广义复数上的函数

Functions of an Generalized Complex Variable

本文用复变函数方法统一讨论既含椭圆型也含双曲型的一类线性偏微分方程。可以看到它们的许多个性寓于其共性之中。

In this paper we brought forward the generalized Cauchy-Riemann equa-tions including the elliptic equations and the hyperbolic equations as the special cases (§1) We introduced the generalized complex numbers including the complex numbers andhyperbolic numbers as the special cases (§ 2). We defined the mapping with operations pre-serving including the analytic mapping and the hyperbolic mapping as the special cases(§3). Using a Dedekind's theorem with respect to commutative algebra as an effectual workwe obtained the necessary and sufficient conditions for the function of an generalized com-plex variable to be a mapping with operations preserving, and showed the characteristicproperties of the hyperbolic mapping and the solution of hyperbolic equations (§3). The theory of power series of generalized complex numbers (§) is an advantageouslyanalytic tool in research.