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. Hue is one of the main properties of a Color described with names such as " Red " " Yellow " etc In Colorimetry and Color theory, colorfulness, chroma, and saturation are related but distinct concepts referring to the perceived intensity

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted It is useful in many branches of mathematics, including number theory and applied mathematics, and in physics. 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 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 Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

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. This article is about both real and complex analytic functions Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic 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, the real numbers may be described informally in several different ways Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

1 History
2 Complex functions
3 Derivatives and the Cauchy-Riemann equations
4 Holomorphic functions
5 Major results
6 See also
7 References
8 External links

History

$The Mandelbrot set, a fractal.$

The Mandelbrot set, a fractal. In Mathematics, the Mandelbrot set, named after Benoît A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole"

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. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician Traditionally, complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytical number theory. In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane 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 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. Complex dynamics the study of Dynamical systems for which the Phase space is a Complex manifold. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" In Mathematics, the Mandelbrot set, named after Benoît Another important application of complex analysis today is in string theory which is a conformally invariant quantum field theory. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings In quantum field theory (QFT the forces between particles are mediated by other particles

Complex functions

A complex function is a function in which the independent variable and the dependent variable are both complex numbers. Dependent variables and independent variables refer to values that change in relationship to each other Dependent variables and independent variables refer to values that change in relationship to each other 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 Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis 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:

$z = x + iy\,$ and

$w = f(z) = u(z) + iv(z)\,$

where $x,y \in \mathbb{R}\,$ and $u(z), v(z)\,$ are real-valued functions. In Mathematics, the real numbers may be described informally in several different ways Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane

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 Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In fact, the definition of the derivative

$f^\prime(z) = \frac{dw}{dz} = \lim_{h \to 0}\frac{f(z+h) - f(z)}{h}\,$

is analogous to the real case but 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.

(The claim that "in real analysis, the limit can only be approached by moving along the one dimensional number line" should not be confused with directional derivatives. In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the It may be explained that in directional derivatives, one still moves along the one dimensional x line but it can be in "discrete" units; that is, if one follows y = x² curve, that does not mean that one is moving on the plane (instead of the one dimensional x line) but means that one is approaching in steps of discrete units. )

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 article is about both real and complex analytic functions 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. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. (See "Holomorphic functions are analytic" for a proof. In Complex analysis, a field of Mathematics, a complex -valued function f of a complex variable z is holomorphic )

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 existence of derivative implies

$f^\prime(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i\frac{\partial u}{\partial y}.\,$

Equating the real and imaginary parts of these two expressions we obtain the traditional formulation of the Cauchy-Riemann Equations:

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\,$ 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

$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \qquad \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0\,$ 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. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner 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 In Mathematics, the Cauchy-Riemann differential equations in Complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Mathematics, Mathematical physics and the theory of Stochastic processes a harmonic function is a twice continuously differentiable function In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties

Holomorphic functions

Main article: Holomorphic function

Holomorphic functions are complex functions defined on an open subset of the complex plane which are differentiable. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change 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. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

See also: analytic function, holomorphic sheaf and vector bundles. This article is about both real and complex analytic functions In Mathematics, more specifically Complex analysis, a holomorphic sheaf (often also called an analytic sheaf) is a natural generalization of the sheaf In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space

Major results

One central tool in complex analysis is the line integral. In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated 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. In Mathematics, the Cauchy integral theorem in Complex analysis, named after Augustin Louis Cauchy, is an important statement about Line integrals 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). In Mathematics, Cauchy's integral formula, named after Augustin Louis Cauchy, is a central statement in Complex analysis. 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). 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 contour integration is a method of evaluating certain Integrals along paths in the complex plane 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 residue theorem in Complex analysis is a powerful tool to evaluate Line integrals of Analytic functions over closed curves and can often be used to compute The remarkable behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theorem. The Casorati-Weierstrass theorem in Complex analysis describes the remarkable behavior of Meromorphic functions near essential singularities. Functions which have only poles but no essential singularities are called meromorphic. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic Laurent series are similar to Taylor series but can be used to study the behavior of functions near singularities. In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

A bounded function which is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. In Complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded Entire function must be constant 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. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients

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. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be The function on the larger domain is said to be analytically continued from its values on the smaller domain. In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. 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. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in 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. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional

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 theory of functions of several complex variables is the branch of Mathematics dealing with functions f ( z1 z2 In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions. In Complex analysis, the Riemann mapping theorem states that if U is a simply connected open subset of the complex number plane

It is also applied in many subjects throughout engineering, particularly in power engineering. Power engineering, also called power systems engineering, is a subfield of Electrical engineering that deals with the generation, transmission

References

Needham T. The theory of functions of several complex variables is the branch of Mathematics dealing with functions f ( z1 z2 In Complex analysis, Runge's theorem, also known as Runge's approximation theorem, named after the German Mathematician Carl Runge, and put forward This is a list of Complex analysis topics, by Wikipedia page Local theory Holomorphic function Antiholomorphic function Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real Tristan Needham is the author of the highly originalbook Visual Complex Analysis in which he uses a geometric approach to develop Complex analysis (which he says was , Visual Complex Analysis (Oxford, 1997).
Henrici P. , Applied and Computational Complex Analysis (Wiley). [Three volumes: 1974, 1977, 1986. ]
Kreyszig, E., Advanced Engineering Mathematics, 9 ed. Erwin Kreyszig (born January 6 1922) was a Professor of Mathematics at Carleton University. , Ch. 13-18 (Wiley, 2006).
Scheidemann, V. , Introduction to complex analysis in several variables (Birkhauser, 2005)
Shaw, W. T. , Complex Analysis with Mathematica (Cambridge, 2006).
Marsden & Hoffman, Basic complex analysis (Freeman, 1999).

External links

Dictionary

-noun

(mathematics) The branch of mathematics that studies holomorphic functions.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents