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15.3 Bessel Functions 553

0.3

0.2

0.1

0
0 5 10 15 20 25
x
–0.1

–0.2

FIGURE 15.10 Interlacing lemma for J 7 (x), J 8 (x),
and J 9 (x).

can be written as

2
ν

xy + y + λx − y = 0
x
which is the Sturm-Liouville equation

2
ν

(xy ) + λx − y = 0
x
2
on (0,1) with r(x) = p(x) = x and q(x) =−ν /x. From equation (15.8) with a = 0, c = 1 and
√

2
λ = b , solutions that are bounded on (0,1) are multiples of J ν λx . Further, for solutions
√
satisfying the boundary condition y(1) = 0, we need λ to be a positive zero of J ν (x). Denote
these positive zeros as j k , ordered so that
0 < j 1 < j 2 < j 3 < ··· .
2
Then the numbers λ n = j are eigenvalues of this problem, with eigenfunctions J ν ( j n x). Notice
n
that ν is fixed here (occurring in the differential equation), so these eigenfunctions are all written
in terms of the same Bessel function J ν .Itis j n that changes to form the eigenfunctions J ν ( j n x).
These eigenfunctions are orthogonal on (0,1) with weight function p(x) = x. This means
that
1
xJ ν ( j n x)J ν ( j m x)dx = 0
0
if n = m.
If f is piecewise smooth on (0,1), then we can write the eigenfunction expansion
∞

c n J ν ( j n x) (15.27)
n=1
in which
1
0 xf (x)J ν ( j n x)dx
c n = . (15.28)
1
2
x(J ν (x)) dx
0
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October 14, 2010 15:20 THM/NEIL Page-553 27410_15_ch15_p505-562