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2 0 1 }{PSTYLE "Maple Outpu t" -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 "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }2 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {SECT 1 {PARA 256 "" 0 "" {TEXT -1 46 "Chapter 11\n\nSeries, A pproximation, and Limits\n" }}{PARA 0 "" 0 "" {TEXT 296 31 "\251 Copyr ight 2003 by Andr\351 Heck." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 257 2 "1 ." }{TEXT -1 76 " Study various numerical approximations of the sine f unction on the segment " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F% " }{TEXT -1 2 ".\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "Taylor Appr oximations" }}{PARA 0 "" 0 "" {TEXT -1 72 "We make an animation of suc cessive Taylor series approximations about 0:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numapprox):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 " for i from 3 to 11 by 2 do \n T[i] := convert(taylor(sin(x), x=0, i+1 ), polynom):\n P[i] := plot([sin(x), T[i]], x=-Pi..Pi, color=[blue,re d] )\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 27 "The Taylor polynomials \+ are:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq(T[2*i+1], i=1..5 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6',&%\"xG\"\"\"*&#F%\"\"'F%*$)F$\" \"$F%F%!\"\",(F$F%*&#F%F(F%F)F%F,*&#F%\"$?\"F%*$)F$\"\"&F%F%F%,*F$F%*& #F%F(F%F)F%F,*&F1F%F3F%F%*&#F%\"%S]F%*$)F$\"\"(F%F%F,,,F$F%*&#F%F(F%F) F%F,*&F1F%F3F%F%*&#F%F " 0 "" {MPLTEXT 1 0 57 "plots[display]([seq(P[2*i+1] , i=1..5)], insequence=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 341 341 341 {PLOTDATA 2 "6$-%(ANIMATEG6'7&-%'CURVESG6$7en7$$!3*****4tk#fTJ!#<$ !3=5KT_Kzzi!#E7$$!3w\"*pr)3PY+$F.$!3UWkbU#y_O\"!#=7$$!3?3E[*ysa)GF.$!3 4`kI;!*GLDF77$$!3gX&))>2g9v#F.$!3mJ^0x.6.QF77$$!3)4577.flh#F.$!3G?))Q' )4U7]F77$$!3u\"QM::*H#[#F.$!3`Zq;6deDhF77$$!3y\\AhcI#yN#F.$!37nDKOFafq 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zSb-\")*F77$Feo$!3!*zbDId64**F77$Fjo$!3'oX\"\\:LXx**F77$F_p$!2_Htul(** ****F.7$Fdp$!3)*\\8Be\\Xw**F77$Fip$!3ww4B5ru1**F77$F^q$!3%os9>b*='z*F7 7$Fcq$!3n,Py!\\KNk*F77$Fhq$!3/&eGL 'zF77$F^w$\"3WWo#fSBcm)F77$Fcw$\"3iqb!H#***4E*F77$Fhw$\"3q[0xT'[Pl*F77 $F]x$\"3N)H#RR582)*F77$Fbx$\"3FW%*Q2=C;**F77$Fgx$\"3;\\FIGn#4d^WFc]oFiilF[jlFaj l-Fg]l6&Fi]lF[^lF[^lF[^l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 26 "The errors on the segment " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F% " }{TEXT -1 43 " for the successive Taylor polynomials are:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "seq(abs(eval(T[2*i+1],x=evalf(Pi))) , i=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"+G,7E?!\"*$\"*9R/C&F% $\"*gh?_(!#5$\"*rq_#p!#6$\"*Y-;X%!#7" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 19 "Pad\351 Approximations" }}{PARA 0 "" 0 "" {TEXT -1 159 "W e make the Pad\351 rational approximations about 0 of degree (3,2) and (5,3), plot the graphs together with the sine plot, and compute the e rrors on the segment " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numapprox):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "P[3,2] := pade(sin(x), x=0, \+ [3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6$\"\"$\"\"#*&,&*&# \"\"(\"#g\"\"\"*$)%\"xGF'F/F/!\"\"F2F/F/,&F/F/*&\"#?F3F2F(F/F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "P[5,2] := pade(sin(x), x=0, \+ [5,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"PG6$\"\"&\"\"#*&,(*&# \"#6\"%?D\"\"\"*$)%\"xGF'F/F/F/*&#F/\"\"(F/*$)F2\"\"$F/F/!\"\"F2F/F/,& F/F/*&\"#UF9F2F(F/F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plo t([P[3,2], P[5,2], sin(x)], x=-Pi..Pi, \n color=[red, green, blue] );" }}{PARA 13 "" 1 "" {GLPLOT2D 341 341 341 {PLOTDATA 2 "6(-%'CURVESG 6$7in7$$!3*****4tk#fTJ!#<$\"3k%*p(GQ*)e=$!#=7$$!35'\\8!o[6tIF*$\"3YZ[d 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$!+aEfTJ!\"*$\"+aEfTJF[fm%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "infnorm(P[3,2] - sin(x), x=-Pi..Pi) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%[*)e=$!