complex variable analysis

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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 functionw=f(z) of a complex variable z is separable into two parts, w = g1(x,y) + ig2(x,y), where g1 and g2 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 z0 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.