Carleman Type Integral Equations

Many problems in physics and engineering which can be reduced to the integral equation 4.1$$ \alpha (\xi )g(\xi ) - \lambda \beta (\xi ){\text{p}}{\text{.v}}{\text{.}}\int\limits_C {\frac{{\gamma (\omega )g(\omega )}}{{\omega - \xi }}} d\omega = f(\xi ) $$, where α(ξ),β(ξ), γ(ξ) and f (ξ) are prescribed functions of a real or complex variable ξ. The range of integration C can be an interval of the real line, a closed or an open contour in the complex plane C. An explicit solution of this equation was first given by Carleman [10] for a real interval and, therefore, the equation bears his name. It has been recognized for several decades that this equation plays a pivotal role in the theory of singular integral equations. When β(ξ) and β(ξ) are constants, equation (4.1) reduces to the Cauchy type integral equation. Subsequent to the analysis of Carleman, many more results have been found and have occurred extensively in the literature [56,62, 70,86,103].