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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
that investigates functions of
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. It is helpful in many branches of mathematics, including
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
,
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, analytic combinatorics, and
applied mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
, as well as in
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, including the branches of
hydrodynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in ...
,
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
,
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear,
aerospace Aerospace is a term used to collectively refer to the atmosphere and outer space. Aerospace activity is very diverse, with a multitude of commercial, industrial, and military applications. Aerospace engineering consists of aeronautics and astron ...
, mechanical and
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s of a complex variable, that is, ''
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
, which deals with the study of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s and functions of a real variable.


History

Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler,
Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
. In modern times, it has become very popular through a new boost from complex dynamics and the pictures of
fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
s produced by iterating holomorphic functions. Another important application of complex analysis is in string theory which examines conformal invariants in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
.


Complex functions

A complex function is a function from
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are generally assumed to have a domain that contains a nonempty open subset of the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. For any complex function, the values z from the domain and their images f(z) in the range may be separated into real and imaginary parts: : z=x+iy \quad \text \quad f(z) = f(x+iy)=u(x,y)+iv(x,y), where x,y,u(x,y),v(x,y) are all real-valued. In other words, a complex function f:\mathbb\to\mathbb may be decomposed into : u:\mathbb^2\to\mathbb \quad and \quad v:\mathbb^2\to\mathbb, i.e., into two real-valued functions (u, v) of two real variables (x, y). Similarly, any complex-valued function on an arbitrary set (is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to, and therefore, in that sense, it) can be considered as an ordered pair of two real-valued functions: or, alternatively, as a
vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
from into \mathbb R^2. Some properties of complex-valued functions (such as continuity) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such as differentiability, are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, every differentiable complex function is analytic (see next section), and two differentiable functions that are equal in a
neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of a point are equal on the intersection of their domain (if the domains are connected). The latter property is the basis of the principle of analytic continuation which allows extending every real
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including the complex exponential function, complex logarithm functions, and
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
.


Holomorphic functions

Complex functions that are differentiable at every point of an open subset \Omega of the complex plane are said to be ''holomorphic on'' In the context of complex analysis, the derivative of f at z_0 is defined to be : f'(z_0) = \lim_ \frac. Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach z_0 in the complex plane. Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions are infinitely differentiable, whereas the existence of the ''n''th derivative need not imply the existence of the (''n'' + 1)th derivative for real functions. Furthermore, all holomorphic functions satisfy the stronger condition of analyticity, meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that functions holomorphic on \Omega can be approximated arbitrarily well by polynomials in some neighborhood of every point in \Omega. This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that are ''nowhere'' analytic; see . Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, extended appropriately to complex arguments as functions are holomorphic over the entire complex plane, making them '' entire functions'', while rational functions p/q, where ''p'' and ''q'' are polynomials, are holomorphic on domains that exclude points where ''q'' is zero. Such functions that are holomorphic everywhere except a set of isolated points are known as ''meromorphic functions''. On the other hand, the functions and z\mapsto \bar are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy–Riemann conditions (see below). An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy–Riemann conditions. If f:\mathbb\to\mathbb, defined by where is holomorphic on a
region In geography, regions, otherwise referred to as areas, zones, lands or territories, are portions of the Earth's surface that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and ...
then for all z_0\in \Omega, :\frac(z_0) = 0,\ \text \frac\partial \mathrel \frac12\left(\frac\partial + i\frac\partial\right). In terms of the real and imaginary parts of the function, ''u'' and ''v'', this is equivalent to the pair of equations u_x = v_y and u_y=-v_x, where the subscripts indicate partial differentiation. However, the Cauchy–Riemann conditions do not characterize holomorphic functions, without additional continuity conditions (see Looman–Menchoff theorem). Holomorphic functions exhibit some remarkable features. For instance, Picard's theorem asserts that the range of an entire function can take only three possible forms: or \ for some In other words, if two distinct complex numbers z and w are not in the range of an entire function then f is a constant function. Moreover, a holomorphic function on a connected open set is determined by its restriction to any nonempty open subset.


