Page 311 - Handbook Of Integral Equations

4.5-6. A Singular Equation

B 2π t – x
58. Ay(x) – cot y(t) dt = f(x), 0 ≤ x ≤ 2π.
2π 0 2
Here the integral is understood in the sense of the Cauchy principal value. Without loss of
2
2
generality we may assume that A + B =1.
Solution:
2π 2 2π
B t – x B
y(x)= Af(x)+ cot f(t) dt + f(t) dt.
2π 2 2πA
0 0
•
Reference: I. K. Lifanov (1996).
4.6. Equations Whose Kernels Contain Inverse
Trigonometric Functions

4.6-1. Kernels Containing Arccosine

b
1. y(x) – λ arccos(βx)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x) = arccos(βx) and h(t)=1.
b
2. y(x) – λ arccos(βt)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = arccos(βt).

b arccos(βx)

3. y(x) – λ y(t) dt = f(x).
a arccos(βt)
1
This is a special case of equation 4.9.1 with g(x) = arccos(βx) and h(t)= .
arccos(βt)
b arccos(βt)
4. y(x) – λ y(t) dt = f(x).
a arccos(βx)
1
This is a special case of equation 4.9.1 with g(x)= and h(t) = arccos(βt).
arccos(βx)
b
k
m
5. y(x) – λ arccos (βx) arccos (µt)y(t) dt = f(x).
a
k
m
This is a special case of equation 4.9.1 with g(x) = arccos (βx) and h(t) = arccos (µt).

b
k
m
6. y(x) – λ t arccos (βx)y(t) dt = f(x).
a
k
m
This is a special case of equation 4.9.1 with g(x) = arccos (βx) and h(t)= t .

b
m
k
7. y(x) – λ x arccos (βt)y(t) dt = f(x).
a
m
k
This is a special case of equation 4.9.1 with g(x)= x and h(t) = arccos (βt).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 290