Page 283 - Handbook Of Integral Equations

∞ y(t)
52. y(x) – λ dt = f(x), 1 ≤ x < ∞, –∞ < πλ <1.
x + t
1
Solution:

∞ τ sinh(πτ) F(τ)
y(x)= P – +iτ (x) dτ,
1
0 cosh(πτ) – πλ 2

∞
F(τ)= f(x)P (x) dx,
1
– +iτ
1 2
1
where P ν (x)= F –ν, ν +1, 1; (1 – x) is the Legendre spherical function of the first kind,
2
for which the integral representation
α
2 cos(τs) ds
P – +iτ (cosh α)= √ (α ≥ 0)
1
2 π 2(cosh α – cosh s)
0
can be used.
•
Reference: V. A. Ditkin and A. P. Prudnikov (1965).
3
λ ∞ a y(t)
2
2
53. (x + b )y(x)= dt.
2
π –∞ a +(x – t) 2
This equation is encountered in atomic and nuclear physics.
We seek the solution in the form
∞
A m x
y(x)= . (1)
2
x +(am + b) 2
m=0
The coefficients A m obey the equations

∞
m +2b
mA m + λA m–1 =0, A m = 0. (2)
a
m=0
Using the first equation of (2) to express all A m via A 0 (A 0 can be chosen arbitrarily),
substituting the result into the second equation of (2), and dividing by A 0 , we obtain

∞ m
(–λ) 1
1+ = 0. (3)
m! (1 + 2b/a)(2 + 2b/a) ... (m +2b/a)
m=1
It follows from the definitions of the Bessel functions of the first kind that equation (3)
can be rewritten in the form
√
λ –b/a J 2b/a 2 λ = 0. (4)
In this sort of problem, a and λ are usually assumed to be given and b, which is proportional
to the system energy, to be unknown. The quantity b can be determined by tables of zeros of
Bessel functions. In some cases, b and a are given and λ is unknown.

•
Reference: I. Sneddon (1951).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 262