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b

7. ln sin(λx) – sin(λt) y(t) dt = f(x).

a
This is a special case of equation 1.8.9 with g(x) = sin(λx).
a sin 1 A
8. ln 1 2 y(t) dt = f(x), –a ≤ x ≤ a.
–a 2 sin |x – t|
2
Solution with 0 < a < A:

1 d a
y(x)= w(t, a)f(t) dt w(x, a)
2M (a) da –a

a ξ

1 d 1 d
– w(x, ξ) w(t, ξ)f(t) dt dξ
2 |x| dξ M (ξ) dξ –ξ

1 d a w(x, ξ) ξ
– w(t, ξ) df(t) dξ,
2 dx |x| M (ξ) –ξ

where the prime stands for the derivative with respect to the argument and
–1
1
1
sin 2 A cos 2 ξ M(ξ)
M(ξ)= ln 1 , w(x, ξ)= √ .
sin ξ π 2 cos x – 2 cos ξ
2
•
Reference: I. C. Gohberg and M. G. Krein (1967).
3.7. Equations Whose Kernels Contain Special
Functions

3.7-1. Kernels Containing Bessel Functions

∞

1
1. tJ ν (xt)y(t) dt = f(x), ν > – .
2
0
Here J ν is the Bessel function of the first kind.
Solution:

∞
y(x)= tJ ν (xt)f(t) dt.
0
The function f(x) and the solution y(t) are the Hankel transform pair.
•
Reference: V. A. Ditkin and A. P. Prudnikov (1965).
b

2. J ν (λx) – J ν (λt) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x)= J ν (λx), where J ν is the Bessel function
of the first kind.
b

3. Y ν (λx) – Y ν (λt) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x)= Y ν (λx), where Y ν is the Bessel function
of the second kind.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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