Computational geometric and boundary value properties of Oblate Spheroidal Quaternionic Wave Functions

This paper introduces the Oblate Spheroidal Quaternionic Wave Functions (OSQWFs), which extend the oblate spheroidal wave functions introduced in the late 1950s by C. Flammer. We show that the theory of the OSQWFs is determined by the Moisil-Teodorescu type operator with quaternionic variable coefficients. We show the connections between the solutions of the radial and angular equations and of the Chebyshev equation, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions on the other. We proceed the paper establishing an analogue of the Cauchy's integral formula as well as analogues of the boundary value properties such as Sokhotski-Plemelj formulas, the Dk-hyperholomorphic extension of a given Hölder function and on the square of the singular integral operator for this version of quaternionic function theory. To progress in this direction, we show how the D0-hyperholomorphic OSQWFs (with a bandwidth parameter c=0) of any order look like, without belabor them. With the help of these functions, we construct a complete orthogonal system for the L2-space consisting of D0-hyperholomorphic OSQWFs. A big breakthrough is that the orthogonality of the basis elements does not depend on the shape of the oblate spheroids, but only on the location of the foci of the ellipse generating the spheroid. As an application, we prove an explicit formula of the quaternionic D0-hyperholomorphic Bergman kernel function over oblate spheroids in R3. In addition, we provide the reader with some plot examples that demonstrate the effectiveness of our approach.