{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 128 0 0 1 0 2 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 0 0 128 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Input" 2 19 "" 0 0 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Non-propor tional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 0 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 24 6 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 13 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 3 0 -1 -1 -1 3 36 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 " " 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "Times" 1 22 0 128 128 1 0 1 0 0 0 0 0 0 0 }3 1 0 -1 14 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "Time s" 1 16 0 0 128 1 0 1 0 0 0 0 0 0 0 }3 1 0 -1 8 0 0 0 0 0 0 0 261 0 } {PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 128 1 1 2 0 0 0 0 0 0 0 }3 1 0 -1 6 -1 0 0 0 0 0 0 263 0 }{PSTYLE "Date" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 128 1 2 0 0 0 0 0 0 0 0 }3 0 0 -1 8 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Address - email" -1 258 1 {CSTYLE "" -1 -1 "Verdana" 1 10 0 0 128 1 2 0 0 0 0 0 0 0 0 }3 0 0 -1 4 -1 0 0 0 0 0 0 262 0 }{PSTYLE "Normal Indented" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 18 -1 -1 0 18 0 0 0 0 -1 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 37 "Series Solutions of Kepler's Equ ation" }}{PARA 19 "" 0 "" {TEXT -1 15 "Marc A. Murison" }}{PARA 256 " " 0 "" {TEXT -1 74 "Astronomical Applications Department\nU.S. Naval O bservatory\nWashington, DC" }}{PARA 258 "" 0 "" {TEXT -1 57 "murison@a a.usno.navy.mil\nhttp://aa.usno.navy.mil/murison/" }}{PARA 257 "" 0 " " {TEXT -1 13 "June 26, 1998" }}{SECT 1 {PARA 3 "" 0 "Contents" {TEXT -1 8 "Contents" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Introduction" 1 "" "Introduction" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "A n Iterative Method" 1 "" "An Iterative Method" }{TEXT -1 1 " " }} {PARA 15 "" 0 "" {HYPERLNK 17 "A Series Expansion Method" 1 "" "A Seri es Expansion Method" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 " A Fourier Sine Series Expansion and Resulting Bessel Function Represen tation for the Coefficients" 1 "" "A Fourier Expansion..." }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "Chebyshev Polynomials" 1 "" "Ch ebyshev Polynomials" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 " A Chebyshev Series Expansion" 1 "" "A Chebyshev Series Expansion" } {TEXT -1 1 " " }}}}{SECT 1 {PARA 3 "" 0 "Introduction" {TEXT -1 12 "In troduction" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 166 "Kepler's equation occurs in the context of the Newtonian two-body problem. The relativ e orbit of one body with respect to the other is easily characterized \+ with the " }{TEXT 258 12 "true anomaly" }{TEXT -1 368 " as the indepen dent variable. The true anomaly is just the angle: pericenter \226 fo cus \227 body, where focus is the ellipse focus around which the body \+ moves. This is adequate for determining the orbit in space \226 its s hape, size, and orientation. However, if one wishes to determine the \+ orbit in time, things get more complicated, and one must solve Kepler' s equation: " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "phi = e*sin*phi+M:" "6#/%$phiG,&*(%\"eG\"\"\"%$sinGF(F$F(F(%\"MGF(" } }{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "phi(t);" "6#-%$ph iG6#%\"tG" }{TEXT -1 8 " is the " }{TEXT 259 17 "eccentric anomaly" } {TEXT -1 33 ", e is the orbital eccentricity, " }{XPPEDIT 18 0 "M = n* (t-t[0]);" "6#/%\"MG*&%\"nG\"\"\",&%\"tGF'&F)6#\"\"!!\"\"F'" }{TEXT -1 8 " is the " }{TEXT 260 12 "mean anomaly" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 40 " is the time of \+ pericenter passage, and " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 69 " is the mean motion. The true and eccentric anomalies are related by " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "cos*nu = (c os*phi-e)/(1-e*cos*phi);" "6#/*&%$cosG\"\"\"%#nuGF&*&,&*&F%F&%$phiGF&F &%\"eG!\"\"F&,&\"\"\"F&*(F,F&F%F&F+F&F-F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 9 " " } {XPPEDIT 18 0 "sin*nu = sqrt(1-e^2)*sin*phi/(1-e*cos*phi);" "6#/*&%$si nG\"\"\"%#nuGF&**-%%sqrtG6#,&\"\"\"F&*$%\"eG\"\"#!\"\"F&F%F&%$phiGF&,& \"\"\"F&*(F/F&%$cosGF&F2F&F1F1" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "Hence, to determine " }{XPPEDIT 18 0 "nu(t);" "6#-%#nuG6# %\"tG" }{TEXT -1 43 " we must first solve Kepler's equation for " } {XPPEDIT 18 0 "phi(t);" "6#-%$phiG6#%\"tG" }{TEXT -1 3 ". " }}{PARA 259 "" 0 "" {TEXT -1 302 "Since Kepler's equation is transcendental, w e require iterative or series expansion approaches. Numerically, one \+ can employ various fast algorithms. Here, we are not concerned with n umerical methods but rather the analytical properties of the solution, so we will consider algebraic approximations. " }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 28 "ellipse cartoon calculations" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "restart;" "6#%(restartG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "with(plottools):" "6#-%%withG6#%*plottoolsG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "with(plots):" "6#-%%withG6#%&plotsG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "a := 5:" "6#>%\"aG\"\"&" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "b := 4:" "6#>%\"bG\"\"%" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "c := sqrt(a^2-b^2):" "6#>%\"cG-%%sqrtG6#,&*$%\"aG\"\"# \"\"\"*$%\"bG\"\"#!\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "e := c/a:" " 6#>%\"eG*&%\"cG\"\"\"%\"aG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "theta[max] := 15*Pi/24:" "6#>&%&thetaG6#%$maxG*(\"#:\"\"\"%#PiGF *\"#C!\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "N := 25:" "6#>%\"NG\"#D" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "x := array(1..N):" "6#>%\"xG- %&arrayG6#;\"\"\"%\"NG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "y := array(1 \+ .. N):" "6#>%\"yG-%&arrayG6#;\"\"\"%\"NG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "for i from 1 to N do\n nu := (i-1)/(N-1)*theta[max];\n r := a*(1-e^2)/(1+e*cos(nu));\n x[i] := r*cos(nu) + c;\n y[i] \+ := r*sin(nu);\nod:" "6#?(%\"iG\"\"\"\"\"\"%\"NG%%trueGC&>%#nuG*(,&F$F& \"\"\"!\"\"F&,&F'F&\"\"\"F/F/&%&thetaG6#%$maxGF&>%\"rG*(%\"aGF&,&\"\" \"F&*$%\"eG\"\"#F/F&,&\"\"\"F&*&F=F&-%$cosG6#F+F&F&F/>&%\"xG6#F$,&*&F7 F&-FC6#F+F&F&%\"cGF&>&%\"yG6#F$*&F7F&-%$sinG6#F+F&" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "p1 := ellipse([0, 0],a,b,filled = true,color = \+ yellow):" "6#>%#p1G-%(ellipseG6'7$\"\"!F)%\"aG%\"bG/%'filledG%%trueG/% &colorG%'yellowG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "p2 := polygon( [ [c ,0], seq([x[i],y[i]],i=1..N), [c,0] ], color=red ):" "6#>%#p2G-%(polyg onG6$7%7$%\"cG\"\"!-%$seqG6$7$&%\"xG6#%\"iG&%\"yG6#F3/F3;\"\"\"%\"NG7$ F*F+/%&colorG%$redG" }}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "display ( p2, p1, axes=normal, scaling=constrained );" "6#-%(displayG6&%#p2G%# p1G/%%axesG%'normalG/%(scalingG%,constrainedG" }}{PARA 13 "" 1 "" {GLPLOT2D 454 377 377 {PLOTDATA 2 "6*-%)POLYGONSG6$7=7$$\"\"$\"\"!F*7$ $\"\"&F*F*7$$\"+)z8e*\\!\"*$\"+)3ukj\"!#57$$\"+>,A$)\\F1$\"+n'4VF$F47$ $\"+&G8@'\\F1$\"+Bz$[\"\\F47$$\"+icJK\\F1$\"+6yKflF47$$\"+*QvN*[F1$\"+ [3&*3#)F47$$\"+\">jb%[F1$\"+\"ySZ')*F47$$\"+n#oyy%F1$\"+,Yv_6F17$$\"+` K**>ZF1$\"+Vsz>8F17$$\"+'G[8k%F1$\"+s8j([\"F17$$\"+2UC^XF1$\"+%eYil\"F 17$$\"+>]))[WF1$\"+N@eD=F17$$\"+_(fLL%F1$\"+*)Q^&*>F17$$\"+XSj.UF1$\"+ e\"Qe;#F17$$\"+()=aeSF1$\"+KF17$$\"+t\\3XSF1$\"+359^BF17$$\"+PJ%[k$F1$\"+C%)=QFF 17$$\"+\\*>r=$F1$\"+rH0#3$F17$$\"+wR8zEF1$\"+-OF17$$\"+t\\3X:F1$\"+kgA/QF17$$\"+5d1p$*F4$\"+.!\\\"HRF17$$\"+jf_ RJF4$\"+9p5#*RF17$$!+jf_RJF4Ffw7$$!+5d1p$*F4Faw7$$!+t\\3X:F1F\\w7$$!+f k*)G@F1Fgv7$$!+wR8zEF1Fbv7$$!+\\*>r=$F1F]v7$$!+PJ%[k$F1Fhu7$$!+t\\3XSF 1Fcu7$$!++M`\"Q%F1F^u7$$!+IC))[YF1Fit7$$!+1e\"H%[F1Fdt7$$!+2Ndg\\F1F_t 7$$!\"&F*F*7$Fjy$!+W$HL,&F47$Fgy$!+)[&fZ**F47$Fdy$!+6#)\\s9F17$Fay$!+' p9q#>F17$F^y$!+359^BF17$F[y$!+C%)=QFF17$Fhx$!+rH0#3$F17$Fex$!+-OF17$F_x$!+kgA/QF17$F\\x$!+.!\\\"HRF17$Fiw$!+9p5#*RF17$Fdw Fa\\l7$F_wF^\\l7$FjvF[\\l7$FevFh[l7$F`vFe[l7$F[vFb[l7$FfuF_[l7$FauF\\[ l7$F\\uFiz7$FgtFfz7$FbtFcz7$F]tF`zF+-Fcs6&FesFfsFfsF*-%+AXESLABELSG6%% !GFd]l-%%FONTG6$%*HELVETICAG\"#=-%*AXESTICKSG6%%(DEFAULTGF]^l-Ff]l6$Fh ]l\"#7-%*AXESSTYLEG6#%'NORMALG-Fcs6&FesF*F*Ffs-%(SCALINGG6#%,CONSTRAIN EDG-%&TITLEG6$Fd]lFe]l" 1 2 0 1 0 2 6 1 4 1 1.000000 45.000000 45.000000 0 }}}}}{SECT 1 {PARA 3 "" 0 "An Iterative Method" {TEXT -1 19 "An Iterative Method" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Make a \+ first approximation, " }{XPPEDIT 18 0 "phi = M;" "6#/%$phiG%\"MG" } {TEXT -1 53 ", and substitute into the rhs of the Kepler equation " } {XPPEDIT 18 0 "phi = e*sin(phi)+M;" "6#/%$phiG,&*&%\"eG\"\"\"-%$sinG6# F$F(F(%\"MGF(" }{TEXT -1 104 " to get a better approximation. Keep do ing this. One finds the following succession of approximations:" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "restart;" "6#%(restartG" }} {PARA 0 "" 0 "" {XPPEDIT 19 1 "alias(phi = phi(e,M)):" "6#-%&aliasG6#/ %$phiG-F'6$%\"eG%\"MG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Keq := phi = e*sin(phi)+M:" "6#>%$KeqG/%$phiG,&*&%\"eG\"\"\"-%$sinG6#F&F*F*% \"MGF*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "N := 6:" "6#>%\"NG\" \"'" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 162 "phi[0]=M;\nfor j to N do \n phi[j] = subs(phi=rhs(%),rhs(Keq)); \n subs( f=phi, expansion ( subs(phi=f,%), e, j ) ); \n collect(combine(%,trig),e); \n print(% ) \nod:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$phiG6#\"\"!%\"MG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$phiG6#\"\"\",&*&%\"eGF'-%$sinG6#% \"MGF'F'F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$phiG6#\"\"#,(%\"MG \"\"\"*&%\"eGF*-%$sinG6#F)F*F**&)F,F'\"\"\"-F.6#,$F)F'F*#F*F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$phiG6#\"\"$,**&,&-%$sinG6#,$%\"MGF '#F'\"\")-F,6#F/#!\"\"F1\"\"\")%\"eGF'\"\"\"F6*&)F8\"\"#F9-F,6#,$F/F-F,6#,$F/F' #\"$D\"\"$%QF7)%\"eGF'\"\"\"F7*&,&-F,6#,$F/\"\"%#F7F0-F,6#,$F/\"\"##! \"\"\"\"'F7)F@FGFAF7*&,&F+#F0\"\")F4#FNFTF7)F@F0FAF7*&)F@FLFAFIF7#F7FL *&F@F7F4F7F7F/F7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%$phiG6#\"\"',0* &,(-%$sinG6#,$%\"MG\"\"##\"\"\"\"#[-F,6#,$F/F'#\"#F\"#!)-F,6#,$F/\"\"% #!\"%\"#:F2)%\"eGF'\"\"\"F2*&,(-F,6#,$F/\"\"$#!#F\"$G\"-F,6#F/#F2\"$#> -F,6#,$F/\"\"&#\"$D\"\"$%QF2)FBFTFCF2*&,&F:#F2FIF+#!\"\"F'F2)FBF=FCF2* &,&FF#FI\"\")FM#FgnF\\oF2)FBFIFCF2*&)FBF0FCF+F2#F2F0*&FBF2FMF2F2F/F2" }}}}{SECT 1 {PARA 3 "" 0 "A Series Expansion Method" {TEXT -1 25 "A Se ries Expansion Method" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Another \+ approach, which yields the same result, is to construct a trial soluti on in the form of a power series in eccentricity, " }{XPPEDIT 19 1 "ph i = sum(C[n]*e^n,n = 0 .. N);" "6#/%$phiG-%$sumG6$*&&%\"CG6#%\"nG\"\" \")%\"eGF,F-/F,;\"\"!%\"NG" }{TEXT -1 2 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$phiG,0&%\"CG6#\"\"!\"\"\"*&&F'6#F*F*%\"eGF*F**&&F'6# \"\"#F*)F.F2\"\"\"F**&&F'6#\"\"$F*)F.F8F4F**&&F'6#\"\"%F*)F.F=F4F**&&F '6#\"\"&F*)F.FBF4F**&&F'6#\"\"'F*)F.FGF4F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Substitute this back into the Kepler equation and solve \+ for the coefficients. Here is a Maple procedure that does this:" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 1314 "#--------------------------- ------------------------------------\n# Solve an equation phi = F(phi, eps) by series expansion, where\n# phi is the solution variable and ep s is a small parameter.\n#-------------------------------------------- -------------------\nxsolve := proc( expr::\{algebraic,algebraic=algeb raic\}, \n soln_var::\{name,function\}, \n \+ small_param::name,\n expansion_order::posint )\n\n \+ local k, eqn, sols, S, trial, C;\n\n if type(expr,`=`) then\n S := lhs(expr) - rhs(expr);\n else\n S := expr;\n fi;\n\n #Create a \+ trial solution of the form \n # phi = C[0] + C[1]*eps + ... + C[N]*e ps^N\n #and substitute that into the equation, then \n #expand into \+ a power series to order N.\n trial := sum( C[k]*small_param^k, k=0..e xpansion_order );\n subs( soln_var=trial, S );\n S := expansion( %, \+ small_param, expansion_order );\n\n #Solve for the coefficients C[k], starting with C[0] \n #and successively working our way up to C[N], \+ by equating\n #coefficients of like powers of eps to zero.\n sols := [];\n for k from 0 to expansion_order do\n coeff( S, small_param, k );\n eqn := isolate( subs(sols,%), C[k] );\n if nargs > 4 the n\n eqn := args[5]( eqn, args[6..nargs] );\n fi;\n sols := \+ [ op(sols), eqn ];\n od;\n\n soln_var = subs( sols, trial );\n\nend: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Applying this to the Kepler e quation, we find" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "xsolve(Keq, phi,e,N,combine,trig);" "6#-%'xsolveG6(%$KeqG%$phiG%\"eG%\"NG%(combine G%%trigG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%$phiG,0%\"MG\"\"\"*&-%$s inG6#F&F'%\"eGF'F'*&-F*6#,$F&\"\"#F')F,F1\"\"\"#F'F1*&,&-F*6#,$F&\"\"$ #F:\"\")F)#!\"\"F\"\"'-F*6#,$F&\"\"%#F'F:F')F,FGF 3F'*&,(F7#!#F\"$G\"F)#F'\"$#>-F*6#,$F&\"\"&#\"$D\"\"$%QF')F,FTF3F'*&,( F.#F'\"#[-F*6#,$F&FC#\"#F\"#!)FD#!\"%\"#:F')F,FCF3F'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "phi6 := %:" "6#>%%phi6G%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Notice that we can regroup this as a Four ier series:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect(%,sin); " "6#-%(collectG6$%\"%G%$sinG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%$ph iG,0*&,(*$)%\"eG\"\"'\"\"\"#\"\"\"\"#[*$)F*\"\"#F,#F.F2*$)F*\"\"%F,#! \"\"F+F.-%$sinG6#,$%\"MGF2F.F.*&,&F4#F.\"\"$F(#!\"%\"#:F.-F:6#,$F=F6F. F.*&)F*\"\"&F,-F:6#,$F=FJF.#\"$D\"\"$%Q*&,(*$)F*FAF,#F8\"\")*$FIF,#F. \"$#>F*F.F.-F:6#F=F.F.*&-F:6#,$F=F+F.F)F,#\"#F\"#!)*&,&FS#FAFVFW#!#F\" $G\"F.-F:6#,$F=FAF.F.F=F." }}}}{SECT 1 {PARA 3 "" 0 "A Fourier Expansi on..." {TEXT -1 97 "A Fourier Sine Series Expansion and Resulting Bess el Function Representation for the Coefficients" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Suppose we expand " }{XPPEDIT 18 0 "e*sin*phi;" "6#* (%\"eG\"\"\"%$sinGF%%$phiGF%" }{TEXT -1 21 " in a Fourier series:" }} {PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 18 0 "e*sin*phi = Sum( 2*b[k]*sin(k*M),k = 1 .. infinity);" "6#/*(%\"eG\"\"\"%$sinGF&%$phiGF& -%$SumG6$*(\"\"#F&&%\"bG6#%\"kGF&-F'6#*&F1F&%\"MGF&F&/F1;\"\"\"%)infin ityG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "where" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 19 1 "Pi*b[k] = Int(e*sin(phi) *sin(k*M),M = 0 .. Pi):" "6#/*&%#PiG\"\"\"&%\"bG6#%\"kGF&-%$IntG6$*(% \"eGF&-%$sinG6#%$phiGF&-F16#*&F*F&%\"MGF&F&/F7;\"\"!F%" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 30 "Integrate this by parts to get" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Pi*b[k] = intparts(rhs(%),sin(ph i));" "6#/*&%#PiG\"\"\"&%\"bG6#%\"kGF&-%)intpartsG6$-%$rhsG6#%\"%G-%$s inG6#%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#PiG\"\"\"&%\"bG6#% \"kGF&,(*&*(-%$sinG6#-%$phiG6$%\"eGF%F&F4F&-%$cosG6#*&F*F&F%\"\"\"F&F9 F*!\"\"!\"\"*&*&-F/6#-F26$F4\"\"!F&F4F9F9F*F:F&-%$IntG6$,$*&**-F66#F2F &-%%diffG6$F2%\"MGF&F4F9-F66#*&F*F9FNF&F&F9F*F:F;/FN;FBF%F;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The first two terms are zero, so w e are left with " }{XPPEDIT 19 1 "Pi*b[k] = select(has,rhs(%),Int);" " 6#/*&%#PiG\"\"\"&%\"bG6#%\"kGF&-%'selectG6%%$hasG-%$rhsG6#%\"%G%$IntG " }{TEXT -1 2 ": " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#PiG\"\"\"&% \"bG6#%\"kGF&,$-%$IntG6$,$*&**-%$cosG6#%$phiGF&-%%diffG6$F5%\"MGF&%\"e GF&-F36#*&F*F&F9F&F&\"\"\"F*!\"\"!\"\"/F9;\"\"!F%F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Now, " }{XPPEDIT 18 0 "diff(phi,M)*e*cos(phi)*dM = d(e*sin(phi));" "6#/**-%%diffG6$%$phiG%\"MG\"\"\"%\"eGF*-%$cosG6#F( F*%#dMGF*-%\"dG6#*&F+F*-%$sinG6#F(F*" }{TEXT -1 34 ". Hence, since we can also write " }{XPPEDIT 18 0 "e*sin*phi = phi-M;" "6#/*(%\"eG\"\" \"%$sinGF&%$phiGF&,&F(F&%\"MG!\"\"" }{TEXT -1 10 ", we have " } {XPPEDIT 18 0 "d(e*sin(phi)) = d*phi-d*M;" "6#/-%\"dG6#*&%\"eG\"\"\"-% $sinG6#%$phiGF),&*&F%F)F-F)F)*&F%F)%\"MGF)!\"\"" }{TEXT -1 5 " and " } }{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "Pi*b[k] = Int( cos(k*M)/k,phi = 0 .. Pi)-Int(cos(k*M)/k,M = 0 .. Pi):" "6#/*&%#PiG\" \"\"&%\"bG6#%\"kGF&,&-%$IntG6$*&-%$cosG6#*&F*F&%\"MGF&F&F*!\"\"/%$phiG ;\"\"!F%F&-F-6$*&-F16#*&F*F&F4F&F&F*F5/F4;F9F%F5" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "T he second integral is zero for integer " }{XPPEDIT 18 0 "k <> 0;" "6#0 %\"kG\"\"!" }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 39 "Int(cos(k*M)/k,M=0..Pi): % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$cosG6#*&%\"kG\"\"\"%\"MGF-\"\"\"F,!\"\"/ F.;\"\"!%#PiG*&-%$sinG6#*&F,F/F4F-F/*$)F,\"\"#F/F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Thus, we are left with " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "Pi*b[k] = Int(cos(k*(phi-e*sin(phi))) /k,phi = 0 .. Pi);" "6#/*&%#PiG\"\"\"&%\"bG6#%\"kGF&-%$IntG6$*&-%$cosG 6#*&F*F&,&%$phiGF&*&%\"eGF&-%$sinG6#F4F&!\"\"F&F&F*F:/F4;\"\"!F%" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "Now, the Bessel functio n of the first kind is defined as" }}{PARA 0 "" 0 "" {TEXT -1 9 " \+ " }{XPPEDIT 18 0 "Pi*J[n](x) = Int(cos(n*theta-x*sin(theta)),theta = 0 .. Pi);" "6#/*&%#PiG\"\"\"-&%\"JG6#%\"nG6#%\"xGF&-%$IntG6$-%$cosG 6#,&*&F+F&%&thetaGF&F&*&F-F&-%$sinG6#F6F&!\"\"/F6;\"\"!F%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "n;" "6#% \"nG" }{TEXT -1 44 " is an integer. Therefore, finally, we have" }} {PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "b[k] = J[k](k*e )/k;" "6#/&%\"bG6#%\"kG*&-&%\"JG6#F'6#*&F'\"\"\"%\"eGF/F/F'!\"\"" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "phi = M+2*sum(J[k](k*e)*sin(k*M )/k,k = 1 .. infinity);" "6#/%$phiG,&%\"MG\"\"\"*&\"\"#F'-%$sumG6$*(-& %\"JG6#%\"kG6#*&F2F'%\"eGF'F'-%$sinG6#*&F2F'F&F'F'F2!\"\"/F2;\"\"\"%)i nfinityGF'F'" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 " Let's compare this with our previous series expansion example, " } {XPPEDIT 19 1 "phi6;" "6#%%phi6G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/% $phiG,0*&,(-%$sinG6#,$%\"MG\"\"##\"\"\"\"#[-F)6#,$F,\"\"'#\"#F\"#!)-F) 6#,$F,\"\"%#!\"%\"#:F/)%\"eGF4\"\"\"F/*&,(-F)6#,$F,\"\"$#!#F\"$G\"-F)6 #F,#F/\"$#>-F)6#,$F,\"\"&#\"$D\"\"$%QF/)F@FRFAF/*&,&F8#F/FGF(#!\"\"F4F /)F@F;FAF/*&,&FD#FG\"\")FK#FenFjnF/)F@FGFAF/*&)F@F-FAF(F/#F/F-*&F@F/FK F/F/F,F/" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "M+2*sum(J[k](k*e)*s in(k*M)/k,k = 1 .. N):" "6#,&%\"MG\"\"\"*&\"\"#F%-%$sumG6$*(-&%\"JG6#% \"kG6#*&F0F%%\"eGF%F%-%$sinG6#*&F0F%F$F%F%F0!\"\"/F0;\"\"\"%\"NGF%F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "series(subs(seq(J[k](k*e) = B esselJ(k,k*e),k = 1 .. N),%),e,N+1);" "6#-%'seriesG6%-%%subsG6$-%$seqG 6$/-&%\"JG6#%\"kG6#*&F1\"\"\"%\"eGF4-%(BesselJG6$F1*&F1F4F5F4/F1;\"\" \"%\"NG%\"%GF5,&F=F4\"\"\"F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+3%\"e G%\"MG\"\"!-%$sinG6#F%\"\"\",$-F(6#,$F%\"\"##\"\"\"F/\"\"#,&-F(6#,$F% \"\"$#F7\"\")F'#!\"\"F9\"\"$,&-F(6#,$F%\"\"%#F1F7F,#F;\"\"'\"\"%,(F4#! #F\"$G\"F'#F1\"$#>-F(6#,$F%\"\"&#\"$D\"\"$%Q\"\"&,(F,#F1\"#[-F(6#,$F%F D#\"#F\"#!)F>#!\"%\"#:\"\"'-%\"OG6#F1\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "simplify(convert(%,polynom)-rhs(phi6));" "6#-%)simplify G6#,&-%(convertG6$%\"%G%(polynomG\"\"\"-%$rhsG6#%%phi6G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Since " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "1 = \+ J[0](x)^2+2*Sum(J[k](x)^2,k = 1 .. infinity);" "6#/\"\"\",&*$-&%\"JG6# \"\"!6#%\"xG\"\"#\"\"\"*&\"\"#F/-%$SumG6$*$-&F)6#%\"kG6#F-\"\"#/F9;\" \"\"%)infinityGF/F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "w e know that the series " }}{PARA 0 "" 0 "" {TEXT -1 9 " " } {XPPEDIT 18 0 "(phi-M)/2 = sum(J[k](k*e)*sin(k*M)/k,k = 1 .. infinity) ;" "6#/*&,&%$phiG\"\"\"%\"MG!\"\"F'\"\"#F)-%$sumG6$*(-&%\"JG6#%\"kG6#* &F3F'%\"eGF'F'-%$sinG6#*&F3F'F(F'F'F3F)/F3;\"\"\"%)infinityG" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "is " }{TEXT 257 10 "absolute ly" }{TEXT -1 14 " convergent. " }}}}{SECT 1 {PARA 3 "" 0 "Chebyshev \+ Polynomials" {TEXT -1 21 "Chebyshev Polynomials" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Definition" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Su ppose we expand a function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " in a Chebyshev series:" }}{PARA 0 "" 0 "" {TEXT -1 9 " \+ " }{XPPEDIT 18 0 "f(x) = sum(a[k]*T[k](x),k = 0 .. infinity); " "6#/-%\"fG6#%\"xG-%$sumG6$*&&%\"aG6#%\"kG\"\"\"-&%\"TG6#F/6#F'F0/F/; \"\"!%)infinityG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "The \+ Chebyshev polynomials of the first kind are defined " }}{PARA 0 "" 0 " " {TEXT -1 9 " " }{XPPEDIT 18 0 "T[k](x) = cos(k*theta);" "6#/ -&%\"TG6#%\"kG6#%\"xG-%$cosG6#*&F(\"\"\"%&thetaGF/" }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "x = cos(theta);" "6#/%\"xG-%$cosG6#%&thetaG" }{TEXT -1 43 ". An explicit polynomial re presentation is" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "2*T[n](x) = n*sum((-1)^k*(n-k-1)!*(2*x)^(n-2*k)/(k!*(n-2*k)!),k \+ = 0 .. `[n/2]`);" "6#/*&\"\"#\"\"\"-&%\"TG6#%\"nG6#%\"xGF&*&F+F&-%$sum G6$**),$\"\"\"!\"\"%\"kGF&-%*factorialG6#,(F+F&F7F6\"\"\"F6F&)*&\"\"#F &F-F&,&F+F&*&\"\"#F&F7F&F6F&*&-F96#F7F&-F96#,&F+F&*&\"\"#F&F7F&F6F&F6/ F7;\"\"!%&[n/2]GF&" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "For example, the first several Chebyshev polynomials are " }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "for j to 12 do value(subs(n = j, T[n](x) = Sum(n*(-1)^k*(n-k-1)!*(2*x)^(n-2*k)/(2*k!*(n-2*k)!),k = 0 .. n/2))) od;" "6#?(%\"jG\"\"\"F%\"#7%%trueG-%&valueG6#-%%subsG6$/%\"nGF $/-&%\"TG6#F/6#%\"xG-%$SumG6$*,F/F%),$\"\"\"!\"\"%\"kGF%-%*factorialG6 #,(F/F%F?F>\"\"\"F>F%)*&\"\"#F%F6F%,&F/F%*&\"\"#F%F?F%F>F%*(\"\"#F%-FA 6#F?F%-FA6#,&F/F%*&\"\"#F%F?F%F>F%F>/F?;\"\"!*&F/F%\"\"#F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG6#\"\"\"6#%\"xGF*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-&%\"TG6#\"\"#6#%\"xG,&*$)F*F(\"\"\"F(!\"\"\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG6#\"\"$6#%\"xG,&*$)F*F(\"\"\" \"\"%F*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG6#\"\"%6#%\"xG, (*$)F*F(\"\"\"\"\")*$)F*\"\"#F.!\")\"\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG6#\"\"&6#%\"xG,(*$)F*F(\"\"\"\"#;*$)F*\"\"$F.!# ?F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG6#\"\"'6#%\"xG,**$)F* F(\"\"\"\"#K*$)F*\"\"%F.!#[*$)F*\"\"#F.\"#=!\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG6#\"\"(6#%\"xG,**$)F*F(\"\"\"\"#k*$)F*\"\" &F.!$7\"*$)F*\"\"$F.\"#cF*!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&% \"TG6#\"\")6#%\"xG,,*$)F*F(\"\"\"\"$G\"*$)F*\"\"'F.!$c#*$)F*\"\"%F.\"$ g\"*$)F*\"\"#F.!#K\"\"\"F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG 6#\"\"*6#%\"xG,,*$)F*F(\"\"\"\"$c#*$)F*\"\"(F.!$w&*$)F*\"\"&F.\"$K%*$) F*\"\"$F.!$?\"F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG6#\"#56# %\"xG,.*$)F*F(\"\"\"\"$7&*$)F*\"\")F.!%!G\"*$)F*\"\"'F.\"%?6*$)F*\"\"% F.!$+%*$)F*\"\"#F.\"#]!\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -&%\"TG6#\"#66#%\"xG,.*$)F*F(\"\"\"\"%C5*$)F*\"\"*F.!%;G*$)F*\"\"(F.\" %;G*$)F*\"\"&F.!%K7*$)F*\"\"$F.\"$?#F*!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG6#\"#76#%\"xG,0*$)F*F(\"\"\"\"%[?*$)F*\"#5F.!%W h*$)F*\"\")F.\"%7p*$)F*\"\"'F.!%%e$*$)F*\"\"%F.\"$S)*$)F*\"\"#F.!#s\" \"\"FD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Chebyshev polynomials s atisfy the following differential equations: " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "(1-x^2)*diff(y(x),x,x)-x*diff(y (x),x)+k^2*y(x) = 0;" "6#/,(*&,&\"\"\"\"\"\"*$%\"xG\"\"#!