complex variable analysis

complex variable analysis,

branch of mathematicsmathematics,
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
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 that deals with the calculuscalculus,
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.
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 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 numbernumber,
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
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). A functionfunction,
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 ").
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 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 seriesseries,
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.
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 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.
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