Cauchy Principal Value Equations on a Finite Interval

Abstract

The first step in solving an integral equation that involves a Cauchy principal value integral is to express the integral and the unknown density in terms of the associated analytic function. The resulting equation poses a Riemann problem. In the physical contexts where such equations arise, the associated analytic function usually has a physical meaning itself, and often the Riemann problem can be posed without ever writing the integral equation explicitly. The methods that we use to solve the Riemann problem, which we begin to explain in the present chapter, are useful even when the problem has not been posed as an integral equation.