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15.3 Bessel Functions 535

In Section 3.7, we used the Laplace transform to solve Bessel’s equation of order n,for n a
nonnegative integer. This led to solutions
∞ k
(−1) 2k+n
J n (x) = x .
2 2k+n k!(n + k)!
k=0
J n (x) is the Bessel function of the first kind of order n. However, we also need solutions for n
not necessarily having integer values, and we need a second, linearly independent, solution for
Bessel’s equation. To these ends we will employ the Frobenius method of solution discussed in
Section 4.2.
Bessel’s equation of order ν can be written

1 ν 2

y + y + 1 − y = 0,
x x 2
from which we see that 0 is a regular singular point. We therefore attempt a Frobenius solution
∞

y(x) = c n x n+r .
n=0
Substitute the proposed Frobenius solution into the differential equation and attempt to solve for
r and the coefficients c n . Begin with the substitution of y into Bessel’s equation (15.7):
∞ ∞

x 2 (n +r)(n +r − 1)c n x n+r−2 + x (n +r)c n x n+r−1
n=0 n=0
∞
n+r
2
2
+ (x − ν ) c n x = 0.
n=0
Write this equation as
∞ ∞
n+r n+r
(n +r)(n +r − 1)c n x + (n +r)c n x
n=0 n=0
∞ ∞

2
+ c n x n+r+2 − ν c n x n+r = 0.
n=0 n=0
Shift indices to write the third summation as
∞ ∞

c n x n+r+2 = c n−2 x n+r
n=0 n=2
to write the last equation as
r
2
2
[r(r − 1) +r − ν ]c 0 x +[r(r + 1) +r + 1 − ν ]c 1 x r+1
∞

2
+ [(n +r)(n +r − 1) + (n +r) − ν ]c n + c n−2 x n+r = 0.
n=2
Set the coefficient of each power of x equal to 0. Since we require in this method that c 0 =0,
r
we obtain from the coefficient of x the indicial equation
2
2
r − ν = 0.
Then r =±ν. Set r = ν in the coefficient of x r+1 to obtain
(2ν + 1)c 1 = 0,
hence c 1 = 0.

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