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15.3 Bessel Functions 537

to obtain the solution usually denoted J ν (x):

∞ n
(−1)

J ν (x) = x 2n+ν .
2 2n+ν n!
(n + ν + 1)
n=0
J ν (x) is called a Bessel function of the first kind of order ν. The series defining J ν (x)
converges for all x.

Because Bessel’s equation is second-order (as a differential equation), there is a second
solution that is linearly independent from J ν . The Frobenius theorem (Theorem 4.2) tells us how
to proceed to find a second solution. Recall that the indicial equation of Bessel’s differential
2
2
equation is r − ν = 0, with roots ν and −ν. The form that a second solution will take depends
on the difference 2ν of these roots. With Theorem 4.2 as a guide, we find the following second
solutions by taking cases on 2ν.
Case 1 If 2ν is not an integer, then J ν and J −ν are linearly independent (neither is a constant
multiple of the other). In this case, the general solution of Bessel’s equation is
y(x) = aJ ν (x) + bJ −ν (x),

with a and b arbitrary constants.
Case 2 If 2ν is an odd positive integer, say 2ν = 2n + 1 for some nonnegative integer n, then
1
ν = n + and J ν and J −ν are again linearly independent, as in Case 1. In this case, the general
2
solution of Bessel’s equation is
y(x) = aJ n+1/2 (x) + bJ −n−1/2 (x).
By manipulating the series for J ν (x), it can be shown that in this case, J n+1/2 (x) and J −n−1/2 (x)
can be expressed in closed form as finite sums of terms involving algebraic combinations of x,
sin(x) and cos(x). For example,

2 2
J 1/2 (x) = sin(x), J −1/2 (x) = cos(x),
πx πx

2 sin(x)
J 3/2 (x) = − cos(x) ,
πx x
and

2 cos(x)
J −3/2 (x) = −sin(x) − .
πx x
Case 3 If 2ν is an integer, but not of the form 2n + 1 for any nonnegative integer n, then J ν and
J −ν are solutions of Bessel’s equation, but are linearly dependent:
ν
J −ν (x) = (−1) J ν (x).
In this case, we cannot manufacture a second linearly independent solution from J ν (x).This
leads us to construct such a second linearly independent solution, leading us to Bessel functions
of the second kind.

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