\(\psi\)-hyperholomorphic functions and a Kolosov-Muskhelishvili formula (Q2795274)

The authors study fundamental problems in linear elasticity in the space, using special hyperholomorphic functions. Hyperholomorphic functions are the basic tool of a hypercomplex function theory, the space analogue of the complex function theory. Elements of classical elasticity and hypercomplex function theory are merged in order to get an alternative Kolosov-Muschelishvili formula in the space. The paper starts with the mathematical and historical deduction of the famous Neuber-Papkovic representation. Then a suitable hypercomplex function theory is introduced using so-called \(\psi\)-hyperholomorphic functions. In reflection of former works by K. Gürlebeck and S. Bock, a generalized Kolosov-Muschelishvili formula in terms of quaternion-valued monogenic functions is presented. Differences between the use of reduced quaternions and the algebra of quaternions and the influence on additive decompositions are discussed. For the displacement field the authors get an additive representation with suitable monogenic, anti-monogenic and \(\psi\)-hyperholomorphic functions. Corresponding approximations with suitable basis polynomials lead to an effective complete polynomial solution representation of the Lamé-Navier equation.