# complex variable analysis

## 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 logical

**.....**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 objects if their

**.....**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*=

*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

**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

*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.

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