Three-dimensional Analogue of Kolosov–Muskhelishvili Formulae

Abstract

In the plane elasticity an effective method of using the holomorphic complex function theory is based on Kolosov–Muskhelishvili formulae. For a three-dimensional case monogenic Clifford functions or regular quaternion functions of a reduced quaternion variable are used. Such functions are solutions of the Moisil–Theodoresco system. In recent papers some variants of three-dimensional Kolosov–Muskhelishvili formulae are obtained but only for star-shaped regions. For applications it is very important to have these formulae for a wider class of domains. We propose the generalized Kolosov– Muskhelishvili formulae in arbitrary simply connected domains with a smooth boundary not only star-shaped, where a notion of harmonic primitive function is used. The method of proving is based on a new theorem about reconstruction of a regular function from a given scalar part.