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552 CHAPTER 15 Special Functions and Eigenfunction Expansions

THEOREM 15.13 Zeros of J ν (x)

Let k > 1 and let ν be a real number.
1. For α sufficiently large, there is a zero of J ν (x) on each of the intervals
(α,α + kπ),(α + kπ,α + 2kπ),(α + 2kπ,α + 3kπ),··· .
2. Every positive zero of J ν (x) is simple.
3. The positive zeros of J ν (x) and J ν+1 (x) are distinct.
4. The positive zeros of J ν (x) and J ν−1 (x) are distinct.
5. If a and b are distinct positive zeros of J ν (x), then J ν−1 (x) and J ν+1 (x) each has a zero
between a and b.

Proof For conclusion (1), we know from the preceding discussion that J ν (x) has a zero between
α and α + kπ for α sufficiently large. But then J ν (x) also has a zero between α + kπ and α +
kπ + kπ = α + 2kπ, and so on.
For conclusion (2), suppose x 0 is a positive zero of J ν (x), but is not simple. Then
J ν (x 0 ) = J (x 0 ) = 0.

ν
But then y = J ν (x) is a solution of the initial value problem
1
ν 2
2
y + y + k − y = 0; y(x 0 ) = y (x 0 ) = 0
x x 2
on some open interval about x 0 .But y(x) = 0 is also a solution of this problem, so J ν (x) would
be identically zero on an open interval, and this is a contradiction.
To verify conclusion (3), suppose x 0 is a positive zero of both J ν (x) and J ν+1 (x). By equation
(15.24), we would also have J (x 0 )=0, so x 0 would be a non-simple zero of J ν (x), contradicting

ν
conclusion (2).
Conclusion (4) is proved by a similar argument, using equation (15.23).
ν
To prove conclusion (5), let f (x) = x J ν (x). Then f (a) = f (b) = 0. By the mean value

theorem, there is some c between a and b such that f (c) = 0. But by equation (15.20),
d
ν ν
f (x) = (x J ν (x)) = x J ν−1 (x)
dx

so f (c) = 0 implies that J ν−1 (c) = 0 also. But then c is a common positive zero of J ν (x) and
J ν−1 (x), contradicting conclusion (4).

Conclusion (5) is called the interlacing lemma. It means that the graphs of J ν−1 (x), J ν (x)
and J ν+1 (x) weave about each other, as can be seen in Figure 15.10 for J 7 (x), J 8 (x) and
J 9 (x).

We are now prepared to look at eigenfunction expansions in terms of Bessel functions.

15.3.11 Fourier-Bessel Expansions
We will write Bessel’s equation in a form to fit within Sturm-Liouville theory and use
equation (15.8) to determine eigenvalues and eigenfunctions. The Bessel equation
2
2
2
x y + xy + (λx − ν )y = 0

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October 14, 2010 15:20 THM/NEIL Page-552 27410_15_ch15_p505-562