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "infnorm(P[5,2] - sin(x), x=-Pi..Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)fx0)Q!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 79 "Errors are abou t two times as small as of Taylor polynomials of degree 5 and 7." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 28 "Chebyshev-Pad\351 Approximation" }}{PARA 0 "" 0 "" {TEXT -1 131 "We make the Chebyshev-Pad\351 approxim ationof degree (3,2) on the segment, plot the error curve, and compute the error on the segment " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\" \"F%" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numapprox):" }} }{EXCHG {PARA 0 "> " 0 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segment " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 93 " for the successive Taylor polynomials are large, especially near the borders of the segment." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 19 "Pad\351 Approximations" }} {PARA 0 "" 0 "" {TEXT -1 163 "We make the Pad\351 rational approximati ons about 0 of degree (2,2) and (4,6), plot the graphs together with t he function plot, and compute the errors on the segment " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numapprox):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f := x -> Integrate(sin(sin(u))/u,u=0..x)/x;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&-%*IntegrateG6$*&-%$sinG6#-F26#%\"uG\"\"\"F6!\"\"/F6;\"\"!9$F7F " 0 "" {MPLTEXT 1 0 86 "Order := 25: ser ies(f(x),x):\n# just to ensure existence of a large series 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"6#$\"*'*>xU\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 51 "Errors are much smaller than of Taylor polynomial s." }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 259 2 "3." }{TEXT -1 65 " Comput e the following limits and check the answers if possible.\n" }{TEXT 270 3 "(a)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Limit(sin(x)/x,x=0)" "6# -%&LimitG6$*&-%$sinG6#%\"xG\"\"\"F*!\"\"/F*\"\"!" }{TEXT -1 2 "\n\n" } {TEXT 271 3 "(b)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Limit(sin(x)^(1/x) , x=0)" "6#-%&LimitG6$)-%$sinG6#%\"xG*&\"\"\"F,F*!\"\"/F*\"\"!" } {TEXT -1 3 "\n\n\n" }{TEXT 272 3 "(c)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Limit((1-cos(x))/x, x=0)" "6#-%&LimitG6$*&,&\"\"\"F(-%$cosG6#%\"xG! \"\"F(F,F-/F,\"\"!" }{TEXT -1 2 "\n\n" }{TEXT 273 3 "(d)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Limit((1+Pi/x)^x, x=infinity)" "6#-%&LimitG6$),& \"\"\"F(*&%#PiGF(%\"xG!\"\"F(F+/F+%)infinityG" }{TEXT -1 2 "\n\n" } {TEXT 274 3 "(e)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Limit(x^sin(x),x=0 )" "6#-%&LimitG6$)%\"xG-%$sinG6#F'/F'\"\"!" }{TEXT -1 2 "\n\n" }{TEXT 275 3 "(f)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((2^x+3^x)^(1/x), x=i nfinity)" "6#-%&LimitG6$),&)\"\"#%\"xG\"\"\")\"\"$F*F+*&F+F+F*!\"\"/F* %)infinityG" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 61 "We comput e the limits and verify them numericaly if possible." }}{SECT 1 {PARA 5 "" 0 "" {TEXT 276 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Limit(sin(x)/x , x=0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&-%$sinG6#%\"xG\"\"\"F+! \"\"/F+\"\"!F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+++++5!\"*$\"\"\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 277 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Limit((sin(x))^(1/x), \+ x=0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$)-%$sinG6#%\"xG*&\"\"\"F-F +!\"\"/F+\"\"!%*undefinedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Limit((sin(x))^(1/x), x=0, left):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%& LimitG6%)-%$sinG6#%\"xG*&\"\"\"F-F+!\"\"/F+\"\"!%%leftG%)infinityG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Limit((sin(x))^(1/x), x=0, \+ right):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6%)-%$sinG6#%\"xG*&\"\"\"F- F+!