Derivation of the solution of certain singular integral equations

Abstract

It is shown that the aplication of the Poincare-Bertrand formula when made in suitable manner produces the solution of certain singular integral equations very quickly, the method of arriving at which, otherwise, is too complicated. Two singular integral equations are considered. One of these equations is with a Cauchy-type kernel and the other is an equation which appears in the waveguide theory and the theory of disloctions.

A different appruach is also made here to solve the singular integral equations of the wave-guide theory and this involves the use of the inversion formula of the Cauchy-type singular integral equation and reduction to a system If Hilbert problems for two unknowns which can be decouplcd very easily to obtain tbe closed form solutien of the iTitegral equation at hand.

The methods of the present paper avoid all the complicated approaches of solving the singular integral equation of the wave.-guide theory known to-date.