On the Theory of Complex Hypersigular Equations

Abstract

Complex hypersingular integral equations (CHSIE) is an effective means to solve plane problems. They can be derived either by application of holomorphity theorems or by differentiation their singular counterparts [1]. Their advantage over real HSIE is simplicity in evaluation hypersingular integrals; their virtue as compared with complex singular equations is zero index in problems for cracks. But due to the novelty of such equations, their theory is not developed to answer questions concerning solution existence and uniqueness. Meanwhile, the well-established theory of singular equations [2] can serve to obtain the necessary results for the theory of hypersingular equations, employing close connection between singular and hypersingular integrals. The results presented below are received following this line. They include theorems of Fredholm ’s type for hypersingular equations which arise in harmonic and biharmonic problems, in particular in plane elasticity.