# On $\psi$-hyperholomorphic functions

Presenter: **Klaus Guerlebeck**, Bauhaus-University Weimar, Germany

Joint work with **Sebastian Bock, Nguyen Manh Hung, Daniel-Weisz-Patrault**

The theory of monogenic quaternion valued or Clifford algebra valued functions can be seen as refinement of harmonic analysis and as generalization of the complex function theory of holomorphic functions. We will follow in the talk the line of generalizing the complex function theory. Main topics are the characterization of hyperholomorphic functions by their geometrical mapping properties, the derivability and the construction of Taylor-type series expansions based on certain polynomial Appell systems. In a second step the relation of $\psi$-hyperholomorphic functions to harmonic functions will be studied in the non-trivial case of mappings from $\mathbb{R}^3\mapsto\mathbb{R}^3$ and an additive decomposition of harmonic functions in terms of $\psi$-hyperholomorphic functions will be proved. As applications we will discuss geometric properties of $\psi$-hyperholomorphic functions and the application to boundary value problems from linear elasticity.