\"\"/F+\"\"!%&rightGF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"\"!F%F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 278 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Limit((1-cos(x))/x, x=0):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&,&\"\"\"F)-%$cosG6#%\"xG!\"\"F)F- F./F-\"\"!F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"\"!F%F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 279 3 "(d)" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Limit((1+Pi/x)^x, x=infinity):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$),&\"\"\"F)*&%#PiGF)%\"xG!\"\"F)F,/F,%)infi nityG-%$expG6#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+j#pSJ#!\")$\"+k#pSJ#F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 280 3 "(e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Limit(x^(sin(x)), x=0) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$)%\"xG-%$sinG6#F(/F(\"\"! \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$)%\"xG-%$sinG6#F(/F(\"\"!$ \"\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(eval(x^sin( x), x=10.0^(-i)), i=1..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"+b(H j%z!#5$\"+!>L*\\&*F%$\"+'\\g6$**F%$\"++Rz!***F%$\"+u([))***F%$\"+Y=')* ***F%$\"+#)Q)*****F%$\"+e\")******F%$\"+$z*******F%$\"+x********F%" }} }}{SECT 1 {PARA 5 "" 0 "" {TEXT 281 3 "(f)" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Limit((2^x+3^x)^(1/x), x=infinity):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%& LimitG6$),&)\"\"#%\"xG\"\"\")\"\"$F+F,*&F,F,F+!\"\"/F+%)infinityGF." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+++++I!\"*$\"\"$\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 260 2 "4." } {TEXT -1 32 " Compute the following limits.\n\n" }{TEXT 282 3 "(a)" } {TEXT -1 3 " " }{XPPEDIT 18 0 "Limit(ln(x)/x, x=infinity)" "6#-%&Lim itG6$*&-%#lnG6#%\"xG\"\"\"F*!\"\"/F*%)infinityG" }{TEXT -1 2 "\n\n" } {TEXT 283 3 "(b)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Limit(ln(x)/exp(x) , x=infinity)" "6#-%&LimitG6$*&-%#lnG6#%\"xG\"\"\"-%$expG6#F*!\"\"/F*% )infinityG" }{TEXT -1 2 "\n\n" }{TEXT 284 3 "(c)" }{TEXT -1 3 " " } {XPPEDIT 18 0 "Limit((x^2+sin(x))/(2*x^2+cos(4*x)), x=infinity)" "6#-% &LimitG6$*&,&*$%\"xG\"\"#\"\"\"-%$sinG6#F)F+F+,&*&F*F+*$F)F*F+F+-%$cos G6#*&\"\"%F+F)F+F+!\"\"/F)%)infinityG" }{TEXT -1 2 "\n\n" }{TEXT 285 3 "(d)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Limit(2/(1+exp(-1/x)), x=0, \+ right)" "6#-%&LimitG6%*&\"\"#\"\"\",&F(F(-%$expG6#,$*&F(F(%\"xG!\"\"F0 F(F0/F/\"\"!%&rightG" }{TEXT -1 2 "\n\n" }{TEXT 286 3 "(e)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Limit(sinh(tanh(x))-tanh(sinh(x)), x=infinity) " "6#-%&LimitG6$,&-%%sinhG6#-%%tanhG6#%\"xG\"\"\"-F+6#-F(6#F-!\"\"/F-% )infinityG" }{TEXT -1 1 "\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 287 3 "(a )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Limit((ln(x))/x, x=infinity):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&-%#lnG6#%\"xG\"\"\"F+!\"\"/F+%)inf inityG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"\"!F%F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 288 3 "(b)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "Limit((ln(x))/exp(x), x=infinity):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&-%#lnG6#%\"xG\"\"\"-%$expGF*!\"\"/ F+%)infinityG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$!\"!\"\"!$F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 289 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Limit((x^2+sin(x))/(2*x^2+co s(4*x)), x=infinity):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% \+ = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&,&*$)%\"x G\"\"#\"\"\"F--%$sinG6#F+F-F-,&*&F,F-F*F-F--%$cosG6#,$*&\"\"%F-F+F-F-F -!\"\"/F+%)infinityG#F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&,&*$)%\"xG \"\"#\"\"\"F--%$sinG6#F+F-F-,&*&F,F-F*F-F--%$cosG6#,$*&\"\"%F-F+F-F-F- !\"\"/F+%)infinityG$\"+++++]!