# 模解析函数及其性质Module Analytic Functions and Its Properties

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In this paper, the finite number is called the module derivative of complex function f(z) . And if f(z) exists module derivative at any z0 point of some field D, then f(z) is module analytic function over field D . Let f(z)=u(x,y)+iv(x,y) be a complex function, then we give a necessary condition, such that f(z) is a module analytic function as follows: which can be called module Cauchy-Riemann equation or shortly by M-C.R. equation. Furthermore, for module analytic function f(z)=u(x,y)+iv(x,y) of field D  , we get the necessary and sufficient conditions: (1) u(x,y)v(x,y) satisfies the M-C.R. equation within the field D. (2) u(x,y)v(x,y) satisfies the equation uxuy=-vxvy within the field D. Finally, the correlations between module analytic function and several preexisting functions are discussed, including analysis function, semi-analytic function, and conjugate analytic function.

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