# 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* = - 1art/square-root-of-negative-1.gif the square root of negative 1 (see number). A function *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;e8;none;0;clt;e8;;;block;;;;no;1;139392n;16544n;;;;;eq8;comptd;;center;stack;;;;;CE5where *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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