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Encyclopedia > Complex analysis  Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). The hue represents the function argument, while the saturation represents the magnitude.

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics. Image File history File links Size of this preview: 600 Ã— 600 pixelsFull resolution (800 Ã— 800 pixel, file size: 59 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links Size of this preview: 600 Ã— 600 pixelsFull resolution (800 Ã— 800 pixel, file size: 59 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... An image with the hues cyclically shifted The hues in the image of this Painted Bunting are cyclically rotated with time. ... It has been suggested that this article or section be merged with Chromaticity. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ...

Complex analysis is particularly concerned with the analytic functions of complex variables, which are commonly divided into two main classes: the holomorphic functions and the meromorphic functions. Because the separable real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics. In mathematics, an analytic function is a function that is locally given by a convergent power series. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...

## Contents

A complex function is a function in which the independent variable and the dependent variable are both complex numbers. More precisely, a complex function is a function whose domain Ω is a subset of the complex plane and whose range is also a subset of the complex plane. In an experimental design, the independent variable (argument of a function, also called a predictor variable) is the variable that is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ... In experimental design, a dependent variable (also known as response variable, responding variable or regressand) is a factor whose values in different treatment conditions are compared. ... In mathematics, the domain of a function is the set of all input values to the function. ... â€œSupersetâ€ redirects here. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, the range of a function is the set of all output values produced by that function. ...

For any complex function, both the independent variable and the dependent variable may be separated into real and imaginary parts: In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... $z = x + iy,$ and $w = f(z) = u(z) + iv(z),$
where $x,y in mathbb{R},$ and $u(z), v(z),$ are real-valued functions.

In other words, the components of the function f(z), $u = u(x,y),$ and $v = v(x,y),,$

can be interpreted as real valued functions of the two real variables, x and y.

The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponentials, logarithms, and trigonometric functions) into the complex domain.

## Derivatives and the Cauchy-Riemann equations

Just as in real analysis, a "smooth" complex function w = f(z) may have a derivative at a particular point in its domain Ω. In fact, the definition of the derivative This article is about derivatives and differentiation in mathematical calculus. ... $f^prime(z) = frac{dw}{dz} = lim_{h to 0}frac{f(z+h) - f(z)}{h},$

is analogous to the real case, with one very important difference. In real analysis, the limit can only be approached by moving along the one-dimensional number line. In complex analysis, the limit can be approached from any direction in the two-dimensional complex plane.

If this limit, the derivative, exists for every point z in Ω, then f(z) is said to be differentiable on Ω. It can be shown that any differentiable f(z) is analytic. This is a much more powerful result than the analogous theorem that can be proved for real-valued functions of real numbers. In the calculus of real numbers, we can construct a function f(x) that has a first derivative everywhere, but for which the second derivative does not exist at one or more points in the function's domain. But in the complex plane, if a function f(z) is differentiable in a neighborhood it must also be infinitely differentiable in that neighborhood. (See "Holomorphic functions are analytic" for a proof.) In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In complex analysis, a complex-valued function f of a complex variable is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a, and is analytic at a if in some open disk centered at a it can be expanded as...

By applying the methods of vector calculus to compute the partial derivatives of the two real functions u(x, y) and v(x, y) into which f(z) can be decomposed, and by considering two paths leading to a point z in Ω, it can be shown that the derivative exists if and only if Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... â†” â‡” â‰¡ logical symbols representing iff. ... $f^prime(z) = frac{partial u}{partial x} + ifrac{partial v}{partial x} = frac{partial v}{partial y} - ifrac{partial u}{partial y}.,$

Equating the real and imaginary parts of these two expressions we obtain the traditional formulation of the Cauchy-Riemann Equations: In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic. ... $frac{partial u}{partial x} = frac{partial v}{partial y} qquad frac{partial u}{partial y} = -frac{partial v}{partial x},$ or, in another common notation, $u_x=v_y qquad u_y=-v_x.,$

By differentiating this system of two partial differential equations first with respect to x, and then with respect to y, we can easily show that In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... $frac{partial^2 u}{partial x^2} + frac{partial^2 u}{partial y^2} = 0 qquad frac{partial^2 v}{partial x^2} + frac{partial^2 v}{partial y^2} = 0,$ or, in another common notation, $u_{xx} + u_{yy} = v_{xx} + v_{yy} = 0.,$

In other words, the real and imaginary parts of a differentiable function of a complex variable are harmonic functions because they satisfy Laplace's equation. In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U â†’ R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ...

