0\"F+7$Fb_n$\"1LT\"3%G\"\\8\"F+7$F^cn $\"1%fV%3O8B7F+7$Ffen$\"1z:#oZNEJ\"F+7$F[fn$\"18I)G,TAS\"F+7$F`fn$\"1W J%f`)>0:F+7$Fefn$\"1aeN\\7#3g\"F+7$Fjfn$\"10h^*Hphq\"F+7$F_gn$\"169\"f MHX!=F+7$Fdgn$\"1H#*efn@:>F+7$Fbin$\"11jF?KPA?F+7$F^]o$\"13mw*Q+v8#F+7 $Fh]o$\"1qYv\"=RJD#F+7$F``o$\"19?mOhYxBF+7$F\\do$\"1W\"[K*RO+DF+7$Fhgo $\"1+hv9xCHEF+7$F\\io$\"1?XsnGEgFF+7$Fd[p$\"1Uj],0Y$)GF+7$Ff^p$\"1)GwP qmz-$F+7$F^ap$\"1[f; t^2, 0 < t < 2" }{TEXT -1 39 " (Extend this f unction periodically)" }}{PARA 0 "" 0 "" {TEXT -1 87 "To extend this f unction periodically to the positive reals we could make the definitio n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := t -> sum((t-2*k)^2*(Heaviside(t-2*k)-Heaviside(t -2*k-2)), k = -infinity .. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"tG6\"6$%)operatorG%&arrowGF(-%$sumG6$*&,&9$\"\"\"%\"kG! \"#\"\"#,&-%*HeavisideG6#F0F2-F86#,(F1F2F3F4F4F2!\"\"F2/F3;,$%)infinit yGF=FAF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "However, MAPLE will just spin its wheels trying to find a clos ed form expression for the sum. Let us just truncate it." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "f := \+ t -> sum((t-2*k)^2*(Heaviside(t-2*k)-Heaviside(t-2*k-2)), k = - 100 .. 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"tG6\"6$%)operato rG%&arrowGF(-%$sumG6$*&,&9$\"\"\"%\"kG!\"#\"\"#,&-%*HeavisideG6#F0F2-F 86#,(F1F2F3F4F4F2!\"\"F2/F3;!$+\"\"$+\"F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Here is its plot over several p eriods:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot(f(t), t = -6 .. 6, color = PLUM, thickness = 2, \+ discont = true);" }}{PARA 13 "" 1 "" {INLPLOT "6*-%'CURVESG6%7S7$$\"1+ ++++++S!#C$\"1+++++++;!#K7$$\"14c*e@J%fV!#<$\"12Dh_SY+>!#=7$$\"1tk0f'[ D:)F1$\"1\\$GQ'\\SYmF47$$\"16gO$fI=C\"!#;$\"1z(>EAV@a\"F17$$\"1)R[;%RB r;F=$\"1Swr()G-$z#F17$$\"1\"Gs2T'f)4#F=$\"1,jI&*o5/WF17$$\"1pS&e7;[\\# F=$\"1.:#=]2TA'F17$$\"1&oz3py]!HF=$\"1CK?+A[R%)F17$$\"1Ue^AGPHLF=$\"15 28RBZ36F=7$$\"1v2\\PiI_PF=$\"1=7**4-)zS\"F=7$$\"1'G<*R_M(=%F=$\"1&zDe, 'Q`+&F=$\"1RqCJ5!