Page 73 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 63

TABLE 4.1: Some Properties of Bessel Functions
ν
ν
1. [x J ν (x)] = x J ν−1 (x)

x

2. [
−ν J ν (x)] =−x −ν J ν+1 (x)
3. J ν−1 (x) + J ν+1 (x) = 2ν/x[J ν (x)]
4. J ν−1 (x) − J ν+1 (x) = 2J ν (x)
ν
ν
5. x J ν−1 (x)dx = x J v + constant
−ν −ν
6. x J ν+1 (x)dx = x J ν (x) + constant
∞ n
(−1) 2(n + ν)
2n+2ν−1
= x (4.92)
2 2n+ν n!(n + ν) (n + ν)
n=0
∞ n
(−1)
ν
= x ν x 2n+2ν−1 = x J ν−1 (x) (4.93)
2 2n+ν−1 n! (n + ν)
n=0
These will prove important when we begin solving partial differential equations in cylindrical
coordinates using separation of variables.
Bessel’s equation is of the form (4.138) of a Sturm–Liouville equation and the func-
tions J n (x) are orthogonal with respect to a weight function ρ (see Eqs. (3.46) and (3.53),
Chapter 3).
Note that Bessel’s equation (4.67) with ν = n is

2
2 2
2
ρ J + ρ J + (λ ρ − n )J n = 0 (4.94)

n
n
which can be written as
d d
2
2 2
2
2
(ρ J ) + (λ ρ − n ) J = 0 (4.95)
dρ n dρ n
Integrating, we find that
1

2
2 1
2
2 2
2
[(ρ J ) + (λ ρ − n )J ] − 2λ 2 ρ J dρ = 0 (4.96)
0
ρ=0
Thus,
1

2
2
2
2
2

2λ 2 ρ J dρ = λ [J (λ)] + (λ − n )[J n (λ)] 2 (4.97)
n
n
ρ=0