Abstract

In this paper we develop a $g$-analytic function and a $g$-harmonic function theory for one-dimensional wave equation in the Minkowski space. In terms of the Minkowskian polar coordinates we can derive a set of complete hyperbolic type Trefftz bases, which can be transformed to polynomials as the bases for a trial solution of wave equation. The Cauchy-Riemann equations and the Cauchy theoremfor $g$-analytic functions are proved, and meanwhilethe existence of Cauchy integral formula is disproved from thenon-uniqueness of the Dirichlet problem for wave equation under the boundary conditions on whole boundary, which isalso known as the backward wave problem (BWP).Examples are used to demonstrate these results.

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