{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Red Emphasis" -1 256 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 259 "" 0 1 0 0 0 0 0 1 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 1 }{CSTYLE "" -1 261 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 263 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 264 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 265 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 266 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 273 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "Magenta Emphasis" -1 280 "Times" 1 12 200 0 200 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 284 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 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 "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{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 "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 43 "A procedure for constructing Four ier series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B. C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "load Fourier series and Fourier transfo rm procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 273 9 "fourier.m" }{TEXT -1 37 " contains the code for the proce dure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 25 " used in this workshe et. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple sess ion by a command similar to the one that follows, where the file path \+ gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \+ \"K:\\\\Maple/procdrs/fourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "A procedure for constructing Four ier series: " }{TEXT 0 13 "FourierSeries" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "FourierSeri es: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 259 2 " " }{TEXT -1 28 " FourierSeries( fx, rng ) " }{TEXT 260 1 "\n" } {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " fx - " } {TEXT -1 81 " an expression involving a single variable, say x, d efining a function f(x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rng - " }{TEXT 261 76 "the ra nge x=a..b for the Fourier series trigonometric approximation of f(x). " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedur e " }{TEXT 0 13 "FourierSeries" }{TEXT -1 77 " attempts to find a trun cated Fourier trigonometric series approximation for " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " of the form" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x)=c+Sum(a[k]*cos(2*Pi*k*x/(b-a) )+b[k]*sin(2*Pi*k*x/(b-a)),k=1..n)" "6#/-%\"FG6#%\"xG,&%\"cG\"\"\"-%$S umG6$,&*&&%\"aG6#%\"kGF*-%$cosG6#*,\"\"#F*%#PiGF*F3F*F'F*,&%\"bGF*F1! \"\"F$\"3?mI(3i,9G$F,7$$\"3'pmm;H[D:)F0$\"3[,;)eS5 \")[$F,7$$\"3LLLLe0$=C\"F,$\"3I#px)e^R!o$F,7$$\"3ILLL3RBr;F,$\"3%R0EZy X^$QF,7$$\"3Ymm;zjf)4#F,$\"3E,]JF!Qk&RF,7$$\"3=LL$e4;[\\#F,$\"3gcj%4e# 4YSF,7$$\"3p****\\i'y]!HF,$\"3,#4`>\\`I7%F,7$$\"3,LL$ezs$HLF,$\"3A>DZN 2*R>%F,7$$\"3_****\\7iI_PF,$\"3eyu8IjxkUF,7$$\"3#pmmm@Xt=%F,$\"3u]8$oD eoM%F,7$$\"3QLLL3y_qXF,$\"3mi!z@%>#RV%F,7$$\"3i******\\1!>+&F,$\"3__Y( 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\\lQ\"a6\"-%&COLORG6&F[\\lF)F)$\"\"\"Fd\\l-Fi]o6%7$Fd[lF\\^oQ\"bF_^oF` ^o-%+AXESLABELSG6%Q\"xF_^oQ!F_^o-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%N ONEG-%%VIEWG6$;Fc\\lFc[oFa_o" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 39 "mode=auto/ formula/ termbyterm/ numeric" }}{PARA 15 "" 0 "" {TEXT -1 35 "With the option \"mode=termbyterm\", " }{TEXT 0 13 "Fo urierSeries" }{TEXT -1 150 " constructs the coefficients of the sine a nd cosine terms in the Fourier series by evaluating the integral expre ssion for each coefficient separately." }}{PARA 15 "" 0 "" {TEXT -1 32 "With the option \"mode=numeric\", " }{TEXT 0 13 "FourierSeries" } {TEXT -1 152 " constructs the coefficients of the terms in the Fourier series by evaluating the integral expressions for the coefficients us ing numerical integration." }}{PARA 15 "" 0 "" {TEXT -1 32 "With the o ption \"mode=formula\", " }{TEXT 0 13 "FourierSeries" }{TEXT -1 327 " \+ attempts to construct a general formula for the coefficients of the co sine terms in the Fourier series, and a general formula for the coeffi cients of the sine terms. If this fails, the integral formulas for all the coefficients are left unevaluated. If the evaluation of a particu lar coefficient using the formula fails, then " }{TEXT 0 13 "FourierSe ries" }{TEXT -1 42 " the integral formula is left unevaluated." }} {PARA 15 "" 0 "" {TEXT -1 29 "With the option \"mode=auto\", " }{TEXT 0 13 "FourierSeries" }{TEXT -1 243 " attempts to construct a general f ormula for the coefficients of the cosine terms in the Fourier series, and a general formula for the coefficients of the sine terms. If the \+ evaluation of a particular coefficient using the formula fails, then \+ " }{TEXT 0 13 "FourierSeries" }{TEXT -1 171 " attempts to evaluate the integral formula for the particular coefficient directly. If this fai ls for the particular coefficient, the integral formula is left uneval uated." }}{PARA 0 "" 0 "" {TEXT -1 27 "The default is \"mode=auto\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "numterm s=n" }}{PARA 0 "" 0 "" {TEXT -1 108 "This option specifies the number \+ of cosine and/or sine terms in the sum for the trigonometric approxima tion." }}{PARA 0 "" 0 "" {TEXT -1 28 "The default is \"numterms=4\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "type=fo urier/ cos/ sin" }}{PARA 15 "" 0 "" {TEXT -1 32 "With the option \"typ e=fourier\", " }{TEXT 0 13 "FourierSeries" }{TEXT -1 95 " constructs a truncated Fourier series which approximates the function over the int erval [a,b]." }}{PARA 15 "" 0 "" {TEXT -1 28 "With the option \"type=c os\", " }{TEXT 0 13 "FourierSeries" }{TEXT -1 137 " constructs a trunc ated Fourier cosine series which approximates the function over the in terval [0,b], that is, a Fourier series for the " }{TEXT 265 4 "even" }{TEXT -1 29 " extension of f(x) to [-b,b]." }}{PARA 15 "" 0 "" {TEXT -1 28 "With the option \"type=sin\", " }{TEXT 0 13 "FourierSeries" } {TEXT -1 135 " constructs a truncated Fourier sine series which approx imates the function over the interval [0,b], that is, a Fourier series for the " }{TEXT 265 3 "odd" }{TEXT -1 29 " extension of f(x) to [-b, b]." }}{PARA 0 "" 0 "" {TEXT 265 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 82 "In the last two cases,the lower bound of the r ange for the 2nd argument must be 0." }}{PARA 15 "" 0 "" {TEXT -1 122 "The alternative expressions \"Fourier\", \"cosine\" and \"sine\" may \+ be used in place of \"fourier\",\"cos\" and \"sin\" respectively." }} {PARA 0 "" 0 "" {TEXT -1 30 "The default is \"type=fourier\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "output=se ries/ coeffs" }}{PARA 15 "" 0 "" {TEXT -1 33 "With the option \"output =series\", " }{TEXT 0 13 "FourierSeries" }{TEXT -1 33 " returns a fini te Fourier series." }}{PARA 15 "" 0 "" {TEXT -1 33 "With the option \" output=coeffs\", " }{TEXT 0 13 "FourierSeries" }{TEXT -1 82 " returns \+ the coefficients in the Fourier series as follows.\nWhen \"type=fourie r\", " }{XPPEDIT 18 0 "[c,[a[1],` . . . `,a[n]],[b[1],` . . . `,b[n]] ]" "6#7%%\"cG7%&%\"aG6#\"\"\"%(~.~.~.~G&F'6#%\"nG7%&%\"bG6#F)F*&F06#F- " }{TEXT -1 21 ". \nWhen \"type=cos\", " }{XPPEDIT 18 0 "[c, [a[1], ` . . . `, a[n]]];" "6#7$%\"cG7%&%\"aG6#\"\"\"%(~.