## complex variable analysis

## 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*=-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
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.