# complex variable analysis

**complex variable analysis,**branch of mathematics that deals with the calculus of functions of a complex variable, i.e., a variable of the form

*z*=

*x*+

*iy,*where

*x*and

*y*are real and

*i*=

*w*=

*f(z)*of a complex variable

*z*is separable into two parts,

*w*=

*g*

_{1}(

*x,y*) +

*ig*

_{2}(

*x,y*), where

*g*

_{1}and

*g*

_{2}are real-valued functions of the real variables

*x*and

*y.*The theory of functions of a complex variable is concerned mainly with functions that have a derivative at every point of a given domain of values for

*z;*such functions are called analytic, regular, or holomorphic. If a function is analytic in a given domain, then it also has continuous derivatives of higher order and can be expanded in an infinite series in terms of these derivatives (i.e., a Taylor's series). The function can also be expressed in the infinite series where

*z*

_{0}is a point in the domain. Also of interest in complex variable analysis are the points in a domain, called singular points, where a function fails to have a derivative. The theory of functions of a complex variable was developed during the 19th cent. by A. L. Cauchy, C. F. Gauss, B. Riemann, K. T. Weierstrass, and others.

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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