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "seq(eval((x^2+sin(x))/(2*x^2+cos(4*x)), x=10.0^(i)), i=1..10); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"+trV*)\\!#5$\"+R\"y)**\\F%$\"+g f++]F%$\"+x******\\F%$\"+++++]F%F,F,F,F,F," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 290 3 "(d)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 28 "Limit(2/(1+exp(-1/x)), x=0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$,$*&\"\"#\"\"\",&F*F*-%$expG6#,$*&F*F*%\"xG !\"\"F2F*F2F*/F1\"\"!%*undefinedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Limit(2/(1+exp(-1/x)), x=0, left):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6%,$*&\"\"#\"\"\",&F*F*-%$expG6#,$*&F*F*%\"xG !\"\"F2F*F2F*/F1\"\"!%%leftGF4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Limit(2/(1+exp(-1/x)), x=0, right):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%&LimitG6%,$*&\"\"#\"\"\",&F*F*-%$expG6#,$*&F*F*%\"xG!\"\"F2F*F2F*/F1 \"\"!%&rightGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+++++?!\"*$\"\"#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 291 3 "(e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Limit(sinh(tanh(x))-ta nh(sinh(x)), x=infinity):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "% = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$,&-%% sinhG6#-%%tanhG6#%\"xG\"\"\"-F,6#-F)F-!\"\"/F.%)infinityG,(*&#F/\"\"#F /-%$expG6#F/F/F/*&#F/F9F/-F;6#F3F/F3F/F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+O>,_ ,_ " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "poly := x^3 - 1 - 4*x^ 2 + 5*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%polyG,**$)%\"xG\"\"$\" \"\"F*F*!\"\"*&\"\"%F*)F(\"\"#F*F+*&\"\"&F*F(F*F*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "taylor(poly, x=1, 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+),&%\"xG\"\"\"F&!\"\"F&\"\"!F'\"\"#F&\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 85 "It looks like a polynomial, but in fact Maple u ses internlly a series data structure." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "whattype(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'ser iesG" }}}{PARA 0 "" 0 "" {TEXT -1 83 "The order of the series expansio n is, as far as Maple concerned, equal to infinity." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "%% - taylor(poly, x=1, infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 262 2 "6." }{TEXT -1 37 " W hat is the asymptotic expansion of " }{XPPEDIT 18 0 "MATRIX([[2*n],[n] ])" "6#-%'MATRIXG6#7$7#*&\"\"#\"\"\"%\"nGF*7#F+" }{TEXT -1 3 " ?\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 27 "asympt(binomial(2*n,n), n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,.*&#\"\"\"\"\"#F'*,%#PiG#!\"\"F(-%$expG6#,&F'F, -%#lnG6#F(F,F,-F.6#F'F'-F.6#F,F(*&F'F'%\"nGF,F&F'F'*&#F'\"#;F'*,F*F+F- F,F4F'F6F(F8#\"\"$F(F'F,*&#F'\"$c#F'*,F*F+F-F,F4F'F6F(F8#\"\"&F(F'F'*& #FE\"%[?F'*,F*F+F-F,F4F'F6F(F8#\"\"(F(F'F'*&#\"#@\"&Ob'F'*,F*F+F-F,F4F 'F6F(F8#\"\"*F(F'F,-%\"OG6#*$)F8#\"#6F(F'F'F')\"\"%F9F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(%, assume=positive);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"&oF$F&**,.*&F'F&)%\"nG\" \"%F&F&*&\"%'4%F&)F,\"\"$F&!\"\"*&\"$c#F&)F,\"\"#F&F&*&\"$g\"F&F,F&F& \"#@F2**F'F&-%\"OG6#*&F&F&*$)F,#\"#6F6F&F2F&%#PiG#F&F6)F,#\"\"*F6F&F&F &)F-F,F&FC#F2F6F,#!\"*F6F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*()\"\"%%\"nG\" \"\"%#PiG#!\"\"\"\"#F'F*F(**\"\")F+F%F(F)F*F'#!\"$F,F+**\"$G\"F+F%F(F) F*F'#!\"&F,F(*,\"\"&F(\"%C5F+F%F(F)F*F'#!\"(F,F(*,\"#@F(\"&oF$F+F%F(F) F*F'#!\"*F,F+*&F%F(-%\"OG6#*&F(F(*$)F'#\"#6F,F(F+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "4^n * map(x->x/4^n, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)\"\"%%\"nG\"\"\",.*&F'F'*&%#PiG#F'\"\"#F& #F'F-!\"\"F'*&F'F'*(\"\")F'F+#F'F-)F&#\"\"$F-F'F/F/*&F'F'*(\"$G\"F'F+# F'F-)F&#\"\"&F-F'F/F'**F=F'\"%C5F/F+#F/F-F&#!\"(F-F'**\"#@F'\"&oF$F/F+ F@F&#!\"*F-F/-%\"OG6#*&F'F'*$)F&#\"#6F-F'F/F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 79 "You can also do the following series expansion and furthe r simplify the result:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "se ries(binomial(2*n,n), n=infinity); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #*&,.