(*^ ::[paletteColors = 128; showRuler; automaticGrouping; currentKernel; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = Left Header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = Left Footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12;] :[font = title; inactive; preserveAspect; startGroup; ] Bessel Functions :[font = subtitle; inactive; preserveAspect; ] Steve Dunbar Math 221H, Fall 1993 :[font = subsubtitle; inactive; preserveAspect; ] Covers Nagle & Saff, Section 8.8 :[font = input; preserveAspect; ] O wonder! How many goodly creatures are there here! How beauteous mankind is! O brave new world That has such people in't! William Shakespeare, The Tempest/5.1 ;[s] 3:0,0;6,1;165,2;180,-1; 3:1,10,8,Times,1,12,0,0,0;1,13,10,Times,0,14,0,0,0;1,13,10,Times,2,14,0,0,0; :[font = section; inactive; preserveAspect; startGroup; ] The Bessel Equation and Functions :[font = text; inactive; preserveAspect; ] The Bessel equation of order n (``nu'') is x2 y'' + x y' + (x2 - n2) y = 0. Notice that 0 is a regular singular point and so may not be able to be solved with the straightforward power series technique that we have developed so far. The solution of the Bessle equation, particularly of integer order and half-integer order is important in physica and engineering. ;[s] 10:0,0;30,1;31,2;48,3;49,4;65,5;67,6;69,7;70,8;71,9;412,-1; 10:1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,11,8,Times,32,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,11,8,Times,32,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,32,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] The Bessel Function of Order 0, J0(x): ;[s] 3:0,0;33,1;34,2;38,-1; 3:1,12,9,Times,1,14,0,0,0;1,12,9,Times,65,14,0,0,0;1,12,9,Times,1,14,0,0,0; :[font = input; preserveAspect; ] DSolve[ x^2 y''[x] + x y'[x] + x^2 y[x] == 0, y[x], x] :[font = input; preserveAspect; ] Plot[BesselJ[0,x], {x,0,10}] :[font = input; preserveAspect; ] Plot[BesselJ[0,x], {x,0,100}, PlotRange -> All] :[font = input; preserveAspect; ] Series[BesselJ[0,x],{x,0,20}] :[font = input; preserveAspect; endGroup; ] Sum[ ( (-1)^n/(n!)^2) (x/2)^(2n), {n,0,10}] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Zeroes of the Bessel Function J0(x) ;[s] 3:0,0;31,1;32,2;35,-1; 3:1,12,9,Times,1,14,0,0,0;1,12,9,Times,65,14,0,0,0;1,12,9,Times,1,14,0,0,0; :[font = input; preserveAspect; ] besselzeroes = Table[ x /. FindRoot[BesselJ[0,x] == 0, {x, guess}], {guess,1,20,2}] :[font = input; preserveAspect; ] besselzeroes[[3]] - besselzeroes[[2]] :[font = input; preserveAspect; ] besselzeroes[[5]] - besselzeroes[[3]] :[font = input; preserveAspect; ] besselzeroes[[6]] - besselzeroes[[5]] :[font = input; preserveAspect; ] besselzeroes[[8]] - besselzeroes[[6]] :[font = text; inactive; preserveAspect; endGroup; ] Exercise: What do you guess is the limit of the distance between successive zeroes of the Bessel function J0(x)? Can you make an intuitive justification of this by rewriting the equation in normal form? ;[s] 4:0,0;8,1;109,2;110,3;206,-1; 4:1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,11,8,Times,64,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] The Complementary Bessel Function of Order 0, Y0(x) ;[s] 3:0,0;47,1;48,2;52,-1; 3:1,12,9,Times,1,14,0,0,0;1,12,9,Times,65,14,0,0,0;1,12,9,Times,1,14,0,0,0; :[font = input; preserveAspect; startGroup; ] Plot[BesselY[0,x], {x,0,10}] :[font = message; inactive; preserveAspect; ] Plot::plnr: CompiledFunction[{x}, <<1>>, -C<<9>>de-][x] is not a machine-size real number at x = 0.. :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.095238 0.459528 0.276104 [ [(2)] 0.21429 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Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect; startGroup; ] yseries = Normal[Series[BesselY[0,x], {x,0,4}]] :[font = output; output; inactive; preserveAspect; endGroup; ] (2*EulerGamma)/Pi + (2*(Log[1/2] + Log[x]))/Pi + x^2*((2 - 2*EulerGamma)/(4*Pi) - (Log[1/2] + Log[x])/(2*Pi)) + x^4*(-(3 - 2*EulerGamma)/(64*Pi) + (Log[1/2] + Log[x])/(32*Pi)) ;[o] 1 2 (Log[-] + Log[x]) 2 EulerGamma 2 ------------ + ------------------- + Pi Pi 1 Log[-] + Log[x] 2 2 - 2 EulerGamma 2 x (---------------- - ---------------) + 4 Pi 2 Pi 1 Log[-] + Log[x] 4 -(3 - 2 EulerGamma) 2 x (------------------- + ---------------) 64 Pi 32 Pi :[font = input; preserveAspect; startGroup; ] Plot[{BesselJ[0,x], BesselY[0,x]}, {x, 0, 10}] :[font = message; inactive; preserveAspect; ] Plot::plnr: CompiledFunction[{x}, <<1>>, -<<11>>e-][x] is not a machine-size real number at x = 0.. :[font = postscript; PostScript; 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