complex variable analysis definition of complex variable analysis in the Free Online Encyclopedia.

complex variable analysis, branch of mathematics mathematics, deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or
..... Click the link for more information. that deals with the calculus calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit —the notion of tending toward, or approaching, an ultimate value.
..... Click the link for more information. 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 number, entity describing the magnitude or position of a mathematical object or extensions of these concepts.

The Natural Numbers

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of
..... Click the link for more information. ). A function function, in mathematics, a relation f that assigns to each member x of some set X a corresponding member y of some set Y; y is said to be a function of x, usually denoted f(x) (read "f of x
..... Click the link for more information. w=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 series, in mathematics, indicated sum of a sequence of terms. A series may be finite or infinite. A finite series contains a definite number of terms whose sum can be found by various methods. An infinite series is a sum of infinitely many terms, e.g.
..... Click the link for more information. 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.