*&#\"\"\"\"\"#F'*,%#PiG#!\"\"F(-%$expG6#,&F'F,-%#lnG6#F(F,F,-F.6# F'F'-F.6#F,F(*&F'F'%\"nGF,F&F'F'*&#F'\"#;F'*,F*F+F-F,F4F'F6F(F8#\"\"$F (F'F,*&#F'\"$c#F'*,F*F+F-F,F4F'F6F(F8#\"\"&F(F'F'*&#FE\"%[?F'*,F*F+F-F ,F4F'F6F(F8#\"\"(F(F'F'*&#\"#@\"&Ob'F'*,F*F+F-F,F4F'F6F(F8#\"\"*F(F'F, -%\"OG6#*$)F8#\"#6F(F'F'F')\"\"%F9F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&# \"\"\"\"&oF$F&**,.*(F'F&*&F&F&%\"nG!\"\"#F&\"\"#)F,\"\"%F&F&*(\"%'4%F& F+F.)F,\"\"$F&F-*(\"$c#F&F+F.)F,F/F&F&*(\"$g\"F&F+F.F,F&F&*&\"#@F&F+F. F-**F'F&-%\"OG6#*&F,!\"&F+F.F&%#PiGF.F0F&F&F&)F1F,F&F,!\"%FC#F-F/F&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT 263 2 "7." }{TEXT -1 61 " What is the power series expans ion of Lambert's W function?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 50 "The first few terms of \+ the series expansion about " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "series(Lamb ertW(x), x, 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+7%\"xG\"\"\"F%!\" \"\"\"##\"\"$F'F)#!\")F)\"\"%#\"$D\"\"#C\"\"&#!#aF0\"\"'#\"&2o\"\"$?( \"\"(#!&%Q;\"$:$\"\")#\"'T9`\"%![%\"\"*-%\"OG6#F%\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 59 "This may bring up the idea that the nth degree term \+ equals " }{XPPEDIT 18 0 "(-n)^(n-1)/n!" "6#*&),$%\"nG!\"\",&F&\"\"\"F) F'F)-%*factorialG6#F&F'" }{TEXT -1 9 " , i.e.\n " }{XPPEDIT 18 0 "Lamb ertW(x) = Sum((-n)^(n-1)/n!*x^n, n=1..infinity" "6#/-%)LambertWG6#%\"x G-%$SumG6$*(),$%\"nG!\"\",&F.\"\"\"F1F/F1-%*factorialG6#F.F/)F'F.F1/F. ;F1%)infinityG" }{TEXT -1 17 ". Let us use the " }{HYPERLNK 17 "powser ies" 2 "" "" }{TEXT -1 53 " package to verify this for a large number \+ of terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(powseries );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#78%(composeG%(evalpowG%(inverseG %*multconstG%)multiplyG%)negativeG%'powaddG%'powcosG%*powcreateG%(powd iffG%'powexpG%'powintG%'powlogG%(powpolyG%'powsinG%)powsolveG%(powsqrt G%)quotientG%*reversionG%)subtractG%)templateG%(tpsformG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "powcreate(f(n)=(-n)^(n-1)/n!, f(0)= 0):" }}}{PARA 0 "" 0 "" {TEXT -1 65 "10 terms to be compared manually \+ with the above series expansion:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "tpsform(f, x, 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+7%\"xG \"\"\"F%!\"\"\"\"##\"\"$F'F)#!\")F)\"\"%#\"$D\"\"#C\"\"&#!#aF0\"\"'#\" &2o\"\"$?(\"\"(#!&%Q;\"$:$\"\")#\"'T9`\"%![%\"\"*-%\"OG6#F%\"#5" }}} {PARA 0 "" 0 "" {TEXT -1 43 "Checking the first 500 terms in the serie s:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "tpsform(f, x, 200) - s eries(LambertW(x), x, 200);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 264 2 "8." }{TEXT -1 108 " In this exercise the Chebyshev p olynomials of the first kind are introduced via their generating funct ion.\n" }{TEXT 292 3 "(a)" }{TEXT -1 40 " Compute the Taylor series ex pansion of " }{XPPEDIT 18 0 "(1-t^2)/(1-2*x*t+t^2)" "6#*&,&\"\"\"F%*$% \"tG\"\"#!\"\"F%,(F%F%*(F(F%%\"xGF%F'F%F)*$F'F(F%F)" }{TEXT -1 7 " abo ut " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 18 " up to degree 8.\n\n " }{TEXT 293 3 "(b)" }{TEXT -1 27 " The Chebyshev polynomials " } {XPPEDIT 18 0 "T[n](x)" "6#-&%\"TG6#%\"nG6#%\"xG" }{TEXT -1 89 " of th e first kind are defined by the generating function\n \+ " }{XPPEDIT 18 0 "(1-t^2)/(1-2*x*t+t^2)=sum(epsilon[n]*T [n](x)*t^n" "6#/*&,&\"\"\"F&*$%\"tG\"\"#!\"\"F&,(F&F&*(F)F&%\"xGF&F(F& F**$F(F)F&F*-%$sumG6#*(&%(epsilonG6#%\"nGF&-&%\"TG6#F66#F-F&)F(F6F&" } {TEXT -1 7 "\nwhere " }{XPPEDIT 18 0 "epsilon[0]=1" "6#/&%(epsilonG6# \"\"!