Relations among spheroidal and spherical harmonics

摘要

A contragenic function in a domain Ω⊆R3 is a reduced-quaternion valued (i.e the last quaternionic coordinate is zero) harmonic function, which is orthogonal in L2(Ω) to all reduced-quaternion monogenic functions and their conjugates. Contragenicity is not a local property. For spheroidal domains of arbitrary eccentricity, we relate standard orthogonal bases of harmonic and contragenic functions for one spheroid to another via computational formulas. This permits us to show that there exist nontrivial contragenic functions common to the spheroids of all eccentricities.