模解析函数及其性质

Module Analytic Functions and Its Properties

摘要: 若复变函数f(z) 在z₀处满足如下极限存在(有限) 称函数f(z) 于点z₀模可导；若f(z) 在z₀的某个邻域内的任一点模可导，则称f(z) 在z₀模解析。如果函数f(z) 在区域D内任一点模解析，则称(z)为区域D内模解析函数。我们给出了一个复变函数w=f(z)=u(x,y)+iv(x,y) 模解析的如上定义，并导出了模解析函数的必要条件： 。我们称此为模柯西–黎曼方程(简称模C.-R.方程)。进一步，我们给出了模解析的充分必要条件：(1) u(x,y) ，v(x,y) 在区域D 内满足模C.-R.方程；(2)u(x,y) ，v(x,y) 在区域 D内满足u_xu_y=-v_xv_y 。最后，我们讨论了模解析函数与已有的各类复变函数，如解析函数，半解析函数，共轭解析函数之间的关系。

Abstract: 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 u_xu_y=-v_xv_y 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.