\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "epsilon[n]=2" "6#/&%( epsilonG6#%\"nG\"\"#" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "n*`>`*0" "6# *(%\"nG\"\"\"%\">GF%\"\"!F%" }{TEXT -1 10 ". Compute " }{XPPEDIT 18 0 "T[2](x)" "6#-&%\"TG6#\"\"#6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "T[10](x)" "6#-&%\"TG6#\"#56#%\"xG" }{TEXT -1 46 ". Check your answe r with the built-in command " }{TEXT 0 10 "ChebyshevT" }{TEXT -1 2 ". \n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 294 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "mtaylor((1-t^2)/(1-2*x*t+t^2), [x=0,t=0], 8);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,6\"\"\"F$*&\"\"#F$)%\"tGF&F$!\"\"*(F&F$%\"xGF$F(F$F$ *(\"\"%F$F'F$)F+F&F$F$*(\"\"'F$)F(\"\"$F$F+F$F)*&F&F$)F(F-F$F$*(\"\")F $F1F$)F+F2F$F$*(\"#;F$F4F$F.F$F)*(\"#5F$)F(\"\"&F$F+F$F$*&F&F$)F(F0F$F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "collect(%,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$*&\"\"#F$)%\"tG\"\"'F$!\"\"* (\"#5F$)F(\"\"&F$%\"xGF$F$*&,&F&F$*&\"#;F$)F/F&F$F*F$)F(\"\"%F$F$*&,&* &F)F$F/F$F**&\"\")F$)F/\"\"$F$F$F$)F(F=F$F$*&,&F&F**&F6F$F4F$F$F$)F(F& F$F$*(F&F$F/F$F(F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 1 {PARA 5 "" 0 "" {TEXT 295 3 "(b)" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "mtaylor((1-t^2)/(1-2*x*t+t^2), [x=0,t=0], 21):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "collect(%, t, sort):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "T[2](x) = coeff(%, t^2)/2, T [10](x) = coeff(%, t^10)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-&%\"T G6#\"\"#6#%\"xG,&*&F(\"\"\")F*F(F-F-F-!\"\"/-&F&6#\"#5F),.*&\"$7&F-)F* F4F-F-*&\"%!G\"F-)F*\"\")F-F/*&\"%?6F-)F*\"\"'F-F-*&\"$+%F-)F*\"\"%F-F /*&\"#]F-F.F-F-F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "T[2 ](x) = simplify(ChebyshevT(2,x), 'ChebyshevT'),\nT[10](x) = simplify(C hebyshevT(10,x), 'ChebyshevT');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-& %\"TG6#\"\"#6#%\"xG,&*&F(\"\"\")F*F(F-F-F-!\"\"/-&F&6#\"#5F),.*&\"$7&F -)F*F4F-F-*&\"%!G\"F-)F*\"\")F-F/*&\"%?6F-)F*\"\"'F-F-*&\"$+%F-)F*\"\" %F-F/*&\"#]F-F.F-F-F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 265 2 "9." }{TEXT -1 87 " Find th e Taylor series expansion up to order 25 of the solution for Kepler's \+ equation " }{XPPEDIT 18 0 "E=u+e*sin(E)" "6#/%\"EG,&%\"uG\"\"\"*&%\"eG F'-%$sinG6#F$F'F'" }{TEXT -1 15 " by use of the " }{TEXT 0 6 "RootOf" }{TEXT -1 99 " procedure. Compare the efficiency with the method descr ibed in the first section of this chapter.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Order := 25:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "settime := time():\nsetbytes := kernelopts(bytesused):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "series(RootOf(E=u+e*sin(E), E), e):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "map(combine, %, 'trig'):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "cputime = (time()-settime) *seconds,\nmemory = evalf((kernelopts(bytesused)-setbytes)/1024*kbytes , 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%(cputimeG,$*&$\"%j?!\"$\"\" \"%(secondsGF*F*/%'memoryG,$*&$\"&>o#\"\"!F*%'kbytesGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "Order := 25:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "settime := time():\nsetbytes := kernelopts(bytesused):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "series(E-u-e*sin(E), E=u):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(%, E-u):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(combine, %[1], 'trig'): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "whattype(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'seriesG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "cputime = (time()-settime)*seconds,\nmemory = evalf((kernelopts(b ytesused)-setbytes)/1024*kbytes, 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$/%(cputimeG,$*&$\"&!49!\"$\"\"\"%(secondsGF*F*/%'memoryG,$*&$\"&R+#F *F*%'kbytesGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT 266 3 "10." }{TEXT -1 33 " Find the seri es expansion about " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 31 " up to order 3 of the function " }{XPPEDIT 18 0 "x->(x^a-a^x)/(x^x -a^a)" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"*&,&)F%%\"aG\"\"\")F.F%! \"\"F/,&)F%F%F/)F.F.F1F1F*F*F*" }{TEXT -1 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 10 "First try:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := x -> (x^a-a^ x)/(x^x-a^a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(*&,&)9$%\"aG\"\"\")F0F/!