Quaternionic Analysis: Application to Boundary Value Problems

Generalizing the complex one-dimensional function theory the class of quaternion-valued functions, defined in domains of ℝ 4 $$\mathbb{R}^{4}$$ , will be considered. The null solutions of a generalized Cauchy–Riemann operator are defined as the ℍ $$\mathbb{H}$$ -holomorphic functions. They show a lot of analogies to the properties of classical holomorphic functions. An operator calculus is studied that leads to integral theorems and integral representations such as the Cauchy integral representation, the Borel–Pompeiu representation, and formulas of Plemelj–Sokhotski type. Also a Bergman–Hodge decomposition in the space of square integrable functions can be obtained. Finally, it is demonstrated how these tools can be applied to the solution of non-linear boundary value problems.