>]# F=7$$\"1?)f#oZ/NaF=$\"1W))Hj6(R&HF=7$$\"1j,fm$fC&eF=$\"1$*)eP1G^U$F=7$ $\"1&G24\"=^JiF=$\"1*pZ\\%R<$)QF=7$$\"1+x$*H=C#o'F=$\"18pP(eN_Y%F=7$$ \"1!R5%y#pS1(F=$\"1\\\"=qZ2,*\\F=7$$\"1)4nCPD#3vF=$\"1oePC[MPcF=7$$\"1 ^61v(y8!zF=$\"1'o)pa'yJC'F=7$$\"1z!p\"p.tK$)F=$\"1ls_S&RM%pF=7$$\"1%3E D4zMu)F=$\"1wS9kE%[k(F=7$$\"1e%y\\H_?<*F=$\"1f>=IVl7%)F=7$$\"1,U!4G;cc *F=$\"1_*\\L[,,:*F=7$$\"1>GP3#G,***F=$\"1KE(>;m-)**F=7$$\"1.4Tzw5V5!#: $\"1)*f%3jt!)3\"Ffs7$$\"1ISU$Q#\\\"3\"Ffs$\"1!>/axD'p6Ffs7$$\"1Q:M:*[H 7\"Ffs$\"1OnYmU,h7Ffs7$$\"1!*oLpvxl6Ffs$\"1F!p6MP!f8Ffs7$$\"1#Hp;0xw? \"Ffs$\"1L'G6'Q[e9Ffs7$$\"1KrDbp@[7Ffs$\"1'Q\"Rnb/e:Ffs7$$\"1#3FjgHKH \"Ffs$\"1**3o9GWs;Ffs7$$\"1l>LvanL8Ffs$\"1U>]t-pymVd*=Ffs7$$\"1m$>zt\"*fT\"Ffs$\"12f&=gK]+#Ffs7$$\"13\\;$)Hx e9Ffs$\"1X=T;'=!G@Ffs7$$\"1f0#4!o-*\\\"Ffs$\"1F\"y)\\83ZAFfs7$$\"1&e&3 RO5T:Ffs$\"1MGSE/+vBFfs7$$\"1,xLR9C#e\"Ffs$\"1RuNsz[.DFfs7$$\"1k([P!*3 `i\"Ffs$\"15]gK!H;k#Ffs7$$\"1#=m1*zym;Ffs$\"12!He?#=yFFfs7$$\"13l*zM1# 4Ffs$\"16r@WUYrOFfs7$$\"1BQ#**>Wl&> Ffs$\"1abD1_1GQFfs7$$\"1+++'*******>Ffs$\"1+++%)******RFfs-%'COLOURG6& %$RGBG$\")1Zw\"*!\")$\")PJ%y'Fb[lF`[l-%*THICKNESSG6#\"\"#-F$6%7S7$$\"1 +++3+++?Ffs$\"1=S`[.++kF-7$$\"1zX)fJ%fV?Ffs$\"1FL;'Qk/!>F47$$\"1jWE!\\ D:3#Ffs$\"1J5'Gc0kk'F47$$\"1p)RG1$=C@Ffs$\"1\"4H'4L9U:F17$$\"1Yj\\(RBr ;#Ffs$\"1t@2**H-$z#F17$$\"1!zPUkf)4AFfs$\"1h6'z-2TS%F17$$\"1\"[(e:;[\\ AFfs$\"1cRh^w5CiF17$$\"1Pf#>(y]!H#Ffs$\"1mO4lB[R%)F17$$\"1N)>\\GPHL#Ff s$\"19z*oNs%36F=7$$\"1b\"[iiI_P#Ffs$\"1GeuG-)zS\"F=7$$\"1\"z;j_M(=CFfs $\"1ruHNgQ`+DFfs$\" 1RqC^5!>]#F=7$$\"1k>lyW]VDFfs$\"1Nu9$=rR&HF=7$$\"1L!=$QfC&e#Ffs$\"1Wv< $3G^U$F=7$$\"1C\")f#=^Ji#Ffs$\"12WtjR<$)QF=7$$\"1t3K%=C#oEFfs$\"1jH60c BlWF=7$$\"17a,HpS1FFfs$\"1Z)4O\\2,*\\F=7$$\"1?MCQD#3v#Ffs$\"1=HMR[MPcF =7$$\"1jbMyy8!z#Ffs$\"1%GkzmyJC'F=7$$\"1;QePIFLGFfs$\"1$fT;bRM%pF=7$$ \"1<_v4zMuGFfs$\"1lJ$HnU[k(F=7$$\"1D!H)H_?Z;\"GWs;Ffs7$$\"1js*RZvOL$Ffs$\"1B=%*p-py^)>2g oP$Ffs$\"1oo#ylVd*=Ffs7$$\"1*Rbit\"*fT$Ffs$\"1!eVrfK]+#Ffs7$$\"1;)H8)H xeMFfs$\"1!)y06'=!G@Ffs7$$\"1_W#*)zE!