\"\"F1,&)F/F/F1)F0F0F3F3F( F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series(f(x), x=a, 4 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+-,&%\"xG\"\"\"%\"aG!\"\"*&,&)F' F'F&-%$expG6#*&F'F&-%#lnG6#F'F&F(F&,&F,F&F+F(F(\"\"!*&,(*&F,F&F0F&F(F+ F&*&F,F&,&F&F&F0F&F&F&F&F3F(F&*&,**&#F&\"\"#F&*&F,F&)F0F>F&F&F(**F>F(F +F&,&F'F&F&F(F&F'F(F&*&#F&F>F&*(F,F&,*F&F&F'F&*(F>F&F'F&F0F&F&*&F'F&F@ F&F&F&F'F(F&F&**,&F+F&F,F&F&F3F(F,F&F9F&F(F&F3F(F>*&,,*&#F&\"\"'F&*&F, F&)F0\"\"$F&F&F(*,FOF(F+F&FBF&F'!\"#,&F'F&F>F(F&F&*&#F&FOF&*(F,F&,0F&F (*&FRF&F'F&F&*(FRF&F'F&F0F&F&*$)F'F>F&F&*(FRF&FgnF&F0F&F&*(FRF&FgnF&F@ F&F&*&FgnF&FQF&F&F&F'FTF&F&*&#F&F>F&*,FJF&F3F(F,F&FFF&F'F(F&F(*&FDF&*, ,0**F>F&F+F&F'F&F,F&F&*&F'F&)F+F>F&F&*(F>F&F+F&F,F&F&*$FcoF&F(*$)F,F>F &F(*&F'F&FgoF&F&*,\"\"%F&F,F&F'F&F0F&F+F&F&F&F'F(F3FTF,F&F9F&F&F&F&F3F (FR-%\"OG6#F&Fjo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplif y(%);" }}{PARA 8 "" 1 "" {TEXT -1 63 "Error, (in simplify/power) numer ic exception: division by zero\n" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Next , we redefine zero recognition:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := x - > (x^a-a^x)/(x^x-a^a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(*&,&)9$%\"aG\"\"\")F0F/!\"\"F1,&)F/F/F1) F0F0F3F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Testzero \+ := proc() evalb( Normalizer(args[1])=0) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Normalizer := proc() normal( simplify(args[1])) \+ end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series(f(x), x=a, 4 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#++,&%\"xG\"\"\"%\"aG!\"\"*(,&)F' F'F&-%$expG6#*&F'F&-%#lnG6#F'F&F(F&F,F(,&F&F&F0F&F(F(*(,&*&F,F&F0F&F(F +F&F&F,F(F3F(\"\"!*(,(*&#F&\"\"#F&*&F,F&)F0FF&F&F&F'F(F&F&F&F,F(F3F(F&-%\"OG6#F&F<" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+), &%\"xG\"\"\"%\"aG!\"\",$*&,&-%#lnG6#F'F&F&F(F&,&F&F&F,F&F(F(\"\"!,$*&F &F&*&)F/\"\"#F&F'F&F(F(F&-%\"OG6#F&F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 268 3 "11." } {TEXT -1 95 " When you study Josephson's junction circuit you may need the series expansion of the function " }{TEXT 267 4 "ptan" }{TEXT -1 12 " defined as " }{XPPEDIT 18 0 "ptan(s)=p" "6#/-%%ptanG6#%\"sG%\"pG " }{TEXT -1 5 ", if " }{XPPEDIT 18 0 "p-tan(p)=s" "6#/,&%\"pG\"\"\"-%$ tanG6#F%!\"\"%\"sG" }{TEXT -1 49 ". Compute the series expansion of th is function.\n" }}{PARA 0 "" 0 "" {TEXT -1 10 "First try:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "Order := 10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ptan := s -> RootOf(p-tan(p)-s, p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ptanGf*6#%\"sG6\"6$%)operatorG%&arrowGF(-%'RootOfG6$ ,(%\"pG\"\"\"-%$tanG6#F0!\"\"9$F5F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "series(ptan(s), s);" }}{PARA 8 "" 1 "" {TEXT -1 51 "Error, (in series/RootOf) unable to compute series\n" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Next, the explicit way" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "series(p-tan(p)-s, p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"pG,$%\"sG!\"\"\"\"!#F'\"\"$F*#!\"#\"#:\"\"&#!#<\"$:$\"\"(#! #i\"%NG\"\"*-%\"OG6#\"\"\"\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(%, p);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%,,*&)\"\"$#\" \"\"F&F()%\"sGF'F(!\"\"*(\"\"#F(\"\"&F+F*F(F(**\"\"*F(\"$v\"F+F&#F-F&F *#F.F&F+**F-F(F1F+F&F'F*#\"\"(F&F(-%\"OG6#*$)F*#\"\")F&F(F(,,**F&F+,&* &F-F(),&*&F-F+F&F'F(*&^##F+F-F()F&#F.\"\"'F(F(F&F(F(F&F+F(FC!\"#F*F'F( *,F-F(\"#XF+FC!\"$,(*&\"#@F(FBF(F(*&F-F()FCFJF(F(\"#=F(F(F*F(F(**\"%v: F+FC!\"%,**&F0F(FSF(F(*&\"#?F()FCF0F(F(*&\"$i\"F(FBF(F+\"$c(F+F(F*F3F( *,F-F(\"&DO#F+FC!