~.~.~G&F'6#%\"nG" } {TEXT -1 21 ". \nWhen \"type=sin\", " }{XPPEDIT 18 0 "[b[1], ` . . . \+ `, b[n]];" "6#7%&%\"bG6#\"\"\"%(~.~.~.~G&F%6#%\"nG" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 31 "The default is \"output=series\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "info=n or info=true/false" }}{PARA 0 "" 0 "" {TEXT -1 94 "This option allows so me information about the computation of the coefficients to be provide d. " }}{PARA 15 "" 0 "" {TEXT -1 36 "info=0: No information is give n. " }}{PARA 15 "" 0 "" {TEXT -1 94 "info=1: Where a general formul a for sine or cosine coefficients is determined, it is given." }} {PARA 15 "" 0 "" {TEXT -1 163 "info=2: The values of the coefficien ts are given as they are calculated. \n In the case wher e a general formula is determined, this is also given. " }}{PARA 15 " " 0 "" {TEXT -1 73 "info=true is equivalent to info=2 and info=false i s equivalent to info=0." }}{PARA 0 "" 0 "" {TEXT -1 29 "The default is \"info=false\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "evalf_coeffs: usage " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 18 "Calling Sequence:\n" }}{PARA 0 " " 0 "" {TEXT 282 2 " " }{TEXT -1 108 " evalf_coeffs( trig_series ) \+ \n\n evalf_coeffs[d]( trig_series ) \n\n evalf_coeffs( trig_se ries, d ) \n" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 18 " trig_serie s - " }{TEXT -1 56 " an trigonometric series involving a variable, say x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 10 " d - " }{TEXT 283 32 "the number of digits to be used " }{TEXT -1 56 "which can be given as an index or as the second \+ argument" }{TEXT 284 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 12 "evalf_coeffs" }{TEXT -1 55 " evaluates the coefficients in a trigonometric series: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c + a[1]*cos(r[1]*x)+b[1]*sin(s[1]*x) + a[2]*cos(r[2]*x)+b[2]*sin(s[2 ]*x)+a[2]*cos(r[2]*x)+b[2]*sin(s[2]*x)+` . . . `+a[n]*cos(r[n]*x)+b[n] *sin(s[n]*x)" "6#,6%\"cG\"\"\"*&&%\"aG6#F%F%-%$cosG6#*&&%\"rG6#F%F%%\" xGF%F%F%*&&%\"bG6#F%F%-%$sinG6#*&&%\"sG6#F%F%F1F%F%F%*&&F(6#\"\"#F%-F+ 6#*&&F/6#F@F%F1F%F%F%*&&F46#F@F%-F76#*&&F;6#F@F%F1F%F%F%*&&F(6#F@F%-F+ 6#*&&F/6#F@F%F1F%F%F%*&&F46#F@F%-F76#*&&F;6#F@F%F1F%F%F%%(~.~.~.~GF%*& &F(6#%\"nGF%-F+6#*&&F/6#F\\oF%F1F%F%F%*&&F46#F\\oF%-F76#*&&F;6#F\\oF%F 1F%F%F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 16 "How to activate: " }}{PARA 0 "" 0 "" {TEXT -1 154 "To make the procedure active open th e subsection, place the cursor anywhere after the prompt [ > and pres s [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 29 "FourierSeries: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14883 "FourierSe ries := proc(alg_expr::algebraic,eq::equation)\nlocal Options,n,a,b,f, prntflg,typ,md,x0,x1,L,k,k_,m_,c,x,rs,\n formulaOK,form,A,B,ak,bk,aj ,bj,i,j,outpt,mx,eps,wasauto,\n nonnumericcoeff,temp,vars,formulaZer o;\n\n if nargs<2 then\n error \"at least 2 arguments are requi red; the basic syntax is: '%1(f(x),x=a..b)'.\",procname;\n end if;\n \n x := op(1,eq);\n if not type(x,symbol) then\n error \"2nd \+ argument equation left side, %1, must be the independent variable\",x; \n end if;\n if not type(indets(alg_expr,name) minus \{x\},set(rea lcons)) then\n error \"the 1st argument, %1, is invalid .. it sho uld depend only on the variable %2\",alg_expr,x;\n end if;\n\n var s := indets(alg_expr,name) minus indets(alg_expr,realcons);\n if nop s(vars)>1 then \n if not has(indets(alg_expr),\{Int,Sum\}) then\n error \"the 1st argument, %1, is invalid .. it should be an e xpression which depends only on a single variable\",alg_expr;\n e nd if;\n end if;\n\n if not member(x,vars) and vars<>\{\} then\n \+ error \"the 1st argument, %1, is invalid .. it should be an expres sion which depends only on the variable %2\",alg_expr,x;\n end if;\n \n rs := op(2,eq);\n if not type(rs,range) then\n error \" the right side of the 2nd argument equation, %1, is incorrect ..it sho uld give a range for the approximation\",rs;\n end if;\n \n # ge t the requested options, but first give defaults\n n := 4;\n typ : = 'fourier';\n md := 'auto';\n outpt := 'series';\n prntflg := 0 ;\n if nargs>2 then\n Options:=[args[3..nargs]];\n if not \+ type(Options,list(equation)) then\n error \"each optional argu ment must be an equation\"\n end if;\n if hasoption(Options, 'numterms','n','Options') then\n if not type(n,posint) then\n \+ error \"\\\"numterms\\\" must be a positive integer\"\n \+ end if;\n end if;\n if hasoption(Options,'type','typ',' Options') then\n if not member(typ,\{'fourier','Fourier','cos' ,'cosine',\n 'sin','sine'\} ) then\n error \"\\\"type\\\" must be 'fourier','Fourier',' cos','cosine','sin' or 'sine'\"\n end if;\n if typ='Fo urier' then typ := 'fourier'\n elif typ='cosine' then typ := ' cos'\n elif typ='sine' then typ := 'sin' end if;\n end if ;\n if hasoption(Options,'mode','md','Options') then\n if not member(md,\{'auto','formula','termbyterm',\n \+ 'numeric'\}) then\n error \"\\\"mode\\\" m ust be 'auto','formula','termbyterm' or 'numeric'\"\n end if; \n end if;\n if hasoption(Options,'output','outpt','Options' ) then\n if not member(outpt,\{'series','coeffs'\}) then\n \+ error \"\\\"output\\\" must be 'series' or 'coeffs'\"\n \+ end if;\n end if;\n if hasoption(Options,'info','prntflg', 'Options') then\n if not member(prntflg,\{true,false,0,1,2\}) \+ then\n error \"\\\"info\\\" must be false=0, 1, or true=2\" \n end if;\n if prntflg=false then prntflg := 0\n \+ elif prntflg=true then prntflg := 2 end if; \n end if;\n \+ if nops(Options)>0 then\n error \"%1 is not a valid option fo r %2\",op(1,Options),procname;\n end if;\n end if;\n\n x0 := \+ op(1,rs);\n x1 := op(2,rs);\n if not type(rs,realcons..realcons) t hen\n if md='auto' then md := 'formula' end if;\n if md='num eric' then\n error \"2nd argument equation right side, %1, mus t be a range of real values\",rs;\n end if;\n end if;\n if md ='auto' or md='numeric' then\n if evalf(x0)>evalf(x1) then\n \+ error \"invalid range\";\n end if;\n end if;\n\n if type( rs,realcons..realcons) then\n if typ='sin' or typ='cos' then\n \+ if x0<>0 or evalf(x1)<=0 then\n error \"2nd argument \+ equation right side, %1, must be a range of the form 0..b, where b is \+ positive\",rs;\n end if;\n end if;\n end if;\n L := x 1-x0;\n #wasauto := false;\n nonnumericcoeff := false;\n\n if ty p<>'sin' then\n if md<>'numeric' then\n c := traperror(si mplify(int(alg_expr,x=x0..x1)/L));\n if c=lasterror or indets( c,'specfunc(anything,int)')<>\{\} then\n # leave integral u nevaluated\n c := Int(alg_expr,x=x0..x1)/L;\n end i f;\n end if;\n if md='numeric' then\n c := traperror (evalf(Int(alg_expr,x=x0..x1)/L));\n if c=lasterror or not typ e(c,numeric) then\n # leave integral unevaluated\n \+ c := Int(alg_expr,x=x0..x1)/L;\n end if;\n end if;\n \+ if prntflg>0 then\n print(`constant coefficient -->`,c);\n end if;\n end if;\n\n if typ<>'fourier' then L := 2*L end if ;\n if typ<>'sin' then\n A := table();\n formulaOK := fals e;\n formulaZero := false;\n if md='auto' or md='formula' th en\n assume(k_>0);\n ak := traperror(2*int(alg_expr*co s(2*k_*Pi*x/L),x=x0..x1)/L);\n formulaOK := evalb(ak<>lasterro r and indets(ak,'specfunc(anything,int)')=\{\});\n formulaZero := evalb(ak=0);\n if prntflg>0 and formulaOK then\n \+ assume(m_,posint);\n form := traperror(eval(subs(k_=m_,ak )));\n if form<>lasterror then\n temp := trap error(simplify(form));\n if temp<>lasterror then form := temp end if;\n form := subs(m_=k,form);\n \+ temp := subs(k_=k,ak);\n if typ<>'fourier' then\n \+ form := 2*form;\n temp := 2*temp;\n \+ end if;\n if temp<>form then\n \+ print(`k th cosine coefficient .. `,temp=form);\n else \n print(`k th cosine coefficient .. `,temp);\n \+ end if;\n end if;\n end if; \n end i f;\n\n if (md='auto' or md='formula') and formulaOK then\n \+ for j from 1 to n do\n aj := traperror(simplify(eval(subs (k_=j,ak))));\n # if error, try direct evaluation of the in tegral\n if aj=lasterror then\n aj := traperr or(simplify(2*int(alg_expr*cos(2*j*Pi*x/L),x=x0..x1)/L));\n \+ if aj=lasterror or indets(aj,'specfunc(anything,int)')<>\{\} then \n #if md='auto' then\n #wasauto \+ := true;\n #md := 'numeric';\n \+ #break;\n #end if;\n # leave integr al unevaluated\n aj := 2*Int(alg_expr*cos(2*j*Pi*x/L) ,x=x0..x1)/L;\n end if;\n end if;\n \+ if typ='cos' then A[j] := 2*aj else A[j] := aj end if;\n \+ if prntflg>1 and not formulaZero then\n print(`coefficie nt of`,cos(2*j*Pi*x/L),` -->`,A[j]);\n end if;\n en d do;\n end if;\n if md='termbyterm' then\n for j fr om 1 to n do\n # direct evaluation of the integral\n \+ aj := traperror(simplify(2*int(alg_expr*cos(2*j*Pi*x/L),x=x0..x1) /L));\n if aj=lasterror or indets(aj,'specfunc(anything,int )')<>\{\} then\n # leave integral unevaluated\n \+ aj := 2*Int(alg_expr*cos(2*j*Pi*x/L),x=x0..x1)/L;\n e nd if;\n if typ='cos' then A[j] := 2*aj else A[j] := aj end if;\n if prntflg>1 and not formulaZero then\n \+ print(`coefficient of`,cos(2*j*Pi*x/L),` -->`,A[j]);\n en d if;\n end do;\n end if;\n if md='numeric' then\n \+ for j from 1 to n do\n # numerical evaluation of the \+ integral\n aj := traperror(evalf(2*Int(alg_expr*cos(2*j*Pi* x/L),\n x=x0..x1)/L)); \n if aj=lasterror or not type(aj,numeric) then\n \+ error \"unable to determine cosine coefficient\"\n end if; \n if typ='cos' then A[j] := 2*aj else A[j] := aj en d if;\n if prntflg>1 and not formulaZero then\n \+ print(`coefficient of`,cos(2*j*Pi*x/L),` -->`,A[j]);\n e nd if;\n end do;\n end if;\n if md='auto' and not fo rmulaOK then\n for j from 1 to n do\n # direct eval uation of the integral\n aj := traperror(simplify(2*int(alg _expr*cos(2*j*Pi*x/L),x=x0..x1)/L));\n if aj=lasterror or i ndets(aj,'specfunc(anything,int)')<>\{\} then\n # leave \+ integral unevaluated\n aj := 2*Int(alg_expr*cos(2*j*Pi*x /L),x=x0..x1)/L;\n nonnumericcoeff := true;\n \+ end if; \n if typ='cos' then A[j] := 2*aj else A[j] := a j end if;\n if prntflg>1 and not formulaZero then\n \+ print(`coefficient of`,cos(2*j*Pi*x/L),` -->`,A[j]);\n \+ end if;\n end do;\n end if;\n if md='formula' and not formulaOK then\n for j from 1 to n do\n # leav e integral unevaluated\n aj := 2*Int(alg_expr*cos(2*j*Pi*x/ L),x=x0..x1)/L;\n if typ='cos' then A[j] := 2*aj else A[j] \+ := aj end if;\n if prntflg>1 and not formulaZero then\n \+ print(`coefficient of`,cos(2*j*Pi*x/L),` -->`,A[j]);\n \+ end if;\n end do;\n end if;\n end if;\n\n #if \+ wasauto then md := 'auto' end if;\n if typ<>'cos' then\n B := t able();\n formulaOK := false;\n formulaZero := false;\n \+ if md='auto' or md='formula' then\n assume(k_>0);\n b k := traperror(2*int(alg_expr*sin(2*k_*Pi*x/L),x=x0..x1)/L);\n \+ formulaOK := evalb(bk<>lasterror and\n indets(bk,'specfu nc(anything,int)')=\{\});\n formulaZero := evalb(bk=0);\n \+ if prntflg>0 and formulaOK then\n assume(m_,posint);\n \+ form := traperror(eval(subs(k_=m_,bk)));\n if fo rm<>lasterror then\n temp := traperror(simplify(form)); \n if temp<>lasterror then form := temp end if;\n \+ form := subs(m_=k,form);\n temp := subs(k_=k,bk) ;\n if typ<>'fourier' then\n form := 2* form;\n temp := 2*temp;\n end if;\n \+ if temp<>form then\n print(`k th sine coef ficient .. `,temp=form);\n else\n print (`k th sine coefficient .. `,temp);\n end if;\n \+ end if;\n end if; \n end if;\n\n if (md='auto' o r md='formula') and formulaOK then\n for j from 1 to n do\n \+ bj := traperror(simplify(eval(subs(k_=j,bk))));\n \+ # if error, try direct evaluation of the integral\n if bj=l asterror then\n bj := traperror(simplify(2*int(alg_expr* sin(2*j*Pi*x/L),x=x0..x1)/L));\n if bj=lasterror or inde ts(bj,'specfunc(anything,int)')<>\{\} then\n #if md=' auto' then\n #wasauto := true;\n \+ #md := 'numeric';\n #break;\n \+ #end if;\n # leave integral unevaluated\n \+ bj := 2*Int(alg_expr*sin(2*j*Pi*x/L),x=x0..x1)/L;\n \+ end if;\n end if;\n if typ='sin' then B[j] := \+ 2*bj else B[j] := bj end if;\n if prntflg>1 and not formula Zero then\n print(`coefficient of`,sin(2*j*Pi*x/L),` --> `,B[j]);\n end if;\n end do;\n end if;\n \+ if md='termbyterm' then\n for j from 1 to n do\n # \+ direct evaluation of the integral\n bj := traperror(simplif y(2*int(alg_expr*sin(2*j*Pi*x/L),\n \+ x=x0..x1)/L));\n if bj=lasterror or indets(bj ,'specfunc(anything,int)')<>\{\} then\n # leave integral unevaluated\n bj := 2*Int(alg_expr*sin(2*j*Pi*x/L),x=x0 ..x1)/L;\n end if;\n if typ='sin' then B[j] := 2 *bj else B[j] := bj end if;\n if prntflg>1 and not formulaZ ero then\n print(`coefficient of`,sin(2*j*Pi*x/L),` -->` ,B[j]);\n end if;\n end do;\n end if;\n i f md='numeric' then\n for j from 1 to n do\n # nume rical evaluation of the integral\n bj := traperror(evalf(2* Int(alg_expr*sin(2*j*Pi*x/L),x=x0..x1)/L));\n if bj=lasterr or or not type(bj,numeric) then\n error \"unable to dete rmine cosine coefficient\"\n end if;\n if typ='s in' then B[j] := 2*bj else B[j] := bj end if;\n if prntflg> 1 and not formulaZero then\n print(`coefficient of`,sin( 2*j*Pi*x/L),` -->`,B[j]);\n end if;\n end do;\n \+ end if;\n if md='auto'and not formulaOK then\n for j fr om 1 to n do\n # direct evaluation of the integral\n \+ bj := traperror(simplify(2*int(alg_expr*sin(2*j*Pi*x/L),x=x0..x1) /L));\n if bj=lasterror or indets(bj,'specfunc(anything,int )')<>\{\} then\n # leave integral unevaluated\n \+ bj := 2*Int(alg_expr*sin(2*j*Pi*x/L),x=x0..x1)/L;\n \+ nonnumericcoeff := true;\n end if; \n if typ ='sin' then B[j] := 2*bj else B[j] := bj end if;\n if prntf lg>1 and not formulaZero then\n print(`coefficient of`,s in(2*j*Pi*x/L),` -->`,B[j]);\n end if;\n end do;\n \+ end if;\n if md='formula' and not formulaOK then \n \+ for j from 1 to n do\n # leave integral unevaluated\n \+ bj := 2*Int(alg_expr*sin(2*j*Pi*x/L),x=x0..x1)/L;\n i f typ='sin' then B[j] := 2*bj else B[j] := bj end if;\n if \+ prntflg>1 and not formulaZero then\n print(`coefficient \+ of`,sin(2*j*Pi*x/L),` -->`,B[j]);\n end if;\n end do ;\n end if;\n end if;\n\n if md='numeric' and not nonnumericc oeff then\n eps := Float(1,-Digits);\n if typ='fourier' then \n mx := max(c,seq(A[i],i=1..n),seq(B[i],i=1..n));\n i f abs(c) " 0 "" {MPLTEXT 1 0 689 "eval f_coeffs := proc(ee::algebraic)\n local i,j,ev,s,t;\n if nargs>1 t hen Digits := args[2] end if;\n if type(op(1,procname),posint) then \+ Digits := op(1,procname) end if;\n ev := proc(z) `if`(type(z,specfun c(algebraic,\{cos,sin\})) and \n nops(indets(op(1,z),name))>=1,z, evalf(z)) end proc;\n if op(0,ee)=`+` then\n s := 0;\n for i to nops(ee) do\n t := op(i,ee);\n if type(t,realcon s) then s := s+evalf(t)\n elif member(op(0,t),\{sin,cos\}) the n s := s+t;\n else s := s+map(ev,t) end if; \n en d do;\n elif member(op(0,ee),\{sin,cos\}) and nops(indets(op(1,ee),n ame))>=1 then s := ee\n else s := map(ev,ee)\n end if; \ne nd proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Examples are given in the next \+ section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "FourierSeri es" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 29 " be the function defined by: " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([-1, -1 <= x and x < 0],[1, 0 <= x and x < 1]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,$\"\"\"!