ψ-hyperholomorphic Functions and an Application to Elasticity Problems

In complex analysis, every harmonic function admits a decomposition as a sum of a holomorphic and an antiholomorphic function. However, this fact does not hold for paravector-valued harmonic functions, or so-called A-valued harmonic functions, in quaternion function theory. In previous articles, the authors proved that by taking into account psi-hyperholomorphic functions, where psi is a structural set different from the standard one and its conjugation, every A-valued harmonic function can be written as a sum of a monogenic, an anti-monogenic and a psi-hyperholomorphic function. In this paper, we revisit such a decomposition and apply it to study problems in elasticity.