\"\"F(-%%diff G6%-%\"yG6#F*F*F*F(F(*&F*F(-F.6$-F16#F*F*F(F,*&%\"kG\"\"#-F16#F*F(F(\" \"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "T[k](x);" "6#-&%\"TG6#%\"kG6# %\"xG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 " " 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "diff(y(theta),theta,thet a)+k^2*y(theta) = 0;" "6#/,&-%%diffG6%-%\"yG6#%&thetaGF+F+\"\"\"*&%\"k G\"\"#-F)6#F+F,F,\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "T[k](cos*t heta);" "6#-&%\"TG6#%\"kG6#*&%$cosG\"\"\"%&thetaGF+" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Here is a graphical view of the first seven polynomials:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "wi th(polynomials):" "6#-%%withG6#%,polynomialsG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "plotpoly([1, 2, 3, 4, 5, 6, 7], T);" "6#-%)plotpolyG 6$7)\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(%\"TG" }}{PARA 13 "" 1 "" {GLPLOT2D 563 328 328 {PLOTDATA 2 "61-%'CURVESG6$7S7$$!\"\"\"\"!F(7$$! 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Pi) = matr ix([[Pi], [Pi/2]]);" "6#/-%$intG6$*$-&%\"TG6#%\"nG6#*&%$cosG\"\"\"%&th etaGF0\"\"#/F1;\"\"!%#PiG-%'matrixG6#7$7#F67#*&F6F0\"\"#!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "matrix([[n = 0], [n <> 0]]);" "6#-%'matri xG6#7$7#/%\"nG\"\"!7#0F)F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "If we write " }{XPPEDIT 18 0 "x = cos*theta;" "6#/%\"xG*&%$cosG \"\"\"%&thetaGF'" }{TEXT -1 14 ", these become" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "int(T[n](x)*T[m](x)/sqrt(1-x^2) ,x = -1 .. 1) = 0;" "6#/-%$intG6$*(-&%\"TG6#%\"nG6#%\"xG\"\"\"-&F*6#% \"mG6#F.F/-%%sqrtG6#,&\"\"\"F/*$F.\"\"#!\"\"F m;" "6#0%\"nG%\"mG" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "int(T[n](x)^2/sqrt(1-x^2),x = - 1 .. 1) = matrix([[Pi], [Pi/2]]);" "6#/-%$intG6$*&-&%\"TG6#%\"nG6#%\"x G\"\"#-%%sqrtG6#,&\"\"\"\"\"\"*$F.\"\"#!\"\"F8/F.;,$\"\"\"F8\"\"\"-%'m atrixG6#7$7#%#PiG7#*&FCF5\"\"#F8" }{TEXT -1 5 " for " }{XPPEDIT 18 0 " matrix([[n = 0], [n <> 0]]);" "6#-%'matrixG6#7$7#/%\"nG\"\"!7#0F)F*" } {TEXT -1 2 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "Least Squares Fit and Determination of Chebyshev Series Coefficients" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Let us write the truncation error for an \+ order " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 18 " approximation of \+ " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "epsilon[N] = f(x)-sum (a[k]*T[k](x),k = 0 .. N);" "6#/&%(epsilonG6#%\"NG,&-%\"fG6#%\"xG\"\" \"-%$sumG6$*&&%\"aG6#%\"kGF--&%\"TG6#F56#F,F-/F5;\"\"!F'!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 90 "One measure of goodness of fi t of a polynomial to a function is the least squares integral" }} {PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "S[N] = Int(w(x) *epsilon[N]^2,x = -1 .. 1);" "6#/&%\"SG6#%\"NG-%$IntG6$*&-%\"wG6#%\"xG \"\"\"*$&%(epsilonG6#F'\"\"#F0/F/;,$\"\"\"!\"\"\"\"\"" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 28 "for some weighting function " } {XPPEDIT 18 0 "w(x);" "6#-%\"wG6#%\"xG" }{TEXT -1 15 ". To minimize \+ " }{XPPEDIT 18 0 "S[N];" "6#&%\"SG6#%\"NG" }{TEXT -1 9 ", we set " } {XPPEDIT 18 0 "diff(S[N],a[k]) = 0;" "6#/-%%diffG6$&%\"SG6#%\"NG&%\"aG 6#%\"kG\"\"!" }{TEXT -1 13 " for all the " }{XPPEDIT 18 0 "a[k];" "6#& %\"aG6#%\"kG" }{TEXT -1 11 ", finding " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "Int(2*w(x)*(f(x)-Sum(a[n]*T[n](x),n = 0 .. N))*T[k](x),x = -1 .. 1) = 0;" "6#/-%$IntG6$**\"\"#\"\"\"-%\"wG6#%\"x GF),&-%\"fG6#F-F)-%$SumG6$*&&%\"aG6#%\"nGF)-&%\"TG6#F96#F-F)/F9;\"\"!% \"NG!\"\"F)-&F<6#%\"kG6#F-F)/F-;,$\"\"\"FC\"\"\"FA" }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 30 "Let the weighting function be " } {XPPEDIT 18 0 "w(x) = 1/sqrt(1-x^2);" "6#/-%\"wG6#%\"xG*&\"\"\"\"\"\"- %%sqrtG6#,&\"\"\"F**$F'\"\"#!\"\"F2" }{TEXT -1 7 ". Then" }}{PARA 0 " " 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "Sum(a[n]*Int(T[n](x)*T[k ](x)/sqrt(1-x^2),x = -1 .. 1),n = 0 .. N) = Int(f(x)*T[k](x)/sqrt(1-x^ 2),x = -1 .. 1);" "6#/-%$SumG6$*&&%\"aG6#%\"nG\"\"\"-%$IntG6$*(-&%\"TG 6#F+6#%\"xGF,-&F36#%\"kG6#F6F,-%%sqrtG6#,&\"\"\"F,*$F6\"\"#!\"\"FC/F6; ,$\"\"\"FC\"\"\"F,/F+;\"\"!%\"NG-F.6$*(-%\"fG6#F6F,-&F36#F:6#F6F,-F=6# ,&\"\"\"F,*$F6\"\"#FCFC/F6;,$\"\"\"FC\"\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 59 "Using the orthogonality relations, we arrive at the results" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "Pi*a[0] = Int(f(x)/sqrt(1-x^2),x = -1 .. 1);" "6#/*&%#PiG\"\"\"&%\"aG 6#\"\"!F&-%$IntG6$*&-%\"fG6#%\"xGF&-%%sqrtG6#,&\"\"\"F&*$F2\"\"#!\"\"F :/F2;,$\"\"\"F:\"\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " and" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "Pi*a[k] \+ = 2*Int(f(x)*T[k](x)/sqrt(1-x^2),x = -1 .. 1);" "6#/*&%#PiG\"\"\"&%\"a G6#%\"kGF&*&\"\"#F&-%$IntG6$*(-%\"fG6#%\"xGF&-&%\"TG6#F*6#F4F&-%%sqrtG 6#,&\"\"\"F&*$F4\"\"#!\"\"FA/F4;,$\"\"\"FA\"\"\"F&" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "k <> 0;" "6#0%\"kG\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "These are equivalent to" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "Pi*a[0] = Int(f(cos*theta),theta = 0 \+ .. Pi);" "6#/*&%#PiG\"\"\"&%\"aG6#\"\"!F&-%$IntG6$-%\"fG6#*&%$cosGF&%& thetaGF&/F3;F*F%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "Pi*a[k] = 2*I nt(f(cos*theta)*cos(k*theta),theta = 0 .. Pi);" "6#/*&%#PiG\"\"\"&%\"a G6#%\"kGF&*&\"\"#F&-%$IntG6$*&-%\"fG6#*&%$cosGF&%&thetaGF&F&-F56#*&F*F &F6F&F&/F6;\"\"!F%F&" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "k <> 0;" "6 #0%\"kG\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "cos*theta = x;" "6#/*&%$cosG\"\"\"%&thetaGF&%\"xG" } {TEXT -1 3 ". " }}}}}{SECT 1 {PARA 3 "" 0 "A Chebyshev Series Expansi on" {TEXT -1 28 "A Chebyshev Series Expansion" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Determination of the Coefficients" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 18 "Suppose we expand " }{XPPEDIT 18 0 "phi-M = e*sin* phi;" "6#/,&%$phiG\"\"\"%\"MG!\"\"*(%\"eGF&%$sinGF&F%F&" }{TEXT -1 23 " in a Chebyshev series:" }}{PARA 0 "" 0 "" {TEXT -1 8 " " } {XPPEDIT 18 0 "e*sin*phi = Sum(2*a[k]*T[k](cos*M),k = 1 .. infinity); " "6#/*(%\"eG\"\"\"%$sinGF&%$phiGF&-%$SumG6$*(\"\"#F&&%\"aG6#%\"kGF&-& %\"TG6#F16#*&%$cosGF&%\"MGF&F&/F1;\"\"\"%)infinityG" }{TEXT -1 3 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 20 "The coefficients are" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 19 1 "Pi*a[k] = Int(e*sin(phi)*cos( k*M),M = 0 .. Pi):" "6#/*&%#PiG\"\"\"&%\"aG6#%\"kGF&-%$IntG6$*(%\"eGF& -%$sinG6#%$phiGF&-%$cosG6#*&F*F&%\"MGF&F&/F8;\"\"!F%" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 30 "Integrate this by parts to get" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Pi*a[k] = intparts(rhs(%),sin(ph i));" "6#/*&%#PiG\"\"\"&%\"aG6#%\"kGF&-%)intpartsG6$-%$rhsG6#%\"%G-%$s inG6#%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#PiG\"\"\"&%\"aG6#% \"kGF&,&*&*(-%$sinG6#-%$phiG6$%\"eGF%F&F4F&-F/6#*&F*F&F%\"\"\"F&F8F*! \"\"F&-%$IntG6$*&**-%$cosG6#F2F&-%%diffG6$F2%\"MGF&F4F8-F/6#*&F*F8FEF& F&F8F*F9/FE;\"\"!F%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The f irst term is zero, so we are left with " }{XPPEDIT 19 1 "Pi*a[k] = sel ect(has,rhs(%),Int);" "6#/*&%#PiG\"\"\"&%\"aG6#%\"kGF&-%'selectG6%%$ha sG-%$rhsG6#%\"%G%$IntG" }{TEXT -1 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#PiG\"\"\"&%\"aG6#%\"kGF&,$-%$IntG6$*&**-%$cosG6#%$phiGF&-%% diffG6$F4%\"MGF&%\"eGF&-%$sinG6#*&F*F&F8F&F&\"\"\"F*!\"\"/F8;\"\"!F%! \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Again, " }{XPPEDIT 18 0 "d iff(phi,M)*e*cos(phi)*dM = d(e*sin(phi));" "6#/**-%%diffG6$%$phiG%\"MG \"\"\"%\"eGF*-%$cosG6#F(F*%#dMGF*-%\"dG6#*&F+F*-%$sinG6#F(F*" }{TEXT -1 22 ". Since we can write " }{XPPEDIT 18 0 "e*sin*phi = phi-M;" "6# /*(%\"eG\"\"\"%$sinGF&%$phiGF&,&F(F&%\"MG!\"\"" }{TEXT -1 16 ", we aga in have " }{XPPEDIT 18 0 "d(e*sin(phi)) = d*phi-d*M;" "6#/-%\"dG6#*&% \"eG\"\"\"-%$sinG6#%$phiGF),&*&F%F)F-F)F)*&F%F)%\"MGF)!\"\"" }{TEXT -1 17 " and, therefore, " }}{PARA 0 "" 0 "" {TEXT -1 9 " " } {XPPEDIT 18 0 "Pi*a[k] = Int(sin(k*M)/k,M = 0 .. Pi)-Int(sin(k*M)/k,ph i = 0 .. Pi);" "6#/*&%#PiG\"\"\"&%\"aG6#%\"kGF&,&-%$IntG6$*&-%$sinG6#* &F*F&%\"MGF&F&F*!\"\"/F4;\"\"!F%F&-F-6$*&-F16#*&F*F&F4F&F&F*F5/%$phiG; F8F%F5" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The fi rst integral is " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 39 "Int(sin(k *M)/k,M=0..Pi): % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ IntG6$*&-%$sinG6#*&%\"kG\"\"\"%\"MGF-\"\"\"F,!\"\"/F.;\"\"!%#PiG,$*&,& -%$cosG6#*&F,F/F4F-F-!\"\"F-F/*$)F,\"\"#F/F0F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Hence, we have " }}{PARA 0 "" 0 "" {TEXT -1 9 " \+ " }{XPPEDIT 18 0 "Pi*a[k] = (1-cos(k*Pi))/(k^2)-Int(sin(k*(phi-e*s in(phi)))/k,phi = 0 .. Pi);" "6#/*&%#PiG\"\"\"&%\"aG6#%\"kGF&,&*&,&\" \"\"F&-%$cosG6#*&F*F&F%F&!\"\"F&*$F*\"\"#F3F&-%$IntG6$*&-%$sinG6#*&F*F &,&%$phiGF&*&%\"eGF&-F;6#F?F&F3F&F&F*F3/F?;\"\"!F%F3" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 37 "Now, the Weber function is defined as " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "Pi*E[nu](x) = Int(sin(nu*theta-x*sin(theta)),theta = 0 .. Pi);" "6#/*&%#PiG\"\"\" -&%\"EG6#%#nuG6#%\"xGF&-%$IntG6$-%$sinG6#,&*&F+F&%&thetaGF&F&*&F-F&-F2 6#F6F&!\"\"/F6;\"\"!F%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "Therefore, we have" }}{PARA 0 "" 0 "" {TEXT -1 9 " " } {XPPEDIT 18 0 "a[k] = (1-cos(k*Pi))/(Pi*k^2)-E[k](k*e)/k;" "6#/&%\"aG6 #%\"kG,&*&,&\"\"\"\"\"\"-%$cosG6#*&F'F,%#PiGF,!