\"\"3 1,$F-F.F'2F'\"\"!7$F-31F3F'2F'F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 29 " is periodic with period 2. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Since the piecewise description of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" } {TEXT -1 36 " is initially given on the interval " }{XPPEDIT 18 0 "-1< =x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" } {TEXT -1 59 ", it is convenient to use this interval with the procedur e " }{TEXT 0 13 "FourierSeries" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "f := x -> p iecewise(x<0,-1,1):\n'f(x)'=f(x);\nf_ := x -> f(x-2*floor((x+1)/2)):\n 'f_(x)'=f_(x);\nFourierSeries(f(x),x=-1..1):\nF := unapply(%,x):\n'F(x )'=F(x);\nplot([f_(x),F(x)],x=-2.2..3.2,color=[black,red],linestyle=[3 ,1],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-% *PIECEWISEG6$7$!\"\"2F'\"\"!7$\"\"\"%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%*PIECEWISEG6$7$!\"\"2,&F'\"\"\"*&\"\"#F /-%&floorG6#,&*&F1F,F'F/F/#F/F1F/F/F,\"\"!7$F/%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,&*(\"\"%\"\"\"%#PiG!\"\"-%$si nG6#*&F,F+F'F+F+F+*&#F*\"\"$F+*&F,F--F/6#,$*(F4F+F,F+F'F+F+F+F+F+" }} {PARA 13 "" 1 "" {GLPLOT2D 628 217 217 {PLOTDATA 2 "6'-%'CURVESG6%7_r7 $$!3;+++++++A!#<$!\"\"\"\"!7$$!3')***\\PoZ69#F*F+7$$!3+++]n`H#3#F*F+7$ $!3y\\ilC=pc?F*F+7$$!3,+D\"=G)3J?F*F+7$$!37D1R5lG=?F*F+7$$!3C](o*QZ[0? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "f := x -> piecewise(x<1,x,2-x):\n' f(x)'=f(x);\n'f_(x)'='f(x-4*floor((x+1)/4))';\nf_ := unapply(rhs(%),x) :\nplot(f_(x),x=-1..8,color=COLOR(RGB,.4,0,.9),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$F'2F'\"\"\"7$ ,&\"\"#F-F'!\"\"%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_ G6#%\"xG-%\"fG6#,&F'\"\"\"*&\"\"%F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,F.F, F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 650 203 203 {PLOTDATA 2 "6'-%'CURVE SG6#7`q7$$!\"\"\"\"!F(7$$!3K++]i!G\">!*!#=F,7$$!3k+++DhDQ!)F.F07$$!37+ ](op/[=(F.F37$$!3g***\\(oKNJjF.F67$$!3W****\\([iF?7$$\"3#****\\7VsmA\"F.FC7$$\"3'))**\\7)R&G2$ F.FF7$$\"3A++D\"ex@)\\F.FI7$$\"3H***\\(oxxLfF.FL7$$\"3Q)**\\i&zP&)oF.F O7$$\"3K+]i:d@kyF.FR7$$\"3F-++vM0V))F.FU7$$\"3[,]i!*Q8u#*F.FX7$$\"3q++ D1V@0(*F.Fen7$$\"3M^i:5W)H\")*F.Fhn7$$\"3'3]iS^a2#**F.F[o7$$\"3=Ec,m&R Y(**F.F^o7$$\"3/vozhC&G+\"!#<$\"3i\\7.#Qv9(**F.7$$\"3Y(=#*p'4C35Fco$\" 3SD\"y+L!f<**F.7$$\"35+v=s%HO,\"Fco$\"3)*)*\\7y_qj)*F.7$$\"3,](oH\\$=N 5Fco$\"3#**\\7.2l\"['*F.7$$\"3#****\\P^Pn0\"Fco$\"3(3++D'[iK%*F.7$$\"3 5+]7.kz`6Fco$\"33***\\(of.i%)F.7$$\"3G++]#Hb3D\"Fco$\"3G(****\\2Z9\\(F .7$$\"3%)****\\P,xX9Fco$\"3q,++D')HUbF.7$$\"3E++]2ngL;Fco$\"3Y(****\\# H$Rm$F.7$$\"30+D\"y]$*)=e>F.7$$\"3&**\\7Gc%f0>Fco$\"3?0+v=PaS%*FA7$$\"3/++]<)3q+#Fco $!3*=/++]<)3q!#?7$$\"3#)***\\P**>H4#Fco$!3'=)***\\P**>H*FA7$$\"3/+++q6 $)y@Fco$!3S++++Wt_&)o!)))*F.7$$\"33Q4@'pL[*HFco$!3!4Q4@'p L[**F.7$$\"3*>`%*o]g3+$Fco$!37!oa5$\\R\"***F.7$$\"3WD\"yvJ()o+$Fco$!3g X(=U#o7J**F.7$$\"3B8`%*Q4%*=IFco$!3lnoa51f5)*F.7$$\"3f+DJgX*4.$Fco$!3; %*\\(oRa+p*F.7$$\"3W]7yX!4#zIFco$!3p&\\(=U&4z?*F.7$$\"3s++DJNUFJFco$!3 z#***\\(okds)F.7$$\"3R+vVGa(f@$Fco$!31'*\\i:dCSyF.7$$\"32+]iDt_/LFco$! 3M***\\PuEZ&pF.7$$\"3x*\\(o4D/+MFco$!3I-]7.\\d**fF.7$$\"3[***\\Ppdb\\$ Fco$!3F0+]iIUW]F.7$$\"3q*\\Pf7rZf$Fco$!3+.]iS()G_SF.7$$\"3#***\\7eX)Rp $Fco$!3s++v=W:gIF.7$$\"37++]U,N!y$Fco$!3y)****\\d)\\'>#F.7$$\"3K+](os: n'QFco$!3'o**\\7tUGL\"F.7$$\"3u***\\77qK0%Fco$\"3ou***\\77qK&FA7$$\"3Y *****\\1**fC%Fco$\"3i%*****\\1**fCF.7$$\"3t***\\itYXV%Fco$\"3K(***\\it YXVF.7$$\"3S*\\(o>:wDXFco$\"3+%*\\(o>:wD&F.7$$\"3'***\\7.j(ph%Fco$\"3e ***\\7.j(phF.7$$\"3D+vVoZD=ZFco$\"3Y-]P%oZD=(F.7$$\"3l***\\PBL&>[Fco$ \"3Y'***\\PBL&>)F.7$$\"31**\\PRk`5\\Fco$\"3j!**\\PRk`5*F.7$$\"3P***** \\kR:+&Fco$\"3K1++]Ng%)**F.7$$\"3)****\\7*\\q)4&Fco$\"3:++](3]H,*F.7$$ \"3e++]P.(e>&Fco$\"3)R****\\i'HT!)F.7$$\"3e**\\7GG'>P&Fco$\"35/+v=Mcd+(Fco$!3 8PMF!eOC%**F.7$$\"3y\\7`WA>6qFco$!3;-voav2)))*F.7$$\"3mCcwae$H.(Fco$!3 Q`PM_9kq'*F.7$$\"3a*****\\YzY0(Fco$!3i/++]`?`%*F.7$$\"3Y**\\P*)>OarFco $!3Y0+D1,Qc%)F.7$$\"3Q***\\P^WSD(Fco$!3K1+]i[bfuF.7$$\"37+](=v,KM(Fco$ !3u)**\\7[#)zc'F.7$$\"3*3+++**eBV(Fco$!37\"******45kn&F.7$$\"3L**\\78% zCi(Fco$!3u1+voe?vPF.7$$\"3a*****\\;kMr(Fco$!3g/++]$e`'GF.7$$\"3v**\\( o\"*[W!yFco$!3[-+DJ3^b>F.7$$\"3))*\\P%eWA-zFco$!3O7+DcTbx(*FA7$$\"\")F *$F*F*-%+AXESLABELSG6$Q\"x6\"Q!Fehl-%*THICKNESSG6#\"\"#-%&COLORG6&%$RG BG$\"\"%F)F`hl$\"\"*F)-%%VIEWG6$;F(F^hl%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "Since the piecewise desc ription of f(" }{TEXT 272 1 "x" }{TEXT -1 37 ") is initially given on \+ the interval " }{XPPEDIT 18 0 "-1<=x" "6#1,$\"\"\"!\"\"%\"xG" } {XPPEDIT 18 0 "`` < 3;" "6#2%!G\"\"$" }{TEXT -1 59 ", it is convenient to use this interval with the procedure " }{TEXT 0 13 "FourierSeries " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 185 "f := x -> piecewise(x<1,x,2-x):\n'f(x)'=f(x); \nFourierSeries(f(x),x=-1..3,numterms=15):\nF := unapply(%,x):\n'F(x)' =F(x);\nplot(F(x),x=-1..8,color=red,ytickmarks=3,thickness=2,numpoints =60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$ 7$F'2F'\"\"\"7$,&\"\"#F-F'!\"\"%*otherwiseG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,2*(\"\")\"\"\"%#PiG!\"#-%$sinG6#,$*(\"\" #!\"\"F,F+F'F+F+F+F+*&#F*\"\"*F+*&F,F--F/6#,$**\"\"$F+F3F4F,F+F'F+F+F+ F+F4*&#F*\"#DF+*&F,F--F/6#,$**\"\"&F+F3F4F,F+F'F+F+F+F+F+*&#F*\"#\\F+* &F,F--F/6#,$**\"\"(F+F3F4F,F+F'F+F+F+F+F4*&#F*\"#\")F+*&F,F--F/6#,$**F 7F+F3F4F,F+F'F+F+F+F+F+*&#F*\"$@\"F+*&F,F--F/6#,$**\"#6F+F3F4F,F+F'F+F +F+F+F4*&#F*\"$p\"F+*&F,F--F/6#,$**\"#8F+F3F4F,F+F'F+F+F+F+F+*&#F*\"$D #F+*&F,F--F/6#,$**\"#:F+F3F4F,F+F'F+F+F+F+F4" }}{PARA 13 "" 1 "" {GLPLOT2D 683 203 203 {PLOTDATA 2 "6(-%'CURVESG6#7`p7$$!\"\"\"\"!$!3;K !\\03Dqu*!#=7$$!3(y%30$)\\+/%)F-$!3q(e4>SV%y$)F-7$$!3UsBa_#Q`,(F-$!3g/ ^By<1VqF-7$$!3K,h'zFPOX&F-$!3$yPw@:s6V&F-7$$!3#*=KfNVe\")QF-$!3k%f(4uj A*)QF-7$$!3lx1W'3.qJ#F-$!3;q!=z*=sBBF-7$$!3%>$fN\"))fVm)!#>$!3\"yGhdvR h`)FJ7$$\"3')z*Q.AAaN'FJ$\"3'f)40`NE)>'FJ7$$\"3g@KfN-*))=#F-$\"3t.\">? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "FourierSeries(cos(x/2),x=-Pi..Pi,numterms=5,info=tru e):\nF := unapply(%,x):\n'F(x)'=F(x);\nplot(F(x),x=-3*Pi..5*Pi,thickne ss=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G, $*&\"\"#\"\"\"%#PiG!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%G ,$*(\"\"%\"\"\"\"\"$!\"\"%#PiGF.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6& %/coefficient~ofG-%$cosG6#,$*&\"\"#\"\"\"%\"xGF*F*%%~-->G,$*(\"\"%F*\" #:!\"\"%#PiGF1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG- %$cosG6#,$*&\"\"$\"\"\"%\"xGF*F*%%~-->G,$*(\"\"%F*\"#N!\"\"%#PiGF1F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*&\"\"% \"\"\"%\"xGF*F*%%~-->G,$*(F)F*\"#j!\"\"%#PiGF0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*&\"\"&\"\"\"%\"xGF*F*%%~-- >G,$*(\"\"%F*\"#**!\"\"%#PiGF1F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%: k~th~sine~coefficient~..~G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"FG6#%\"xG,.*&\"\"#\"\"\"%#PiG!\"\"F+*&#\"\"%\"\"$F+*&F,F--%$cosGF&F+ F+F+*&#F0\"#:F+*&F,F--F46#,$*&F*F+F'F+F+F+F+F-*&#F0\"#NF+*&F,F--F46#,$ *&F1F+F'F+F+F+F+F+*&#F0\"#jF+*&F,F--F46#,$*&F0F+F'F+F+F+F+F-*&#F0\"#** F+*&F,F--F46#,$*&\"\"&F+F'F+F+F+F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 552 225 225 {PLOTDATA 2 "6&-%'CURVESG6$7]v7$$!3^***H>%zxC%*!#<$\"3ud*o gZ_uy&!#>7$$!3U2dS$QAyG*F*$\"3%)G?jmSv*Q(F-7$$!3M:9)[#o'3:*F*$\"3ADr)p !Rz*=\"!#=7$$!3DBrNm7\"R,*F*$\"3-C>;t9/^=F87$$!3QHG$yqbp())F*$\"3\\wT. ![)3>RF87$$!3tV!4.\"GY>&)F*$\"3]gv,[#><[%F87$$!3q\\W86&)H+%)F*$\"3+)*=k !zg*z\\F87$$!3]Fwq$z&Gm#)F*$\"3%3_(4(H(z([&F87$$!3a.3GwIFK\")F*$\"3+4 \"e*HryufF87$$!3czR&)e.E)*zF*$\"3U$fFXA%*zY'F87$$!3gbrUTwCkyF*$\"3!)H; &y'y%R(pF87$$!3_&R=2gY$HxF*$\"3W5C!pGs5[(F87$$!3WN'4+cXWf(F*$\"3#*zc,i 'G\"ezF87$$!3Ov3I>XafuF*$\"3u7!*=W.Fy$)F87$$!3H:@fyMkCtF*$\"3DvJ%)ppmF ()F87$$!3#GSr$>P7cqF*$\"31*p5i$RCT#*F87$$!3M!p],'Rg(y'F*$\"31EpYSF+L'* F87$$!3I85Hly7jmF*$\"3)RV$f&\\bCz*F87$$!3RN8Vq'[(R>*f#**F8 7$$!3)f\\,Is8kZ'F*$\"3oQ*3#oI!z(**F87$$!3Zd;dvc<9kF*$\"3LzETcpr,5F*7$$ !3%*==9Gw$>N'F*$\"3>7Go!*Q@/5F*7$$!3W!)>r!e*p*G'F*$\"3#R(p#etq^+\"F*7$ $!3%>ymvob_A'F*$\"3-Xw\\BI\\/5F*7$$!3W$e@Wz63;'F*$\"33E()pr5:-5F*7$$!3 #eQw7!zO'4'F*$\"3))fv%[rjD)**F87$$!3M(=J\"3S#>.'F*$\"3J%Qma+e)H**F87$$ !3C\"zS=AOI!fF*$\"3i'p'\\o\"3Az*F87$$!3E%R]bV[Tx&F*$\"3i-,N0ewE'*F87$$ !3srbjplb2bF*$\"3XkD#pSKpB*F87$$!3I[2s.Z'4C&F*$\"3Rd:WJ<*es)F87$$!3@U3 ^_i43^F*$\"3K&RBAy(*=Q)F87$$!3+P4I,yAv\\F*$\"3m?^#fb%=pzF87$$!3!=.\"4] $fB%[F*$\"37W\")R')*42](F87$$!3gE6)))*3\\4ZF*$\"36IqD*H@>+(F87$$!3u\"4 ._O>Gd%F*$\"3ch)Rg0)e&['F87$$!3(o0D:$y9OWF*$\"3w\"*4)[0SA)fF87$$!3+Aq% yHw%*H%F*$\"3_r!y$zZc&[&F87$$!3/))*oTw/G;%F*$\"3QiCJxD,n\\F87$$!3N3N\" HSCC/%F*$\"3kXrobpzhWF87$$!36H!e;/W?#RF*$\"3c-.b+([0*QF87$$!3()\\DS!oj ;!QF*$\"3qo`aY$\\?D$F87$$!3=qq9>LG\"o$F*$\"3!4yw2\\(fqDF87$$!3a\"F87$$!3Jg:;Cb(4t#F*$\" 37-VS)43(\\=F87$$!3agK=#H**[f#F*$\"3#y1\"=3Mj5EF87$$!39P&f$GYwjCF*$\"3 g*eEXGA)\\LF87$$!3G8e`k*HEL#F*$\"3>\")Q%\\3jA.%F87$$!3V*372I&\\,AF*$\" 3aI2d$*HK=F*$\"3el hCbaDOgF87$$!3Wl$zhGHSf\"F*$\"3c_0eU>rCpF87$$!3!)eCCu#GCX\"F*$\"3?[\" \\CMXzX(F87$$!38_bIis#3J\"F*$\"3s!pX73!efzF87$$!3YX'o.DE#p6F*$\"3#4'o \"*G\"e&)R)F87$$!3\")Qu@*F87$$!3wBC[Yr1yaF8$\"3c%HD*Hcrt&*F87$$!3M$Gd:i4F3%F8$\"3 )[ZgT)e!zv*F87$$!3#H9Km4_to#F8$\"3>?!yt$zZ8**F87$$!3qs&pTLt'*)>F8$\"3o 7'e=(oht**F87$$!3[-qqrX*>H\"F8$\"3O%4kxX4=+\"F*7$$!3\"GKWC4eJ%fF-$\"3] x@i$4dW+\"F*7$$\"3Kz8=K&HO.\"F-$\"3]h:j?v:05F*7$$\"3ZgN9^(o#4sF-$\"3xv i$)z#=T+\"F*7$$\"3;u0,(z!\\Q8F8$\"3JDv;Voc,5F*7$$\"3G#z1*=Z0c>F8$\"3_J pY=29w**F87$$\"3S5I!3k=Od#F8$\"3)QLVWYLU#**F87$$\"3iYaf%[Y(3QF8$\"3-1E #RrQ8z*F87$$\"3&G)yQGV(Q/&F8$\"3Xu5US[/L'*F87$$\"3SjL8Kj8axF8$\"3%*=&H $\\vFP#*F87$$\"3S%)yeL)Rk/\"F*$\"3sT,b-$Ggr)F87$$\"3sixXo3_/8F*$\"33EY PG8l!)zF87$$\"3/TwK.>gi:F*$\"3nmR4W3-WqF87$$\"3!>yJg9Usp\"F*$\"3].YLzL rMlF87$$\"3zAft)Q#)=$=F*$\"3XM=V\\URPgF87$$\"3oj+WJE_m>F*$\"3SS>T*fp%[ bF87$$\"3b/U9uG;,@F*$\"3\"Q>.z3*oU]F87$$\"3Bf:3ZY![A#F*$\"3W5!GG_xA`%F 87$$\"3!R\"*=+UY%[BF*$\"3cO7,Bg8aRF87$$\"3eoi&H>)3sCF*$\"3!Qt\\pO@TI$F 87$$\"3EBO*e'*Hdf#F*$\"3KTF3vO)eg#F87$$\"3%f*)H9K%4HFF*$\"3CX3e#[n(f=F 87$$\"3ioh'pneC'GF*$\"3_dR*e*4y67F87$$\"3ITC]KI#e*HF*$\"3gg30YEw)f(F-7 $$\"3*RrQ!)Q(=HJF*$\"37*[R]A53!eF-7$$\"3]6;2sarnKF*$\"3!eY'[i\\i\\rF-7 $$\"3-4X5cNC1MF*$\"3QtF>')Qd^6F87$$\"3`1u8S;xWNF*$\"3%)40UO)H,\"=F87$$ \"3//.CZ&F87 $$\"3&fWPW@NhU%F*$\"3)QrUa$4'e%fF87$$\"3CeJ\"3vujb%F*$\"3:A9i4!GUU'F87 $$\"3_q))=(G9mo%F*$\"3\"=ds4kh]\"pF87$$\"3x&y7AZk6#[F*$\"3#=0&G%*pUAuF 87$$\"37+nBdYrb\\F*$\"3'*G*)Q#=mL!zF87$$\"3[91EU[E!4&F*$\"3Nphk7!)QI$) F87$$\"3tHXGF]\"[A&F*$\"3=5CJ\"4uxo)F87$$\"3W%>!eRs2)[&F*$\"3&Hz67gm`? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "f := x -> piecewise(xG*&\"\"\"F%%#PiG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%G\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*&\"\" #\"\"\"%\"xGF*F*%%~-->G,$*(F)F*\"\"$!\"\"%#PiGF0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*&\"\"$\"\"\"%\"xGF*F*%%~ -->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6 #,$*&\"\"%\"\"\"%\"xGF*F*%%~-->G,$*(\"\"#F*\"#:!\"\"%#PiGF1F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*&\"\"&\" \"\"%\"xGF*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%:k~th~si ne~coefficient~..~G/,$*(,&*$)%\"kG\"\"#\"\"\"F,F,!\"\"F--%$sinG6#*&%#P iGF,F*F,F,F2F-F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient ~ofG-%$sinG6#%\"xG%%~-->G#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$*&\"\"#\"\"\"%\"xGF*F*%%~-->G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$*&\"\"$\" \"\"%\"xGF*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coeffic ient~ofG-%$sinG6#,$*&\"\"%\"\"\"%\"xGF*F*%%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$*&\"\"&\"\"\"%\"xGF*F* %%~-->G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,**&\"\" \"F*%#PiG!\"\"F**&#F*\"\"#F*-%$sinGF&F*F**&#F/\"\"$F**&F+F,-%$cosG6#,$ *&F/F*F'F*F*F*F*F,*&#F/\"#:F**&F+F,-F76#,$*&\"\"%F*F'F*F*F*F*F," }} {PARA 13 "" 1 "" {GLPLOT2D 592 196 196 {PLOTDATA 2 "6&-%'CURVESG6$7]z7 $$!3)****>YH&=$G'!#<$\"3_^^l^$)>mj!#>7$$!3A%zEdio/='F*$\"3N(fr\\qV#H7! 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[X\\]!)GFbw7$$\"3Tdqz7&y7C\"F[fn$!3YQN#e3LPw'Fbw7$$\"3mr8 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }{TEXT 268 47 " .. an example involving numeric al integration " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 29 " be the function \+ defined by: " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = exp(sin(x));" "6#/-%\"fG6#%\"xG-%$expG6#-%$sinG6#F'" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "This function has period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 38 ", so we can use any interval of width " }{XPPEDIT 18 0 " 2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 20 " with the procedure " } {TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f := x -> exp(sin (x)):\n'f(x)'=f(x);\nFourierSeries(f(x),x=-Pi..Pi,numterms=4):\nF := u napply(%,x):\n'F(x)'=F(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6 #%\"xG-%$expG6#-%$sinGF&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#% \"xG,,*&#\"\"\"\"\"#F+,$-%$IntG6$-%$expG6#-%$sinGF&/F';,$%#PiG!\"\"F9* $F9F:F+F+,$-F/6$*&F1F+F4F+F6*&F9F:F4F+F+,$-F/6$*&F1F+-%$cosG6#,$*&F,F+ F'F+F+F+F6*&F9F:FEF+F+,$-F/6$*&F1F+-F56#,$*&\"\"$F+F'F+F+F+F6*&F9F:FOF +F+,$-F/6$*&F1F+-FF6#,$*&\"\"%F+F'F+F+F+F6*&F9F:FYF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Integrals for which Maple's symbolic integration procedures cannot obtain analytical expr essions are left unevaluated. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The pro cedure " }{TEXT 0 11 "evalf_coeff" }{TEXT -1 123 " can be used to eval uate the coefficients numerically. This involves numerical evaluation \+ of all the unevaluated integrals." }}{PARA 0 "" 0 "" {TEXT -1 84 "The \+ number of digits to be used can be given as an index or as the second \+ argument. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 130 "evalf_coeffs[10](F(x));\nG := unapply(%,x):\n'G(x) '=G(x);\nplot([f(x),G(x)],x=-3..10,color=[red,green],thickness=[1,3],y tickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,$\"+ye1m7!\"*\"\"\" *&$\"+3#=.8\"F&F'-%$sinG6#%\"xGF'F'*&$\"+&R`\\r#!#5F'-%$cosG6#,$*&\"\" #F'F.F'F'F'!\"\"*&$\"+&)\\oLW!#6F'-F,6#,$*&\"\"$F'F.F'F'F'F9*&$\"+U/Cu a!#7F'-F46#,$*&\"\"%F'F.F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"GG6#%\"xG,,$\"+ye1m7!\"*\"\"\"*&$\"+3#=.8\"F+F,-%$sinGF&F,F,*&$\"+& R`\\r#!#5F,-%$cosG6#,$*&\"\"#F,F'F,F,F,!\"\"*&$\"+&)\\oLW!#6F,-F16#,$* &\"\"$F,F'F,F,F,F<*&$\"+U/Cua!#7F,-F76#,$*&\"\"%F,F'F,F,F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 641 271 271 {PLOTDATA 2 "6'-%'CURVESG6%7cq7$$!\"$ \"\"!$\"3m&oSB#4&Qo)!#=7$$!3SLL$e*pj;F!#<$\"3s(oB1(fQ@mF-7$$!3um;/hV3q CF1$\"3+-'3**)pzn`F-7$$!3ELL3P,\"G>#F1$\"3s;o?9fbOWF-7$$!3CLLL[SD`?F1$ \"3(*fqGm\\gBTF-7$$!3ALLefzp8>F1$\"3sa@p/&>$**QF-7$$!3/+Dcc^![x\"F1$\" 3%Qy_.y!)ev$F-7$$!3(omTNN7fj\"F1$\"3)G)=9Y\"*f'o$F-7$$!3B+]iX492:F1$\" 3o63'op_io$F-7$$!3eL$3x`p$y8F1$\"3eY;\"oJCtu$F-7$$!30n;a.U.X7F1$\"3m7M ')f5XxQF-7$$!3H+]Pp))p66F1$\"39;hq&)R9!3%F-7$$!3VpmTgF.Q(*F-$\"3k%3Y#[ RYtVF-7$$!3+OL3Fo2f$)F-$\"3==ACun0iZF-7$$!3P++v='4+h&F-$\"3]qS)*G'*3ue F-7$$!3Komm\"4cAy#F-$\"3E#\\C)e2Q)f(F-7$$!3@,+voE\"p`\"F-$\"37LbdH8b!e )F-7$$!3,TLLeCp:H!#>$\"3U!o\\g\\!o7(*F-7$$\"3#RL$e*[#R56F-$\"3@]r*G(R= <6F17$$\"3%>++]AaB^#F-$\"3!HY[r/IAG\"F17$$\"3********\\D2?RF-$\"3U5NA9 oFl9F17$$\"3/)****\\(3zF`F-$\"3%[`+=GY=m\"F17$$\"39*****\\K)Q%o'F-$\"3 ]e>]S[[e=F17$$\"3A+++vd)4/)F-$\"3[lQ*=db[0#F17$$\"3UI$eRA1HF*F-$\"3Jvo &=zMbA#F17$$\"31m;HnE[]5F1$\"3aDUaO)R8Q#F17$$\"3c*\\iS#*pp>\"F1$\"3\\w 6H8Z)o`#F17$$\"34LL$3=dMM\"F1$\"3%*46t09B\\EF17$$\"38L$e99/bS\"F1$\"3+ Wf\">'Q[\"o#F17$$\"3=LL3-6bn9F1$\"3/Civ)HXQq#F17$$\"3=LeR#eu&)\\\"F1$ \"3'o5\\`)[?6FF17$$\"3@L$3F1)fH:F1$\"3Ww`Gni(fr#F17$$\"3AL3-V:ig:F1$\" 3)*RmeE69=FF17$$\"3CLLLB]k\"f\"F1$\"3!)>LyQ6phqKxi\"F1 $\"3l#[5()e!)Qr#F17$$\"3s\\i!*)Q?Qm\"F1$\"3:(4fYQalq#F17$$\"3'zq#pr!3* *p\"F1$\"3C>#=u^]dp#F17$$\"3@m\"zWv&*ft\"F1$\"3I2NrwZ_\"o#F17$$\"3p#3_ +7r\"3=F1$\"3M45*3K;Jk#F17$$\"3=**\\i&[Y.)=F1$\"3g3n]ug5#f#F17$$\"3#f; 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := x -> exp(sin(x)):\n'f(x)'=f(x) ;\nFourierSeries(f(x),x=-Pi..Pi,numterms=4,mode=numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expG6#-%$sinGF&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,,$\"+ye1m7!\"*\"\"\"*&$\"+3#=.8\"F&F'-%$sinG6#% \"xGF'F'*&$\"+&R`\\r#!#5F'-%$cosG6#,$*&\"\"#F'F.F'F'F'!\"\"*&$\"+&)\\o LW!#6F'-F,6#,$*&\"\"$F'F.F'F'F'F9*&$\"+U/Cua!#7F'-F46#,$*&\"\"%F'F.F'F 'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The convergence is quite rapid as can be seen from the following erro r curve " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "f := x -> exp(sin(x)):\n'f(x)'=f(x);\nFourierSeries( f(x),x=-Pi..Pi,numterms=8,mode=numeric);\nH := unapply(%,x):\n'H(x)'=H (x);\nplot(f(x)-H(x),x=0..2*Pi,color=blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expG6#-%$sinGF&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4$\"+ye1m7!\"*\"\"\"*&$\"+3#=.8\"F&F'-%$sinG6#%\"xGF'F '*&$\"+&R`\\r#!#5F'-%$cosG6#,$*&\"\"#F'F.F'F'F'!\"\"*&$\"+&)\\oLW!#6F' -F,6#,$*&\"\"$F'F.F'F'F'F9*&$\"+U/Cua!#7F'-F46#,$*&\"\"%F'F.F'F'F'F'*& $\"+>JEHa!#8F'-F,6#,$*&\"\"&F'F.F'F'F'F'*&$\"+$GKx\\%!#9F'-F46#,$*&\" \"'F'F.F'F'F'F9*&$\"+hkV)>$!#:F'-F,6#,$*&\"\"(F'F.F'F'F'F9*&$\"+Ah7#*> !#;F'-F46#,$*&\"\")F'F.F'F'F'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-% \"HG6#%\"xG,4$\"+ye1m7!\"*\"\"\"*&$\"+3#=.8\"F+F,-%$sinGF&F,F,*&$\"+&R `\\r#!#5F,-%$cosG6#,$*&\"\"#F,F'F,F,F,!\"\"*&$\"+&)\\oLW!#6F,-F16#,$*& \"\"$F,F'F,F,F,F<*&$\"+U/Cua!#7F,-F76#,$*&\"\"%F,F'F,F,F,F,*&$\"+>JEHa !#8F,-F16#,$*&\"\"&F,F'F,F,F,F,*&$\"+$GKx\\%!#9F,-F76#,$*&\"\"'F,F'F,F ,F,F<*&$\"+hkV)>$!#:F,-F16#,$*&\"\"(F,F'F,F,F,F<*&$\"+Ah7#*>!#;F,-F76# ,$*&\"\")F,F'F,F,F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 591 182 182 {PLOTDATA 2 "6&-%'CURVESG6#7]al7$$\"\"!F)$!38W)3LXAyJ)!#F7$$\"3Hjqb\"[ W>r\"!#>$\"3w![iU#f\"=l)F,7$$\"3eET6j*))QU$F0$\"3WJ2R3M/QD!#E7$$\"3_*= rYWLe8&F0$\"3#zOmZE(3ZTF87$$\"3;`#Gi#zxZoF0$\"3uB%e#4I-acF87$$\"3!oJ&y 2Csf&)F0$\"3Tid!e?)*G-(F87$$\"3\"zBM*)omr-\"!#=$\"3HNB-%\\c4A)F87$$\"3 EW**3P6O)>\"FK$\"31Vkn#=B%>#*F87$$\"3i]cC&eb&p8FK$\"3)e1`p(y>%***F87$$ \"3W!>9#f4^=:FK$\"3e_pg\"e_r/\"!#D7$$\"3_IF=LjYn;FK$\"3sua#*))[iv5Fgn7 $$\"3c+q;?S%>u\"FK$\"3u3x0sSX#3\"Fgn7$$\"3gq7:263x`'>FK$\"39PotF8'H2 \"Fgn7$$\"3e!*o0HyGjAFK$\"3#z^\\#y;[8**F87$$\"3YqR*pd)>hDFK$\"31*zgm#e pj$)F87$$\"3\\LyxtWrGFFK$\"3]AKmG=\")3sF87$$\"3'fph0PIi*GFK$\"3cRZq%GQ 9)eF87$$\"3WebMniujIFK$\"3zlF&4)z[6WF87$$\"3Y@%HT;i7B$FK$\"3O2C(\\c)=K GF87$$\"3[%G841y()R$FK$\"3Ud]Sr0Lz6F87$$\"3'p9(pdRHmNFK$!35*3.)G!*R&4& F,7$$\"3W45[a)4Qt$FK$!3+5kL4h(f>#F87$$\"3Zs[E^dK,RFK$!3=9`bnS]TQF87$$ \"3'y:^@0_*pSFK$!3W&*4H!RY&=aF87$$\"3CVu.`$y&QUFK$!3?K.$[h)RzoF87$$\"3 iGP#Rl/sS%FK$!3'>2w:Y_+>)F87$$\"3+9+\"[&4$ed%FK$!39;KqQX!*>$*F87$$\"3y %e#ecN38\\FK$!3?Qq[R8g$4\"Fgn7$$\"3ab^NehL]_FK$!3Wq16\\Zdd6Fgn7$$\"3]9 8uIF87$$\"3J-7h\">DT!pFK$!3;Y.wBCX,:F87$$\"3O3h= 5.sfqFK$\"3iB#G\")Qktx*!#G7$$\"3]:5wGaJ:sFK$\"3[3.U<&3?p\"F87$$\"3c@fL Z0\"4P(FK$\"3)p!Q5=Ew\\KF87$$\"3fF3\"fm0l_(FK$\"3[SU1Cy@SZF87$$\"3uMd[ %y+@o(FK$\"3,.r)*Rc$Q8'F87$$\"3zS11.fpPyFK$\"3:.Zk6$[HS(F87$$\"3'QF(yo `\"*f\")FK$\"3&**oUqE'pH&*F87$$\"3#e!R^M[8#[)FK$\"3i>qw$[]O3\"Fgn7$$\" 3!QcX4q*oi&)FK$\"3$[GBF6C?5\"Fgn7$$\"3!=AxtcWKk)FK$\"3.n=3V0V96Fgn7$$ \"3zz)3QV*zB()FK$\"3$3\"HeG-!37\"Fgn7$$\"3yP0C+VN/))FK$\"3?h?**Qc4@6Fg n7$$\"3w`Q5LSYl*)FK$\"3Pc[$yquM5\"Fgn7$$\"3upr'fwtl7*FK$\"3GzF0zA\">1 \"Fgn7$$\"3ylU;#o$>$H*FK$\"3)3CL\\btl%**F87$$\"3#=Oh$)f8)f%*FK$\"3p*z) pyG3U!*F87$$\"3'yXeX^Lki*FK$\"3[6/fdC\"f#zF87$$\"3!Rbb2V`Iz*FK$\"3&ern iIiJi'F87$$\"3%*\\E&pMt'f**FK$\"31OGrB\"3K;&F87$$\"3gu\\JE$HE,\"!#<$\" 3?6wr5#Q!zNF87$$\"34%oMzJ\"HH5F\\^l$\"3(GG*4`n]1>F87$$\"3!QRa&4L&f/\"F \\^l$\"3xnt(=x=_$=F,7$$\"3/lc&f'=ci5F\\^l$!3)\\8!['[v_a\"F87$$\"31OpNA /KGT'F87$$\"3w\\2c\"4'**G6F\\^l$!3VD[1`Wg1yF87$ $\"3+@?'zk/c9\"F\\^l$!37%p1T!>(3.*F87$$\"3C#HjV?8A;\"F\\^l$!3TQ7iD?y05 Fgn7$$\"3YjXwg<#)y6F\\^l$!39^'=E.2k3\"Fgn7$$\"3y^VZ-d!f>\"F\\^l$!3v`(* 4J&yV9\"Fgn7$$\"38ST=W'*)H@\"F\\^l$!3gS&=r6Yc<\"Fgn7$$\"3HM!R]hJ:A\"F \\^l$!3?T3e,++\"=\"Fgn7$$\"3YGR*eet+B\"F\\^l$!3)=)GN9.Wz6Fgn7$$\"3iA)[ nb:'Q7F\\^l$!3mKyIcQ(4<\"Fgn7$$\"3!or.w_drC\"F\\^l$!3]Z;)Hb[c:\"Fgn7$$ \"3W$HB5TD8G\"F\\^l$!3$=ME9XX\"G5Fgn7$$\"36qGW%H$\\:8F\\^l$!35b9FS<4^! 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Pi) = Int(f( x)*cos*k*x,x = -Pi .. 0);" "6#/-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF,% \"kGF,F+F,/F+;\"\"!%#PiG-F%6$**-F)6#F+F,F-F,F.F,F+F,/F+;,$F2!\"\"F1" } {XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 7 ", when " }{TEXT 276 1 "k" }{TEXT -1 12 " is odd and " }{XPPEDIT 18 0 "Int(f(x)*sin*k*x,x = 0 .. Pi) = Int(f(x)*sin*k*x,x = -Pi .. 0);" "6#/-%$IntG6$**-%\"fG6#%\"x G\"\"\"%$sinGF,%\"kGF,F+F,/F+;\"\"!%#PiG-F%6$**-F)6#F+F,F-F,F.F,F+F,/F +;,$F2!\"\"F1" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 7 ", when " }{TEXT 277 1 "k" }{TEXT -1 10 " is even. " }}{PARA 0 "" 0 "" {TEXT -1 32 "It follows that the coefficient " }{XPPEDIT 18 0 "a[k]=1/Pi" "6 #/&%\"aG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(f(x)*cos*k*x,x=-Pi..Pi)" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$cosGF+% \"kGF+F*F+/F*;,$%#PiG!\"\"F1" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "cos*k *x" "6#*(%$cosG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 26 " in the Fourier ser ies of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " } {TEXT 265 4 "zero" }{TEXT -1 6 " when " }{TEXT 278 1 "k" }{TEXT -1 4 " is " }{TEXT 265 3 "odd" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "Similarly, the coefficient " }{XPPEDIT 18 0 "b[k] = 1/Pi;" "6#/&% \"bG6#%\"kG*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f (x)*sin*k*x,x = -Pi .. Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$sinGF+ %\"kGF+F*F+/F*;,$%#PiG!\"\"F1" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "sin* k*x;" "6#*(%$sinG\"\"\"%\"kGF%%\"xGF%" }{TEXT -1 26 " in the Fourier s eries of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 265 4 "zero" }{TEXT -1 6 " when " }{TEXT 279 1 "k" }{TEXT -1 4 " is " }{TEXT 265 4 "even" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Note that the graph of " } {XPPEDIT 18 0 "g(x)=exp(sin*x)*cos*9*x" "6#/-%\"gG6#%\"xG**-%$expG6#*& %$sinG\"\"\"F'F.F.%$cosGF.\"\"*F.F'F." }{TEXT -1 32 " is symmetrical a bout the points" }{XPPEDIT 18 0 "``(-Pi/2,0)" "6#-%!G6$,$*&%#PiG\"\"\" \"\"#!\"\"F+\"\"!" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(Pi/2,0)" "6#- %!G6$*&%#PiG\"\"\"\"\"#!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "g := x -> e xp(sin(x))*cos(9*x):\n'g(x)'=g(x);\np1 := plot(g(x),x=-Pi-.4..Pi+.4): \np2 := plot([[[-Pi,-3.3],[-Pi,3.3]],[[Pi,-3.3],[Pi,3.3]]],color=navy, linestyle=2):\np3 := plot([[[-Pi/2,0],[Pi/2,0]]$3],color=black,\n \+ style=point,symbol=[circle,diamond,cross]):\nplots[display](p1,p2,p 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&-%$expG6#-%$sin GF&\"\"\"-%$cosG6#,$*&\"\"*F.F'F.F.F." }}{PARA 13 "" 1 "" {GLPLOT2D 595 313 313 {PLOTDATA 2 "6*-%'CURVESG6$7[\\l7$$!3')***4pk#fTN!#<$\"3j_ l85[sB8F*7$$!3[$p!zmI%>`$F*$\"3u>2ZJ#*>j8F*7$$!34(Qrm[$HANF*$\"3)RaYDD O;R\"F*7$$!3D!3_l!Rk7NF*$\"3KB0xDT,49F*7$$!3'QxKkK%*H]$F*$\"3w5MNXpR:9 F*7$$!3[nMJYZM$\\$F*$\"3ulRIgD$4T\"F*7$$!3kgT>m^p$[$F*$\"3-(Q$F*$\"3gWANC%=Bh(!#= 7$$!3Sgzts^h`LF*$\"3Kj;s1]w&3%Fin7$$!3.C[Z!)4.?LF*$\"3)z64YE_m>%!#>7$$ !3o(o6#)yYkG$F*$!3s(HOsf5$\\IFin7$$!3K^&[ffiGD$F*$!3K:)>PN2T-'Fin7$$!3 pMwGfO4:KF*$!3;jQ/Ep\">\\)Fin7$$!31=niAZKxJF*$!3)*HC8d?SK)*Fin7$$!3o)[ h%)[#)y;$F*$!3o'3wy/=.)**Fin7$$!3ufiHa-WeJF*$!3]]L%>+@`+\"F*7$$!3!3.J, -)**[JF*$!3SKX%z5'>05F*7$$!3V,e'fyb&RJF*$!3=z\\#yDvz(**Fin7$$!3cV`j<8n ?JF*$!3ziej)p())>'*Fin7$$!3C&)[I\\oy,JF*$!3/n.sCE[d-$F*$!3]W#y=Y['*[%Fin7$$!3yLaBo*Gx)HF* $!3A!>h#G4s'e\"Fin7$$!3i;cau'4(\\HF*$\"3-;@k-(\\^G\"Fin7$$!3UP')H&[KS( GF*$\"31UFyomN-dFin7$$!3me;0'Hb$)z#F*$\"3mrY#*ptcKrFin7$$!3)Q=lU&G>GFF *$\"3,VsAnp&4g&Fin7$$!334(yCTI!eEF*$\"3'e)=5&)G$[@#Fin7$$!3%)[g(zE\"Q& e#F*$!3$34')*p=R1l^Z#F*$!3W..ba'G8<&Fin7$$!3'Rk&fO '\\@N&HQ;@F*$\"3%) o**yd$p0<%Fin7$$!3'=^%y;Q7(4#F*$\"3H=YKBko5UFin7$$!3cFv/!oky2#F*$\"3'y 6$)y3()f7%Fin7$$!3oV0JVbge?F*$\"3))Q7gX\\I@RFin7$$!3eMp#3r^2*>F*$\"3AQ =#z\\v**Q#Fin7$$!3rDLMyy*G#>F*$\"3O4f3:_bo5Fdo7$$!37'H.T`5l%=F*$!3z?% \\!zk%GM#Fin7$$!3MmK')*=B,x\"F*$!3[(Qfh#[KfOFin7$$!3A5'f.%z%4v\"F*$!3b *G#\\u].MPFin7$$!34af&3psAMp\"F*$!3@2JY)Rm$4LFin7$$!3/I8%Gpr]l\"F*$!3,z; 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%'CIRCLEG-Fcio6&FeioFjioFjioFjio-%&STYLEG6#%&POINTG-F$6&Fj[p-Fb\\p6#%( DIAMONDGFe\\pFg\\p-F$6&Fj[p-Fb\\p6#%&CROSSGFe\\pFg\\p-%+AXESLABELSG6%Q \"x6\"Q!Fi]p-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+aEfTN!\"*$\"+aEfTNFe^pF^ ^p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "h := x -> ex p(sin(x))*sin(8*x):\n1/Pi*Int(h(x),x=-Pi..Pi);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-F,6# ,$*&\"\")F/F.F/F/F//F.;,$%#PiG!\"\"F8*$F8F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The fir st omitted term (with a non-zero coefficient) in the previous truncate d Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " is the term in " }{XPPEDIT 18 0 "sin*9*x;" "6#*(%$sinG\"\"\"\" \"*F%%\"xGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "The coef ficient of this term is given by " }{XPPEDIT 18 0 "1/Pi;" "6#*&\"\"\"F $%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin*9*x,x = -Pi \+ .. Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$sinGF+\"\"*F+F*F+/F*;,$%#P iG!\"\"F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 70 "The coeffic ient can be calculated by numerical integration as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f \+ := x -> exp(sin(x)):\n1/Pi*Int(f(x)*sin(9*x),x=-Pi..Pi);\nb9 := evalf[ 12](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%$expG6#-%$si nG6#%\"xG\"\"\"-F,6#,$*&\"\"*F/F.F/F/F//F.;,$%#PiG!\"\"F8*$F8F9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b9G$\"-hTrn.6!#>" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The following picture s hows that the error " }{XPPEDIT 18 0 "f(x)-H(x)" "6#,&-%\"fG6#%\"xG\" \"\"-%\"HG6#F'!\"\"" }{TEXT -1 13 ", plotted in " }{TEXT 263 4 "blue" }{TEXT -1 81 ", differs very little from the first omitted term in the Fourier series, namely, " }{XPPEDIT 18 0 "b[9]*sin*9*x" "6#**&%\"bG6# \"\"*\"\"\"%$sinGF(F'F(%\"xGF(" }{TEXT -1 13 ", plotted in " }{TEXT 280 7 "magenta" }{TEXT -1 2 ". 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gn7$F^aq$\"3SH.+\"z?L5\"Fgn7$Fcaq$\"3;')z7h0d_5Fgn7$Fhaq$\"3k?S2Z$3y>* F87$F]bq$\"3[HcbOgCc#)F87$Fbbq$\"3%H:y(Q^,`rF87$Fgbq$\"3+\"[9LU=(4fF87 $F\\cq$\"3+-$y*R8q]XF87$Facq$\"3h+rY\"peX2]F87$F_eq$!3jC\"QuY7\\T'F87$Fdeq$!39J6/GVgzwF87$Fieq$!3>9k(e^ aIg*F87$F^fq$!39\"R\\^cBZ2\"Fgn7$Fcfq$!3oK\"*H7>\"**3\"Fgn7$Fhfq$!3u32 _tq`*4\"Fgn7$F]gq$!3zA!e&>*\\N5\"Fgn7$Fbgq$!3mf/$y'*H>5\"Fgn7$Fggq$!3_ m'F\\^l$!3]F]e$4Zmm(F87$Ffhq$!3M _?C,g#=O'F87$F[iq$!3;Rd#*oPB2\\F87$F`iq$!3;[t)Hk9rL$F87$Feiq$!3aNHlAEV )o\"F87$Fjiq$!3!eR=ufbvC\"!#K-F_jq6&FajqFbjqF(Fbjq-%+AXESLABELSG6$Q\"x 6\"Q!Fh_v-%%VIEWG6$;F($\"+3`=$G'!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }{TEXT 269 28 " .. a fi nite Fourier series " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 " The Fourier series for " }{XPPEDIT 18 0 "f(x) = sin^6 *x;" "6#/-%\"fG6#%\"xG*&%$sinG\"\"'F'\"\"\"" }{TEXT -1 12 " is finite ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "FourierSeries(sin(x)^6,x=0..2*Pi,numterms=6);\n\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,*#\"\"&\"#;\"\"\"*&#\"#:\"#KF'-%$cosG 6#,$%\"xG\"\"#F'!\"\"*&#\"\"$F&F'-F-6#,$F0\"\"%F'F'*&#F'F+F'-F-6#,$F0 \"\"'F'F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 7 "combine" }{TEXT -1 26 " produ ces the same result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "combine(sin(x)^6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*#\"\"&\"#;\"\"\"*&#\"#:\"#KF'-%$cosG6#,$%\"xG\"\"#F'! \"\"*&#\"\"$F&F'-F-6#,$F0\"\"%F'F'*&#F'F+F'-F-6#,$F0\"\"'F'F2" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }{TEXT 270 29 " .. a symbolic cosine series " }}{PARA 0 "" 0 "" {TEXT -1 31 " The Fourier cosine series for " }{XPPEDIT 18 0 "f(x)=x^2,0<=x" "6$/-%\"fG 6#%\"xG*$F'\"\"#1\"\"!F'" }{XPPEDIT 18 0 "``<=L" "6#1%!G%\"LG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "L := 'L':\nFourierSeries(x^2,x=0..L,type=cos);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\"$!\"\"%\"LG\"\"#\"\"\"**\"\"%F )F'F(%#PiG!\"#-%$cosG6#*(F,F)%\"xGF)F'F&F)F&*(F'F(F,F--F/6#,$**F(F)F,F )F2F)F'F&F)F)F)*&#F+\"\"*F)*(F'F(F,F--F/6#,$**F%F)F,F)F2F)F'F&F)F)F)F& *&#F)F+F)*(F'F(F,F--F/6#,$**F+F)F,F)F2F)F'F&F)F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }{TEXT 271 41 " .. gene rating formulas for coefficients " }}{PARA 0 "" 0 "" {TEXT -1 85 " We \+ can use an undefined function to generate formulas for the Fourier coe fficients. " }}{PARA 0 "" 0 "" {TEXT -1 27 "(a) The Fourier series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the inter val " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f := 'f':\nFourierSeries(f(x),x=-Pi..Pi,numterms=3,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G,$*&#\"\" \"\"\"#F',$-%$IntG6$-%\"fG6#%\"xG/F0;,$%#PiG!