Abstract

Let , where λ is a positive real constant. In this paper, by using the methods from quaternion calculus, we investigate the -regular vector functions, that is, the complex vector solutions of the equation , and work out a systematic theory analogous to quaternionic regular functions. Differing from that, the component functions of quaternionic regular functions are harmonic, the component functions of -regular functions satisfy the modified Helmholtz equation, that is , . We give out a distribution solution of the inhomogeneous equation and study some properties of the solution. Moreover, we discuss some boundary value problems for -regular functions and solutions of equation .

MSC: 30G35, 35J05.

Keywords:

quaternion calculus; -regular vector function; modified Helmholtz equation; Riemann-Hilbert type boundary value problem