\"\"F,*&F1F,*$F'\"\"#F, F2F,*&-&%\"EG6#F'6#*&F'F,%\"eGF,F,F'F2F2" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 9 " " } {XPPEDIT 18 0 "phi = M+Sum(2*((1-cos(k*Pi))/(Pi*k^2)-E[k](k*e)/k)*T[k] (cos*M),k = 1 .. infinity);" "6#/%$phiG,&%\"MG\"\"\"-%$SumG6$*(\"\"#F' ,&*&,&\"\"\"F'-%$cosG6#*&%\"kGF'%#PiGF'!\"\"F'*&F6F'*$F5\"\"#F'F7F'*&- &%\"EG6#F56#*&F5F'%\"eGF'F'F5F7F7F'-&%\"TG6#F56#*&F2F'F&F'F'/F5;\"\"\" %)infinityGF'" }{TEXT -1 2 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "The Weber Function" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The firs t few terms of the series expansion of " }{XPPEDIT 18 0 "E[k](x);" "6# -&%\"EG6#%\"kG6#%\"xG" }{TEXT -1 4 " are" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "E[k](x) = series(WeberE(k,x),x,8);" "6#/-&%\"EG6#%\"kG6 #%\"xG-%'seriesG6%-%'WeberEG6$F(F*F*\"\")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-&%\"EG6#%\"kG6#%\"xG+5F**&-%$sinG6#,$*&F(\"\"\"%#PiGF 2#F2\"\"#\"\"\"*&-%&GAMMAG6#,&F2F2F(#!\"\"F5\"\"\"-F96#,&F2F2F(F4\"\" \"!\"\"\"\"!,$*&-%$cosGF/F6*&-F96#,&#\"\"$F5F2F(F<\"\"\"-F96#,&FMF2F(F 4\"\"\"FCF4\"\"\",$*&F-F6*&-F96#,&F5F2F(F<\"\"\"-F96#,&F5F2F(F4\"\"\"F C#F=\"\"%\"\"#,$*&FGF6*&-F96#,&#\"\"&F5F2F(F<\"\"\"-F96#,&FcoF2F(F4\" \"\"FC#F=\"\")\"\"$,$*&F-F6*&-F96#,&FNF2F(F<\"\"\"-F96#,&FNF2F(F4\"\" \"FC#F2\"#;\"\"%,$*&FGF6*&-F96#,&#\"\"(F5F2F(F<\"\"\"-F96#,&FaqF2F(F4 \"\"\"FC#F2\"#K\"\"&,$*&F-F6*&-F96#,&F[oF2F(F<\"\"\"-F96#,&F[oF2F(F4\" \"\"FC#F=\"#k\"\"',$*&FGF6*&-F96#,&#\"\"*F5F2F(F<\"\"\"-F96#,&F_sF2F(F 4\"\"\"FC#F=\"$G\"\"\"(-%\"OG6#F2\"\")" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "limit(convert(rhs(%),polynom),k = 6);" "6#-%&limitG6$-% (convertG6$-%$rhsG6#%\"%G%(polynomG/%\"kG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"xG\"\"\",**$)F&\"\"#\"\"\"!$V\"*$)F&\"\"%F,!#8 !%hQF'*$)F&\"\"'F,F'F'F,%#PiG!\"\"#F+\"'N^8" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 28 "Compare with the definition:" }}}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 19 1 "series(Int(sin(nu*theta-x*sin(theta)),theta = 0 .. Pi )/Pi,x,8);" "6#-%'seriesG6%*&-%$IntG6$-%$sinG6#,&*&%#nuG\"\"\"%&thetaG F0F0*&%\"xGF0-F+6#F1F0!\"\"/F1;\"\"!%#PiGF0F:F6F3\"\")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+5%\"xG,$*&,&-%$cosG6#*&%#PiG\"\"\"%#nuGF-F-!\"\"F -\"\"\"*&F.\"\"\"F,\"\"\"!\"\"F/\"\"!*&,&F(F-F-F-F0*(,&F.F-F-F-\"\"\", &F.F-F/F-\"\"\"F,\"\"\"F4\"\"\",$*&F'F0**F.\"\"\",&F.F-\"\"#F-\"\"\",& F.F-!\"#F-\"\"\"F,\"\"\"F4F/\"\"#*&F7F0*,,&F.F-\"\"$F-\"\"\",&F.F-!\"$ F-\"\"\"F9\"\"\"F;\"\"\"F,\"\"\"F4\"\"$,$*&F'F0*.F.\"\"\",&F.F-\"\"%F- \"\"\",&F.F-!\"%F-\"\"\"FC\"\"\"FF\"\"\"F,\"\"\"F4F/\"\"%*&F7F0*0,&F.F -\"\"&F-\"\"\",&F.F-!\"&F-\"\"\"FM\"\"\"FP\"\"\"F9\"\"\"F;\"\"\"F,\"\" \"F4\"\"&,$*&F'F0*2F.\"\"\",&F.F-\"\"'F-\"\"\",&F.F-!\"'F-\"\"\"Fen\" \"\"Fhn\"\"\"FC\"\"\"FF\"\"\"F,\"\"\"F4F/\"\"'*&F7F0*4,&F.F-\"\"(F-\" \"\"F;\"\"\",&F.F-!\"(F-\"\"\"Fao\"\"\"Fdo\"\"\"FM\"\"\"FP\"\"\"F9\"\" \"F,\"\"\"F4\"\"(-%\"OG6#F-\"\")" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "limit(convert(%,polynom),nu = 6);" "6#-%&limitG6$-%(convertG6$% \"%G%(polynomG/%#nuG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\" xG\"\"\",**$)F&\"\"#\"\"\"!$V\"*$)F&\"\"%F,!#8!%hQF'*$)F&\"\"'F,F'F'F, %#PiG!\"\"#!\"#\"'N^8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "It appea rs Maple's definition differs by a minus sign, so we'll define our own Weber function." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "W := fn(Int (sin(nu*theta-x*sin(theta)),theta = 0 .. Pi)/Pi,nu,x):" "6#>%\"WG-%#fn G6%*&-%$IntG6$-%$sinG6#,&*&%#nuG\"\"\"%&thetaGF2F2*&%\"xGF2-F-6#F3F2! \"\"/F3;\"\"!%#PiGF2F\"\"'-F*6#,$F&\"\"%#F'F:F')F,FG F3F'*&,(F7#!#F\"$G\"F)#F'\"$#>-F*6#,$F&\"\"&#\"$D\"\"$%QF')F,FTF3F'*&, (F.#F'\"#[-F*6#,$F&FC#\"#F\"#!)FD#!\"%\"#:F')F,FCF3F'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "f_taylor := fn(rhs(phi6),e,M):" "6#>%)f _taylorG-%#fnG6%-%$rhsG6#%%phi6G%\"eG%\"MG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "M+sum(2*((1-cos(k*Pi))/(Pi*k^2)-E[k](k*e)/k)*orthopoly[ T](k,cos(M)),k = 1 .. N):" "6#,&%\"MG\"\"\"-%$sumG6$*(\"\"#F%,&*&,&\" \"\"F%-%$cosG6#*&%\"kGF%%#PiGF%!\"\"F%*&F4F%*$F3\"\"#F%F5F%*&-&%\"EG6# F36#*&F3F%%\"eGF%F%F3F5F5F%-&%*orthopolyG6#%\"TG6$F3-F06#F$F%/F3;\"\" \"%\"NGF%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(seq(E[k](k*e) = WeberE(k,k*e),k = 1 .. N),%):" "6#-%%subsG6$-%$seqG6$/-&%\"EG6#%\"k G6#*&F.\"\"\"%\"eGF1-%'WeberEG6$F.*&F.F1F2F1/F.;\"\"\"%\"NG%\"%G" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "f_weber := fn(%,e,M):" "6#>%(f_w eberG-%#fnG6%%\"%G%\"eG%\"MG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(seq(E[k](k*e) = W(k,k*e),k = 1 .. 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