\"\"F4*$F4F5F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#%\"xG%%~-->G ,$-%$IntG6$*&-%\"fGF&\"\"\"F$F0/F';,$%#PiG!\"\"F4*$F4F5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*&\"\"#\"\"\"%\"xGF* F*%%~-->G,$-%$IntG6$*&-%\"fG6#F+F*F$F*/F+;,$%#PiG!\"\"F8*$F8F9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$*&\"\"$\" \"\"%\"xGF*F*%%~-->G,$-%$IntG6$*&-%\"fG6#F+F*F$F*/F+;,$%#PiG!\"\"F8*$F 8F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#%\"xG %%~-->G,$-%$IntG6$*&-%\"fGF&\"\"\"F$F0/F';,$%#PiG!\"\"F4*$F4F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$*&\"\"#\" \"\"%\"xGF*F*%%~-->G,$-%$IntG6$*&-%\"fG6#F+F*F$F*/F+;,$%#PiG!\"\"F8*$F 8F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$*& \"\"$\"\"\"%\"xGF*F*%%~-->G,$-%$IntG6$*&-%\"fG6#F+F*F$F*/F+;,$%#PiG!\" \"F8*$F8F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "(b) The Fourier series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[alpha, alpha+T ];" "6#7$%&alphaG,&F$\"\"\"%\"TGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "unassign(' f','alpha','T'):\nFourierSeries(f(x),x=alpha..alpha+T,numterms=3,info= true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G, $-%$IntG6$-%\"fG6#%\"xG/F+;%&alphaG,&F.\"\"\"%\"TGF0*$F1!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$**\"\"#\" \"\"%#PiGF*%\"xGF*%\"TG!\"\"F*%%~-->G,$,$-%$IntG6$*&-%\"fG6#F,F*F$F*/F ,;%&alphaG,&F;F*F-F**&F)F*F-F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/ coefficient~ofG-%$cosG6#,$**\"\"%\"\"\"%#PiGF*%\"xGF*%\"TG!\"\"F*%%~-- >G,$,$-%$IntG6$*&-%\"fG6#F,F*F$F*/F,;%&alphaG,&F;F*F-F**&\"\"#F*F-F.F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$**\"\" '\"\"\"%#PiGF*%\"xGF*%\"TG!\"\"F*%%~-->G,$,$-%$IntG6$*&-%\"fG6#F,F*F$F */F,;%&alphaG,&F;F*F-F**&\"\"#F*F-F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$**\"\"#\"\"\"%#PiGF*%\"xGF*%\"TG!\" \"F*%%~-->G,$,$-%$IntG6$*&-%\"fG6#F,F*F$F*/F,;%&alphaG,&F;F*F-F**&F)F* F-F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$ **\"\"%\"\"\"%#PiGF*%\"xGF*%\"TG!\"\"F*%%~-->G,$,$-%$IntG6$*&-%\"fG6#F ,F*F$F*/F,;%&alphaG,&F;F*F-F**&\"\"#F*F-F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$**\"\"'\"\"\"%#PiGF*%\"xGF* %\"TG!\"\"F*%%~-->G,$,$-%$IntG6$*&-%\"fG6#F,F*F$F*/F,;%&alphaG,&F;F*F- F**&\"\"#F*F-F.F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "(c) The sine series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,L]" "6#7$ \"\"!%\"LG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "unassign('f','L'):\nFourierSeries(f (x),x=0..L,type=sine,numterms=3,info=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#*(%#PiG\"\"\"%\"xGF)%\"LG!\" \"%%~-->G,$,$-%$IntG6$*&-%\"fG6#F*F)F$F)/F*;\"\"!F+*&\"\"#F)F+F,F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$sinG6#,$**\"\"#\" \"\"%#PiGF*%\"xGF*%\"LG!\"\"F*%%~-->G,$,$-%$IntG6$*&-%\"fG6#F,F*F$F*/F ,;\"\"!F-*&F)F*F-F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient ~ofG-%$sinG6#,$**\"\"$\"\"\"%#PiGF*%\"xGF*%\"LG!\"\"F*%%~-->G,$,$-%$In tG6$*&-%\"fG6#F,F*F$F*/F,;\"\"!F-*&\"\"#F*F-F.F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "(d) The cosine series of \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interv al " }{XPPEDIT 18 0 "[0,L]" "6#7$\"\"!%\"LG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "un assign('f','L'):\nFourierSeries(f(x),x=0..L,type=cosine,numterms=3,inf o=true):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~--> G,$-%$IntG6$-%\"fG6#%\"xG/F+;\"\"!%\"LG*$F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#*(%#PiG\"\"\"%\"xGF)%\"LG!\" \"%%~-->G,$,$-%$IntG6$*&-%\"fG6#F*F)F$F)/F*;\"\"!F+*&\"\"#F)F+F,F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#,$**\"\"#\" \"\"%#PiGF*%\"xGF*%\"LG!\"\"F*%%~-->G,$,$-%$IntG6$*&-%\"fG6#F,F*F$F*/F ,;\"\"!F-*&F)F*F-F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coefficient ~ofG-%$cosG6#,$**\"\"$\"\"\"%#PiGF*%\"xGF*%\"LG!\"\"F*%%~-->G,$,$-%$In tG6$*&-%\"fG6#F,F*F$F*/F,;\"\"!F-*&\"\"#F*F-F.F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q \+ 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 63 "Find the first few terms of the Fourier series of the fun ction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 20 " defined as follows:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) \+ = PIECEWISE([1, -1 <= x and x < 0],[1-x, 0 <= x and x < 1]);" "6#/-%\" fG6#%\"xG-%*PIECEWISEG6$7$\"\"\"31,$F,!\"\"F'2F'\"\"!7$,&F,F,F'F031F2F '2F'F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 28 " is periodic with period 2. " }}{PARA 0 "" 0 "" {TEXT -1 41 "Draw the graphs of some truncated series." }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "f := x -> p iecewise(x<0,1,1-x):\n'f(x)'=f(x);\nf_ := x -> f(x-2*floor((x+1)/2)); \nplot(f_(x),x=-3.3..5.3,color=COLOR(RGB,.4,0,1),\n \+ thickness=2,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"\"2F'\"\"!7$,&F,F,F'!\"\"%*otherwise G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f_Gf*6#%\"xG6\"6$%)operatorG%& arrowGF(-%\"fG6#,&9$\"\"\"*&\"\"#F1-%&floorG6#,&F0#F1F3F8F1F1!\"\"F(F( F(" }}{PARA 13 "" 1 "" {GLPLOT2D 529 130 130 {PLOTDATA 2 "6(-%'CURVESG 6#7is7$$!3#)*************H$!#<$\"3?)*************H!#=7$$!3?LL3PAF1KF*$ \"3aFL$3PAF1#F-7$$!3dmm;uWa7JF*$\"3vlmmTZWD6F-7$$!3/$e*[h%o<2$F*$\"3k/ $e*[h%o<(!#>7$$!3'**\\7)[C*4.$F*$\"3!=&*\\7)[C*4$F=7$$!33HKkX%)z?IF*$ \"3+3HKkX%)z?F=7$$!3keRZUWg5IF*$\"3=keRZUWg5F=7$$!3)HK*)3W2b+$F*$\"3ry 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{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________ ______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q 2" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 63 "Find the first fe w terms of the Fourier series of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " defined as follows: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = 1-abs(x);" "6#/-%\"fG6#%\"x G,&\"\"\"F)-%$absG6#F'!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "-1 <= \+ x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is periodic with period 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "Draw the graphs of some truncated series. 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G\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 28 " is periodic with period 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Draw the graphs of some truncated \+ series." }}{PARA 0 "" 0 "" {TEXT -1 34 "______________________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "f := x -> 4-x^2;\nf_ := x -> f(x-4*floor((x+2)/4)); \nplot(f_(x),x=-2.3..10.3,color=COLOR(RGB,.4,0,1),thickness=2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&\"\"%\"\"\"*$)9$\"\"#F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f_Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%\"fG6#,&9$\"\"\"*&\" \"%F1-%&floorG6#,&F0#F1F3#F1\"\"#F1F1!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 430 133 133 {PLOTDATA 2 "6'-%'CURVESG6#7gt7$$!3#)*********** **H#!#<$\"3V************46F*7$$!3!***\\Pk*Q8B#F*$\"3]?!>1T\"Q=()!#=7$$ !3)****\\(Gzni@F*$\"33*4w\\1wCC'F27$$!3g**\\7$*o,%4#F*$\"3'y?rI'RGsOF2 7$$!3p****\\deND?F*$\"3z'Q/\\50y+\"F27$$!3&>\"y]d/U5?F*$\"3#zad'4WKdT! #>7$$!3mCc^d][&*>F*$\"3C5k=B&QR!=FG7$$!3;PM_d'\\0)>F*$\"3Qo&z$3aIUxFG7 $$!3k\\7`dUhl>F*$\"3aT()*e!fgj8F27$$!3guoadMuN>F*$\"388ZklE(*GDF27$$!3 !)*\\ivlse!>F*$\"3[UlbJT\\wOF27$$!3'*\\Pfd58Y=F*$\"3JBy(==,!=fF27$$!3! 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