Bergman kernels for monogenic and contragenic functions in the interior and exterior of a sphere

TL;DR

This paper introduces Bergman kernels for contragenic functions in quaternionic analysis, revealing a duality with monogenic functions and providing numerical validation of kernel truncations.

Contribution

It defines contragenic functions in quaternionic analysis, derives their Bergman kernels for interior and exterior domains, and uncovers a duality with monogenic functions.

Findings

Bergman kernels for contragenic functions are explicitly constructed.

Numerical examples confirm the accuracy of kernel truncations.

A duality between interior contragenic and exterior monogenic functions is observed.

Abstract

Contragenic functions are defined to be reduced-quaternion-valued harmonic functions which are orthogonal to all monogenic and antimonogenic functions in the $L^2$ norm of a given domain. The parallelism between the spaces of contragenic functions in the interior and exterior of the unit sphere in $\R^3$ is described in detail. Bergman reproducing kernels for the spaces of contragenic functions are given, mirroring the corresponding kernels for the spaces of vector parts of monogenic functions. Numerical examples are given showing the accuracy of truncations of the integral kernels. A striking duality is observed between the basic interior contragenic functions and the vector parts of exterior monogenic functions, and vice versa.