Page 571 - Advanced_Engineering_Mathematics o'neil

15.3 Bessel Functions 551

√
Assume that k > 1 and substitute u(x) = kx J ν (kx) to obtain

2 1
ν −
2 4
u (x) + k − u(x) = 0. (15.25)
x 2
Our intuition is that, as x increases, the term (ν −1/4)/x exerts less influence, and this differen-
2
2
tial equation for u more closely approximates u + k u = 0, with solutions cos(kx) and sin(kx).
2

√ √
This suggests that, for large x, J ν (kx) is approximated by cos(kx)/ kx or sin(kx)/ kx. Since
these functions have infinitely many positive zeros, we suspect that J ν (kx) does also.
While not a proof, this informal argument suggests an approach to the question of zeros of
J ν (kx). Consider the equation

v (x) + v(x) = 0, (15.26)
which has solution v(x) = sin(x − α) for any positive number α. Multiply equation (15.25) by v
and equation (15.26) by u and subtract to get

2 1
ν −
2 4
uv − vu = k − uv − uv.
x 2
Write this as

2 1
ν −
2 4
(uv − vu ) = k − 1 − uv.
x 2
For any positive number α, compute
α+π

(uv − vu )dx = u(α + π)v (α + π) − u(α)v (α) − v(α + π)u (α + π) + v(α)u (α)

α
=−u(α + π) − u(α)
2
α+π
ν − 1
2 4
= k − 1 − u(x)v(x)dx
x 2
α
2
α+π
ν − 1
2 4
= k − 1 − u(x)sin(x − α)dx
x 2
α
By the mean value theorem for integrals, there is some number τ between α and α + π such that
α+π ν −

2 1
2
−u(α + π) − u(α) = u(τ) k − 1 − 4 sin(x − α)dx.
x 2
α
Now sin(x − α) > 0for α< x <α + π. Further, we can choose α large enough (depending on k
and ν) so that
2 1
ν −
2 4
k − 1 − > 0for α ≤ x ≤ α + π.
x 2
Therefore the integral on the right in the last equation is positive. This means that u(α +π), u(α)
and u(τ) cannot all have the same sign. Since u is continuous, u(x) must equal zero for some x
√
between α and α + π.But u(x) = kx J ν (kx), so wherever u(x) vanishes, J ν (kx) must be zero
also. This proves that J ν (kx) has at least one zero between α and α + π.
We conclude that, if α is sufficiently large, then J ν (x) has a zero between α and α + kπ.
With this as background, we will state a fundamental result on positive zeros of Bessel
functions, their distribution on the half-line x > 0, and a relationship between zeros of J ν−1 (x),
J ν (x) and J ν+1 (x).
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October 14, 2010 15:20 THM/NEIL Page-551 27410_15_ch15_p505-562