Page 74 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

64 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
Thus, we note from that if the eigenvalues are λ j , the roots of J ν (λ j ρ) = 0 the orthogonality
condition is, according to Eq. (3.53) in Chapter 3

1

ρ J n (λ j ρ)J n (λ k ρ)dρ = 0, j = k
0
1
2
= [J n+1 (λ j )] , j = k (4.98)
2
On the other hand, if the eigenvalues are the roots of the equation

HJ n (λ j ) + λ j J (λ j ) = 0
n
1

ρ J n (λ j ρ)J n (λ k ρ)dρ = 0, j = k
0
2
2
2
(λ − n + H )[J n (λ j )] 2
j
= 2 , j = k (4.99)
2λ j
Using the equations in the table above and integrating by parts it is not difficult to show that
x x

n
n
s J 0 (s )ds = x J 1 (x) + (n − 1)x n−1 J 0 (x) − (n − 1) 2 s n−2 J 0 (s )ds (4.100)
s =0 s =0
4.2.2 Fourier–Bessel Series
Owing to the fact that Bessel’s equation with appropriate boundary conditions is a Sturm–
Liouville system it is possible to use the orthogonality property to expand any piecewise
continuous function on the interval 0 < x < 1 as a series of Bessel functions. For example,
let
∞

f (x) = A n J 0 (λ n x) (4.101)
n=1

Multiplying both sides by xJ 0 (λ k x)dx and integrating from x = 0to x = 1 (recall that the
weighting function x must be used to insure orthogonality) and noting the orthogonality
property we find that

∞ 1
x=0
xf (x)J 0 (λ j x)dx
f (x) = J 0 (λ j x) (4.102)
1 2
j=1 x=0 x[J 0 (λ j x)] dx