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15.3 Bessel Functions 561

17. Let α be a positive zero of J 0 . Show that Hint: Use the first integral from Problem 20 in the
definition of I n,k .
1

1
J 1 (αx)dx = . (c) Show that
0 α
2k
18. Let u(x) = J 0 (αx) and v(x) = J 0 (βx), with α and β I n,k = I n+1,k−1 .
α
positive constants.
This provides a recurrence relation for the quantities
2
(a) Show that xu + u + α xu = 0, with a similar I n,k . Hint: Integrate by parts in (b).

equation for v. (d) Show that
(b) Multiply the differential equation for u by v and k
the equation for v by u and subtract to show that I n,k = 2 k! I n+k,0 .
α k
2
2

[x(u v − v u)] = (β − α )xuv.
Hint: Apply part (c) in repetition.
(c) Use the conclusion of part (b) to show that (e) Show that
k
1
2
(k + 1)
2 k
2
2
(β − α ) xJ 0 (αx)J 0 (βx)dx (1 − x ) x n+1 J n (αx)dx = k+1 J n+k+1 (α).
0 α

= x αJ (αx)J 0 (βx) − β J (βx)J 0 (αx) . Hint:Use theresultofpart(d).

0 0
(f) Show that
This is one of a class of integrals known as
Lommel’s integrals. x k+1 1 n+1 2 k
J n+k+1 (x) = t (1 − t ) J n (xt)dt.
k
2
(k + 1)
19. Show that, for any positive integer n, 0
(g) Show that, if n is a nonnegative integer and m is a

n
n
x J n−1 (x)dx = x J n (x) positive integer with n < m,then
2x m−n 1
and J m (x)= t n+1 (1−t ) J n (xt)dt.
2 m−n−1
2 m−n
(m − n)
J n+1 (x) J n (x)
0
dx =− . Hint:Let m = n + k + 1 in the result of part (f).
x n x n
The integral expressions in parts (e), (f), and (g) are
(Here we have omitted the constants of integration). called Sonine’s integrals.
Hint: Use Theorem 15.10. Alternatively, one can inte- 22. Use the fact that
grate the series for J n+1 (x)/x term by term.

20. Show that, for any positive integer n and any nonzero 2
J −1/2 (xt) = cos(xt)
number α, πxt
1

n n to show that, if n is a positive integer, then
x J n−1 (αx)dx = x J n (αx)
α
x n
and J n (x) = √
2 n−1 π
(n + 1/2)

J n+1 (αx) J n (αx)
dx =− . 1
2 n−1/2
x n αx n (1 − t ) cos(xt)dt.
0
Hint: The result of Problem 19 can be used, or the
series for J n (x) can be used. This is called Hankel’s integral. Hint: Use Sonine’s
21. For α any nonzero number, and n and k nonnegative integral, Problem 21 (g).
23. Show that, if m is a positive integer, then
integers, define
x m
1 J m (x) = √

2 k
I n,k = (1 − x ) x n+1 J n (αx)dx. 2 m−1 π
(m + 1/2)
0
π/2
2m
(a) Show that cos (θ)cos(x sin(θ))dθ.
0
1
I n,0 = J n+1 (α). This expression is called Poisson’s integral. Hint:Put
α
t = sin(θ) in Hankel’s integral, Problem 22.
Hint: Use the first integral in Problem 20.
(b) Show that
In each of Problems 24 through 29, find (approximately)
1 d x n+1 the first five terms in the Fourier-Bessel expansion of f (x)

2 k
I n,k = (1 − x ) J n+1 (αx) dx. on (0,1) in a series of the functions J 1 ( j n x),where j n is
0 dx α
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