Integral Equations of Curvilinear Convolution Type with Hypergeometric Function in a Kernel

Abstract

We study the class of integral equations of first kind over the circumference in the complex plane. The kernels of the equations contain a Gaussian hypergeometric function and depend on the arguments ratio. This class includes such specific cases as the equations with power and logarithmic kernels. To set the Noetherian property of equations correctly the method of operator normalization with a non-closed image is applied. The space of the right-hand sides of equations is described as the space of fractional integrals of curvilinear convolution type. The solutions of equations in explicit form are obtained as a result of consequent solving characteristic singular equations with a Cauchy kernel and inversion of a curvilinear convolution operator by means of Laurent transform of the functions defined on the circumference.