*\\$Ffs$\"1)o$*QM\"3ZAFfs7$$\"1r6 #pj.6a$Ffs$\"1f;t>/+vBFfs7$$\"1N(3qVTAe$Ffs$\"1mu)\\'z[.DFfs7$$\"1HvC, *3`i$Ffs$\"1\\WZC!H;k#Ffs7$$\"1H!**z)zymOFfs$\"1`y$p>#=yFFfs7$$\"1#of^ M1#4PFfs$\"1>TLIjQ@HFfs7$$\"1zk!pXt7v$Ffs$\"1q^'3sep1$Ffs7$$\"1q0oj(G* *y$Ffs$\"1a50z\\%Q?$Ffs7$$\"14G\\2@BMQFfs$\"18v:CuSkLFfs7$$\"1tCMYv&Q( QFfs$\"1KY)R5U8^$Ffs7$$\"1[6#*Gl5;RFfs$\"1]UWl&RFf s$\"1&Q$G\">l!GQFfs7$$\"1+++#*******RFfs$\"1*****z'******RFfsF\\[lFe[l -F$6%7S7$$\"1+++7+++SFfs$\"1[%H^-++W\"!#J7$$\"1--\")>VfVSFfs$\"1rTr>ZY +>F47$$\"1h$QR\\D:3%Ffs$\"1+Q*=;1kk'F47$$\"1PJMmI=CTFfs$\"1*QQmRV@a\"F 17$$\"1`y#3SBr;%Ffs$\"1;nU5J-$z#F17$$\"1^$)RZ'f)4UFfs$\"15ghgr5/WF17$$ \"1a&*e=;[\\UFfs$\"1ujS,y5CiF17$$\"11Rwuy]!H%Ffs$\"11T)*HD[R%)F17$$\"1 '3)e(GPHL%Ffs$\"1>^muBZ36F=7$$\"1KsuG1BvVFfs$\"1K/]Z-)zS\"F=7$$\"1`=kG Xt=WFfs$\"1X\"pZ0'Q` +XFfs$\"1QqCr5!>]#F=7$$\"1ZzZ![/Na%Ffs$\"1Gg*H?rR&HF=7$$\"1\\q(*RfC&e% Ffs$\"1#>'f-\"G^U$F=7$$\"1>b5%=^Ji%Ffs$\"136_#)R<$)QF=7$$\"1wzk&=C#oYF fs$\"18!\\GiN_Y%F=7$$\"1$y*=IpS1ZFfs$\"1M:?5v5!*\\F=7$$\"1H,CRD#3v%Ffs $\"1c*4V&[MPcF=7$$\"17]=zy8!z%Ffs$\"1$*)H7oyJC'F=7$$\"1C2DQIFL[Ffs$\"1 ?fvi&RM%pF=7$$\"1DyD5zMu[Ffs$\"1aAs\"oU[k(F=7$$\"1/-;I_?<\\Ffs$\"1DBLU Vl7%)F=7$$\"1FzVG;cc\\Ffs$\"1\">)***[,,:*F=7$$\"1!=X3#G,**\\Ffs$\"1L/8 ihE!)**F=7$$\"1Ug1zw5V]Ffs$\"1Wl7IO2)3\"Ffs7$$\"1\"4sFQ#\\\"3&Ffs$\"1c S*RxD'p6Ffs7$$\"1ZzN9*[H7&Ffs$\"1\"pdUE95E\"Ffs7$$\"1p1,ovxl^Ffs$\"1qo 2Qt.f8Ffs7$$\"1wy+]qn2_Ffs$\"1Dd6dQ[e9Ffs7$$\"1'RrK&p@[_Ffs$\"1FTVib/e :Ffs7$$\"1X7)RgHKH&Ffs$\"1([8'3GWs;Ffs7$$\"1hDmsanL`Ffs$\"11T2WO$Ffs7$ $\"1Vq%GadQ(eFfs$\"1dZ)34U8^$Ffs7$$\"1AnDDl5;fFfs$\"1)QJh@k9n$Ffs7$$\" 1q9F#>Wl&fFfs$\"1>7Jw^1GQFfs7$$\"1+++))******fFfs$\"1+++_******RFfsF\\ [lFe[l-F$6%7S7$$!1+++))******fFfsFg[m7$$!1)z*=!o0k&fFfsF]\\m7$$!1R;11X Z=fFfsFb\\m7$$!1jolLp\"e(eFfsFg\\m7$$!1Z@<*fwG$eFfsF\\]m7$$!1\\;g_.9!z &FfsFa]m7$$!1Y/T\"Q=0v&FfsFf]m7$$!1%4O_7#\\4dFfsF[^m7$$!19>T7F1ncFfsF` ^m7$$!1oFDr$pZi&FfsFe^m7$$!1Z\"e8Zl7e&FfsFj^m7$$!1+8l7s%Ha&FfsF__m7$$! 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