{VERSION 4 0 "IBM INTEL LINUX" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# BESSEL FUNCTIONS" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "# What does Maple know about Bessel functions?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "?Bessel" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "?BesselZeros" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "# The exercises occupying pp. 379-386 of D. Betounes," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# Partial Differential Equations for Computat ional Science" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# (Springe r, 1998)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "# show lots of \+ things that computer algebra systems can do with Bessel functions." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 77 "# For example, Ex. 5 challenges your CAS to evaluat e the integrals that arise" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# as normalization constants in Fourier-Bessel series:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Int(BesselJ(0, w*r)^2 * r, r);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Int(BesselJ(3, w*r)^2 * r, r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Int(BesselJ(n, r)^2 *r, r); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "# Let's try to expand something in \+ Bessel functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "h := x -> 1/(1 + x^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "# Use t he notation for the 0th-order Fourier-Bessel series on p. 96 of notes. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "# Take r_0 = 1." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "c := k-> num(k)/den(k);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "den := k -> int( BesselJ(0, \+ BesselJZeros(0,k)*r)^2 * r, r=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "den(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "e valf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# (It works as \+ expected.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "num := k -> i nt( BesselJ(0, BesselJZeros(0,k)*r) * h(r) * r, r=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "num(1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "partialsum := (n,r) -> evalf (sum( c(k)*BesselJ(0,BesselJZeros(0,k)*r), k=1..n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "plot(h, 0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(partialsum(1,r), r=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(\{partialsum(2,r), h(r)\}, r=0..1);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(\{partialsum(6,r), h(r)\}, r=0..1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(\{partialsum(20,r), h(r )\}, r=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "# This loo ks much like a typical Fourier series, complete with Gibbs phenomenon " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# at the end where all \+ the eigenfunctions vanish." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "# Let's try a differ ent value of the Bessel index. This will cause trouble at" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# the origin, too." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "den := k -> int( BesselJ(3, BesselJZeros(3,k)*r)^2 * \+ r, r=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "num := k -> \+ int( BesselJ(3, BesselJZeros(3,k)*r) * h(r) * r, r=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "partialsum := (n,r) -> evalf(sum( c (k)*BesselJ(3,BesselJZeros(3,k)*r), k=1..n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(\{partialsum(1,r), h(r)\}, r=0..1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(\{partialsum(10,r), h(r )\}, r=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(\{par tialsum(50,r), h(r)\}, r=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "54" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }