An eigenvalue relation in spheroidal wave functions related to potential theory and its applications to contact problems

Eigenvalue and related integral relations for an integral operator with a symmetrical kernel, defined in a circular region, in the form of the ratio of an exponential function of the distance between two points to its distance, are established by methods of generalized potential theory, related to the Helmholtz equation, in an orthogonal system of coordinates of an oblate spheroid, where one of the coordinate surfaces is degenerate in a plane doubly covered circular disc. These relations extend corresponding relations for the integral operator with a symmetric kernel in the form of the Weber–Sonin integral and contain spheroidal wave functions. Using the integral relations obtained, an exact solution of the integral equation of the contact problem of the indentation of a punch, circular in plan, into a linearly deformed base of an elastic half-space with a kernel identical with a kernel, which varies exponentially with distance, is constructed. Other applications of the eigenvalue and related integral relations obtained are also indicated.