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536 CHAPTER 15 Special Functions and Eigenfunction Expansions

From the coefficient of x n+r ,wehave
2
[(n +r)(n +r − 1) + (n +r) − ν ]c n + c n−2
for n = 2,3,···. Since r = ν, this reduces to
1
c n =− c n−2
n(n + 2ν)
for n = 2,3,···.But c 1 = 0, so
c 3 = c 5 = c = 0.
odd
All of the odd-indexed coefficients are zero.
For the even-indexed coefficients, write
1 1
c 2n =− c 2n−2 =− c 2n−2
2n(2n + 2ν) 2 n(n + ν)
2
1 −1
=− c 2n−4
2
2 n(n + ν) 2(n − 1)[2(n − 1) + 2ν]
1
= c 2n−4
2 n(n − 1)(n + ν)(n + ν − 1)
4
(−1) n
= ··· = c 0
2 n(n − 1)···(2)(1)(n + ν)(n − 1 + ν)···(1 + ν)
2n
(−1) n
= c 0 .
2n
2 n!(1 + ν)(2 + ν)···(n + ν)
One Frobenius solution of Bessel’s equation of order ν is therefore
∞ n
(−1) 2n+ν
y(x) = c 0 x .
2n
2 n!(1 + ν)(2 + ν)···(n + ν)
n=0
For any c 0 = 0, this function is a solution of Bessel’s equation of order ν ≥ 0. These solutions
occur in many applications, including the analysis of radiation from cylindrical containers and
vibrations of circular membranes.
Consider the factor
(1 + ν)(2 + ν)···(n + ν)
in the denominator of y(x). Using the factorial property of the gamma function, write

(n + ν + 1) = (n + ν)
(n + ν) = (n + ν)(n + ν − 1)
(n + ν − 1)

= ··· = (n + ν)(n + ν − 1)···(n + ν − (n − 1))
(n + ν − (n − 1))
= (1 + ν)(2 + ν)···(n − 1 + ν)(n + ν)
(ν + 1).
Then

(n + ν + 1)
(1 + ν)(2 + ν)···(n + ν) = .

(ν + 1)
This enables us to write the solution of Bessel’s equation as
∞ n
(−1)
(ν + 1) 2n+ν
y(x) = c 0 x .
2 n!
(n + ν + 1)
2n
n=0
It is customary to choose
1
c 0 =
ν
2
(ν + 1)

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October 14, 2010 15:20 THM/NEIL Page-536 27410_15_ch15_p505-562