\"&,,*&\"#8F(FfnF(F(*&\"$;#F(FSF(F+*&\"%)=\"F(FBF(F(* &FenF()FC\"#7F(F(\"%CIF(F(F*F5F(F7F(,,**F&F+,&*&F-F(),&*&F-F+F&F'F(*&^ ##F(F-F(FHF(F(F&F(F(F&F+F(F]pFKF*F'F(*,F-F(FMF+F]pFN,(*&FQF(F\\pF(F(*& F-F()F]pFJF(F(FTF(F(F*F(F(**FVF+F]pFW,**&F0F(FfpF(F(*&FenF()F]pF0F(F(* &FhnF(F\\pF(F+FinF+F(F*F3F(*,F-F(F[oF+F]pF\\o,,*&F_oF(F[qF(F(*&FaoF(Ff pF(F+*&FcoF(F\\pF(F(*&FenF()F]pFfoF(F(FgoF(F(F*F5F(F7F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&)\"\"$#\"\"\"F&F()%\"sGF'F(!\"\"*(\"\"#F(\"\"&F+F*F(F(**\"\" *F(\"$v\"F+F&#F-F&F*#F.F&F+**F-F(F1F+F&F'F*#\"\"(F&F(-%\"OG6#*$)F*#\" \")F&F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&$\"+q&\\AW\"!\"*\"\"\")%\"sG#F(\" \"$F(!\"\"*&$\"+++++S!#5F(F*F(F(*&$\"+&Rd(p5F1F()F*#\"\"&F,F(F-*&$\"+B _G[;!#6F()F*#\"\"(F,F(F(-%\"OG6#*$)F*#\"\")F,F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 269 3 "12 ." }{TEXT -1 50 " Compare the series solution of Kepler's equation " } {XPPEDIT 18 0 "E=u+e*sin(E)" "6#/%\"EG,&%\"uG\"\"\"*&%\"eGF'-%$sinG6#F $F'F'" }{TEXT -1 113 ", which was computed in the first section of thi s chapter, with the following exact solution of Bessel functions " } {XPPEDIT 18 0 "E=u+2*sum(J[n](n*e)*sin(n*u)/n, n=1..infinity)" "6#/%\" EG,&%\"uG\"\"\"*&\"\"#F'-%$sumG6$*(-&%\"JG6#%\"nG6#*&F2F'%\"eGF'F'-%$s inG6#*&F2F'F&F'F'F2!\"\"/F2;F'%)infinityGF'F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "First the series solution:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Order := 10:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "series( RootOf(E=u+e*sin(E), E), e ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "approximate := map(combine, %, 'trig');" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,approx imateG+9%\"eG%\"uG\"\"!-%$sinG6#F'\"\"\",$*&#F,\"\"#F,-F*6#,$*&F0F,F'F ,F,F,F,F0,&*&#\"\"$\"\")F,-F*6#,$*&F8F,F'F,F,F,F,*&#F,F9F,F)F,!\"\"F8, &*&#F,F8F,-F*6#,$*&\"\"%F,F'F,F,F,F,*&#F,\"\"'F,F1F,F@FH,(*&#\"$D\"\"$ %QF,-F*6#,$*&\"\"&F,F'F,F,F,F,*&#\"#F\"$G\"F,F:F,F@*&#F,\"$#>F,F)F,F,F U,(*&#FX\"#!)F,-F*6#,$*&FKF,F'F,F,F,F,*&#F,\"#[F,F1F,F,*&#FH\"#:F,FDF, F@FK,**&#F,\"%;#*F,F)F,F@*&#\"&2o\"\"&!3YF,-F*6#,$*&\"\"(F,F'F,F,F,F,* &#\"%DJFhoF,FQF,F@*&#\"$V#\"%?^F,F:F,F,Fap,**&#FH\"#XF,FDF,F,*&#F,\"$? (F,F1F,F@*&#FY\"$:$F,-F*6#,$*&F9F,F'F,F,F,F,*&#Fgp\"$g&F,F[oF,F@F9,,*& #\"'VN#)\"(gXZ\"F,F]pF,F@*&#\"&D\"y\"''4;&F,FQF,F,*&#Fgp\"&g4%F,F:F,F@ *&#F,\"'!GP(F,F)F,F,*&#\"'T9`\"(!)o9\"F,-F*6#,$*&\"\"*F,F'F,F,F,F,Fas- %\"OG6#F,\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 105 "We shall show that thi s series solution is equal to the series expansion of the following ex act solution:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "exact := u \+ + 2*sum(BesselJ(n,n*e)*sin(n*u)/n, n=1..infinity);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&exactG,&%\"uG\"\"\"*&\"\"#F'-%$sumG6$*(-%(BesselJG 6$%\"nG*&F1F'%\"eGF'F'-%$sinG6#*&F1F'F&F'F'F1!\"\"/F1;F'%)infinityGF'F '" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "expansion := \nseries( u + 2*sum(BesselJ(n,n*e)*sin(n*u)/n, n=1..10), e );" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%*expansionG+9%\"eG%\"uG\"\"!-%$sinG6#F'\"\"\",$*&#F ,\"\"#F,-F*6#,$*&F0F,F'F,F,F,F,F0,&*&#\"\"$\"\")F,-F*6#,$*&F8F,F'F,F,F ,F,*&#F,F9F,F)F,!\"\"F8,&*&#F,F8F,-F*6#,$*&\"\"%F,F'F,F,F,F,*&#F,\"\"' F,F1F,F@FH,(*&#\"$D\"\"$%QF,-F*6#,$*&\"\"&F,F'F,F,F,F,*&#\"#F\"$G\"F,F :F,F@*&#F,\"$#>F,F)F,F,FU,(*&#FX\"#!)F,-F*6#,$*&FKF,F'F,F,F,F,*&#F,\"# [F,F1F,F,*&#FH\"#:F,FDF,F@FK,**&#F,\"%;#*F,F)F,F@*&#\"&2o\"\"&!3YF,-F* 6#,$*&\"\"(F,F'F,F,F,F,*&#\"%DJFhoF,FQF,F@*&#\"$V#\"%?^F,F:F,F,Fap,**& #FH\"#XF,FDF,F,*&#F,\"$?(F,F1F,F@*&#FY\"$:$F,-F*6#,$*&F9F,F'F,F,F,F,*& #Fgp\"$g&F,F[oF,F@F9,,*&#\"'VN#)\"(gXZ\"F,F]pF,F@*&#\"&D\"y\"''4;&F,FQ F,F,*&#Fgp\"&g4%F,F:F,F@*&#F,\"'!GP(F,F)F,F,*&#\"'T9`\"(!)o9\"F,-F*6#, $*&\"\"*F,F'F,F,F,F,Fas-%\"OG6#F,\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "expansion - approximate;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}}{MARK "0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }