Towards a quaternionic function theory linked with the Lamé's wave functions (Q2795437)

The authors introduce the so-called ``Lamé's quaternionic wave function'' as a non-commutative extension of the classical Lamé's wave function, which appears when using elliptical coordinates and the method of separation of variables.NEWLINENEWLINEThose wave functions were introduced by Lamé in 1837. They are 1-1 correspondences between Cartesian and ellipsoidal coordinates. Oblate and prolate spheroidal wave functions have to be distinguished. At first, relations between the representation of the Lamé operator in different coordinates are described in detail. It can be shown that the operator that arises in the Helmholtz equation after the ellipsoidal change of variables is a combination of three Lamé operators. The kernel of the Lamé operators in a parallelepiped can be explicitly given by elementary tensors. Then the authors study the relations to an \(\alpha\)-hyperholomorphic function theory. The quaternionic wave functions are explicitely deduced. A suitable function theory with a quaternionic Borel-Pompeiu formula and a new analogue to Cauchy's integral theorem are obtained. Boundary value properties of the Lamé quaternionic wave functions are discussed (formulae of Plemelj-Sokhotzki type). The special case of spheroids is computed on the last pages of the paper.