Page 650 - Handbook Of Integral Equations

Chapter 13

Methods for Solving Complete

Singular Integral Equations

13.1. Some Definitions and Remarks

13.1-1. Integral Equations With Cauchy Kernel

A complete singular integral equation with Cauchy kernel has the form
1 M(t, τ) 2
a(t)ϕ(t)+ ϕ(τ) dτ = f(t), i = –1, (1)
πi L τ – t

where the integral, which is understood in the sense of the Cauchy principal value, is taken over a
closed or nonclosed contour L and t and τ are the complex coordinates of points of the contour. It is
assumed that the functions a(t), f(t), and M(t, τ) given on L and the unknown function ϕ(t) satisfy
the H¨ older condition (see Subsection 12.2-2), and M(t, τ) satisfies this condition with respect to
both variables.
The integral in Eq. (1) can also be written in a frequently used equivalent form. To this end, we
consider the following transformation of the kernel:

M(t, τ) M(t, τ)– M(t, t) M(t, t)
= + , (2)
τ – t τ – t τ – t
where we set
1 M(t, τ)– M(t, t)
M(t, t)= b(t), = K(t, τ). (3)
πi τ – t
In this case Eq. (1), with regard to (2) and (3), becomes
b(t) ϕ(τ)
a(t)ϕ(t)+ dτ + K(t, τ)ϕ(τ) dτ = f(t). (4)
πi L τ – t L

It follows from formulas (3) that the function b(t) satisfies the H¨ older condition on the entire
contour L and K(t, τ) satisfies the H¨ older condition everywhere except for the points τ = t, at which
one has the estimate
A
|K(t, τ)| < , A = const < ∞, 0 ≤ λ <1.
|τ – t| λ
Naturally, Eq. (4) is also called a complete singular integral equation with Cauchy kernel. The
1
functions a(t) and b(t) are called the coefficients of Eq. (4), is called the Cauchy kernel, and
τ – t
the known function f(t) is called the right-hand side of the equation. The first and the second terms

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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