## Holomorphic functions

Main article: Holomorphic function

Holomorphic functions are complex functions defined on an open subset of the complex plane which are differentiable. Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, holomorphic functions are infinitely differentiable, a fact that is far from true for real differentiable functions. Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, are holomorphic. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... This article is about derivatives and differentiation in mathematical calculus. ... The exponential function is one of the most important functions in mathematics. ... Sine redirects here. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

See also: analytic function, holomorphic sheaf and vector bundles. In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In mathematics, more specifically complex analysis, a holomorphic sheaf (often also called an analytic sheaf) is a natural generalization of the sheaf of holomorphic functions on a complex manifold. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

## Major results

One central tool in complex analysis is the line integral. The integral around a closed path of a function which is holomorphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem. The values of a holomorphic function inside a disk can be computed by a certain path integral on the disk's boundary (Cauchy's integral formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues among others is useful (see methods of contour integration). If a function has a pole or singularity at some point, that is, at that point its values "blow up" and have no finite value, then one can compute the function's residue at that pole, and these residues can be used to compute path integrals involving the function; this is the content of the powerful residue theorem. The remarkable behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theorem. Functions which have only poles but no essential singularities are called meromorphic. Laurent series are similar to Taylor series but can be used to study the behavior of functions near singularities. This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin Louis Cauchy, is an important statement about path integrals for holomorphic functions in the complex plane. ... In mathematics, Cauchys integral formula, named after Augustin Louis Cauchy, is a central statement in complex analysis. ... In complex analysis, the residue is a complex number which describes the behavior of line integrals of a meromorphic function around a singularity. ... In complex analysis, the evaluation of integrals of real-valued functions along intervals on the real line, is not readily found with certain integrands and methods involving only real variables. ... The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ... The Weierstrass-Casorati theorem in complex analysis describes the remarkable behavior of holomorphic functions near essential singularities. ... A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ... A Laurent series is defined with respect to a particular point c and a path of integration Î³. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ... Series expansion redirects here. ...

A bounded function which is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the field of complex numbers is algebraically closed. Liouvilles theorem in complex analysis states that every bounded (i. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ...

An important property of holomorphic functions is that if a function is holomorphic throughout a simply connected domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be analytically continued from its values on the smaller domain. This allows the extension of the definition of functions such as the Riemann zeta function which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane. Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...

All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more than one complex dimension where the analytic properties such as power series expansion still remain true whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality) are no longer true. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions. The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the...

It is also applied in many subjects throughout engineering, particularly in power engineering. This article or section does not cite its references or sources. ...

## History

Complex analysis is one of the classical branches in mathematics with its roots in the 19th century and some even before. Important names are Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Traditionally, complex analysis, in particular the theory of conformal mappings, has many applications in engineering, but it is also used throughout analytical number theory. In modern times, it became very popular through a new boost of complex dynamics and the pictures of fractals produced by iterating holomorphic functions, the most popular being the Mandelbrot set. Another important application of complex analysis today is in string theory which is a conformally invariant quantum field theory. Image File history File links Download high-resolution version (1280x960, 380 KB) Mandelbrot set. ... Image File history File links Download high-resolution version (1280x960, 380 KB) Mandelbrot set. ... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane that forms a fractal. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Bernhard Riemann. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 &#8211; May 23, 1857) was a French mathematician. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 &#8211; February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ... In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Complex dynamics is the study of dynamical systems for which the phase space is a complex manifold. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane that forms a fractal. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory This box:      String theory is a model of fundamental physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero... Quantum field theory (QFT) is the quantum theory of fields. ...

The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... Given a holomorphic function f on the blue compact set and a point in each of the holes, one can approximate f as well as desired by rational functions having poles only at those three points. ... This is a list of complex analysis topics, by Wikipedia page. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Results from FactBites:

 Complex analysis - Wikipedia, the free encyclopedia (821 words) Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. Complex analysis is particularly concerned with analytic functions of complex variables, known as holomorphic functions. Complex analysis is one of the classical branches in mathematics with its roots in the 19th century and some even before.
 Category:Complex analysis - Wikipedia, the free encyclopedia (175 words) Complex analysis is the branch of mathematics investigating holomorphic functions, i.e. functions which are defined in some region of the complex plane, take complex values, and are differentiable as complex functions. Complex differentiability has much stronger consequences than usual (real) differentiability.
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