Conformal map


Major results

One of the central tools in complex analysis is the
line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integr ...
. The line integral around a closed path of a function that is holomorphic everywhere inside the area bounded by the closed path is always zero, as is stated by the Cauchy integral theorem. The values of such a holomorphic function inside a disk can be computed by a path integral on the disk's boundary (as shown in 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 applicable (see methods of contour integration). A "pole" (or isolated singularity) of a function is a point where the function's value becomes unbounded, or "blows up". If a function has such a pole, then one can compute the function's residue there, which 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 Picard's theorem. Functions that have only poles but no essential singularities are called meromorphic.
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
are the complex-valued equivalent to Taylor series, but can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials. A bounded function that 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. If a function is holomorphic throughout a 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 The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
, 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. 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 in which the analytic properties such as power series expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality) do not carry over. The
Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane \mathbb which is not all of \mathbb, then there exists a biholomorphic mapping f (i.e. a bijective hol ...
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. A major application of certain complex spaces is in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
as
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
s.


See also

* Complex geometry * Hypercomplex analysis * Vector calculus * List of complex analysis topics * Monodromy theorem * Riemann–Roch theorem * Runge's theorem


References


Sources

* Ablowitz, M. J. & A. S. Fokas, ''Complex Variables: Introduction and Applications'' (Cambridge, 2003). * Ahlfors, L., ''Complex Analysis'' (McGraw-Hill, 1953). * Cartan, H., ''Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes.'' (Hermann, 1961). English translation, ''Elementary Theory of Analytic Functions of One or Several Complex Variables.'' (Addison-Wesley, 1963). * Carathéodory, C., ''Funktionentheorie.'' (Birkhäuser, 1950). English translation, ''Theory of Functions of a Complex Variable'' (Chelsea, 1954). volumes.* Carrier, G. F., M. Krook, & C. E. Pearson
''Functions of a Complex Variable: Theory and Technique.''
(McGraw-Hill, 1966). * Conway, J. B., ''Functions of One Complex Variable.'' (Springer, 1973). * Fisher, S., ''Complex Variables.'' (Wadsworth & Brooks/Cole, 1990). * Forsyth, A.
''Theory of Functions of a Complex Variable''
(Cambridge, 1893). * Freitag, E. & R. Busam, ''Funktionentheorie''. (Springer, 1995). English translation, ''Complex Analysis''. (Springer, 2005). * Goursat, E.
''Cours d'analyse mathématique, tome 2''
(Gauthier-Villars, 1905). English translation
''A course of mathematical analysis, vol. 2, part 1: Functions of a complex variable''
(Ginn, 1916). * Henrici, P., ''Applied and Computational Complex Analysis'' (Wiley). hree volumes: 1974, 1977, 1986.* Kreyszig, E., ''Advanced Engineering Mathematics.'' (Wiley, 1962). * Lavrentyev, M. & B. Shabat, ''Методы теории функций комплексного переменного.'' (''Methods of the Theory of Functions of a Complex Variable''). (1951, in Russian). * Markushevich, A. I., ''Theory of Functions of a Complex Variable'', (Prentice-Hall, 1965). hree volumes.* Marsden & Hoffman, ''Basic Complex Analysis.'' (Freeman, 1973). * Needham, T., ''Visual Complex Analysis.'' (Oxford, 1997). http://usf.usfca.edu/vca/ * Remmert, R., ''Theory of Complex Functions''. (Springer, 1990). * Rudin, W., ''Real and Complex Analysis.'' (McGraw-Hill, 1966). * Shaw, W. T., ''Complex Analysis with Mathematica'' (Cambridge, 2006). * Stein, E. & R. Shakarchi, ''Complex Analysis.'' (Princeton, 2003). * Sveshnikov, A. G. & A. N. Tikhonov, ''Теория функций комплексной переменной.'' (Nauka, 1967). English translation
''The Theory Of Functions Of A Complex Variable''
(MIR, 1978). * Titchmarsh, E. C.
''The Theory of Functions.''
(Oxford, 1932). * Wegert, E., ''Visual Complex Functions''. (Birkhäuser, 2012). * Whittaker, E. T. & G. N. Watson, '' A Course of Modern Analysis.'' (Cambridge, 1902)
3rd ed. (1920)


External links



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