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538 CHAPTER 15 Special Functions and Eigenfunction Expansions

15.3.3 Bessel Functions of the Second Kind
We are in Case 3 of the preceding subsection. Begin with the case that ν = 0. The Frobenius
theorem (4.2) tells us in this case to look for a second solution
∞
n
∗
y 2 (x) = J ν (x)ln(x) + c x .
n
n=1
Substitute this into Bessel’s equation of order 0 and obtain, after a computation like that used to
derive J ν (x),
∞ n+1
(−1)
y 2 (x) = J 0 (x)ln(x) + φ(n)x 2n
2 (n!) 2
2n
n=1
where
1 1
φ(n) = 1 + + ··· + .
2 n
Instead of using this solution, it is customary to use a linear combination of y 2 (x) and J 0 (x),
which will also be a solution. This leads to the second solution (in this case ν = 0)
2
Y 0 (x) = [y 2 (x) + (γ − ln(2))J 0 (x)],
π
for x > 0. Here γ is the Euler constant, defined to be

γ = lim(φ(n) − ln(n)) ≈ 0.577215664901533··· .
n→∞
Because of the logarithm term, Y 0 and J 0 are linearly independent solutions of Bessel’s equation
of order 0, which therefore has the general solution
y(x) = aJ 0 (x) + bY 0 (x)
for x > 0. Y 0 is the Bessel function of the second kind of order zero. With the choice of constants
used to define Y 0 , this function is also called Neumann’s function of order zero.
In many applications, we can immediately choose b=0 in solving Bessel’s equation of order
zero, because the logarithm term in Y 0 (x) tends to −∞ as x → 0. This means that a bounded
solution requires b = 0. As we will see later, this reasoning applies when we analyze the motion
of a vibrating circular membrane, since in polar coordinates the center of the membrane is r = 0
and the amplitudes of the vibration must be bounded.
If ν is a positive integer, say ν = n, the second solution of Bessel’s equation of order ν is the
Bessel function of the second kind of order n, defined by

∞ k+1
2 (−1) [φ(k) + φ(k + 1)] 2k+n
Y n (x) = J n (x)[γ + ln(x/2)]+ x
π 2 2k+n+1 k!(k + n)!
k=1
n−1
2 n − k − 1!
− x 2k−n .
π 2 2k−n+1 k!
k=0
This agrees with Y 0 (x) if n =0 with the understanding that in this case the last (finite) summation
is omitted.
The general solution of Bessel’s equation of positive integer order ν = n is therefore
y(x) = aJ n (x) + bY n (x).

It is also possible to define Bessel functions of the second kind of noninteger order by setting
1
Y ν (x) = [J ν (x)cos(νπ) − J −ν (x)].
sin(νπ)

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October 14, 2010 15:20 THM/NEIL Page-538 27410_15_ch15_p505-562