Joao Morais

Harmonic functions in three-dimensional Euclidean space conceal a subtle and unexpected structure. Unlike their counterparts in the complex plane or in quaternionic space, they cannot, in general, be decomposed even locally into the sum of a monogenic and an antimonogenic function. This failure gives rise to “contragenics”—harmonic functions lying in the $$L^2$$ L 2 -orthogonal complement of the space of all such sums. This survey traces the development of the theory of contragenics in stages. It begins with the spheroidal harmonics, both internal and external, on spheroidal domains of arbitrary eccentricity. Building on these, we construct graded one-parameter orthogonal bases of spheroidal monogenics and ambigenics, emphasizing their algebraic structure and limiting behavior that reflect the geometry of the underlying spheroid. Contragenics then emerge, first within the interior of spheroidal domains, and later in both the interior and exterior of the unit sphere, where they furnish a richer $$L^2$$ L 2 -orthogonal decomposition of harmonic functions in quaternionic analysis on three-dimensional, simply connected, unbounded Euclidean domains. Finally, we extend the framework to the exterior of spheroidal domains, where logarithmic terms intervene and further enrich the theory.

Contragenic functions are defined to be reduced-quaternion-valued harmonic functions which are orthogonal to all monogenic and antimonogenic functions in the $$L^2$$ 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 $$\mathbb {R}^3$$ 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.

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.