A special relationship in spheroidal wave functions and its application to contact problems (Q1070905)

A spectral and kindred relationship are set up by methods of the theory of the generalized potential for an integral operator generated by a symmetric difference kernel in the form of a Macdonald function in two identical semi-infinite intervals \(\{\) (-\(\infty,-a),\quad (a,\infty)\}\) that contain spheroidal wave functions. The formula for the expansion of an arbitrary function in these functions is also set up by a well-known method. On the basis of the results obtained, a solution is then constructed for the integral equation of the contact problem of the impression of two identical stamps with half-plane bases into a half- space being deformed in a power-law form. This contact problem can be described by the same integral equation when the elastic modulus of a linearly elastic half-space changes with depth accordng to a power law.