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{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 }