Harmonic and Monogenic Functions on Toroidal Domains

A standard technique for producing monogenic functions is to apply the adjoint quaternionic Fueter operator to harmonic functions. We will show that this technique does not give a complete system in \(L^2\) of a solid torus, where toroidal harmonics appear in a natural way. One reason is that this index-increasing operator fails to produce monogenic functions with zero index. Another reason is that the non-trivial topology of the torus requires taking into account a cohomology coefficient associated with monogenic functions, apparently not previously identified because it vanishes for simply connected domains. In this paper, we build a reverse-Appell basis of harmonic functions on the torus expressed in terms of classical toroidal harmonics. This means that the partial derivative of any element of the basis with respect to the axial variable is a constant multiple of another basis element with subindex increased by one. This special basis is used to construct respective bases in the real \(L^2\)-Hilbert spaces of reduced quaternion and quaternion-valued monogenic functions on toroidal domains.