Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, traditionally known as the theory of functions of a complex variable, is the branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
that investigates
functions of
complex numbers. It is useful in many branches of mathematics, including
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
and
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
; as well as in
physics
Physics is the natural science that studies matter, its 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 which r ...
, including
hydrodynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
,
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
, and
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
.
Overview
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Complex numbers
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Complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
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Complex functions
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
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Complex derivative
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 derivativ ...
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Holomorphic functions
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Harmonic functions
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
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Elementary functions
In mathematics, an elementary function is a function (mathematics), function of a single variable (mathematics), variable (typically Function of a real variable, real or Complex analysis#Complex functions, complex) that is defined as taking addit ...
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Polynomial function
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s
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Exponential functions
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
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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 ...
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Hyperbolic functions
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Logarithmic functions
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Inverse trigonometric functions
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Inverse hyperbolic functions
In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions.
For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The s ...
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Residue theory
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Isometries in the complex plane
Related fields
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Number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
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Hydrodynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
*
Thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
*
Electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
Local theory
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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 derivativ ...
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Antiholomorphic function
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Cauchy–Riemann equations
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Conformal mapping
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Conformal welding In mathematics, conformal welding (sewing or gluing) is a process in geometric function theory for producing a Riemann surface by joining together two Riemann surfaces, each with a disk removed, along their boundary circles. This problem can be redu ...
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Power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
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Radius of convergence
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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 expansion c ...
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Meromorphic function
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Entire function
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Pole (complex analysis)
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Zero (complex analysis)
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Residue (complex analysis)
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Isolated singularity
In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number ''z0'' is an isolated singularity of a function ''f'' if there exists an open disk ''D'' c ...
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Removable singularity
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Essential singularity
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Branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
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Principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane.
Examples
Trigonometric inverses
Principal branches are used ...
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Weierstrass–Casorati theorem
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Landau's constants
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Holomorphic functions are analytic
In complex analysis, a complex-valued function f of a complex variable z:
*is said to be holomorphic at a point a if it is differentiable at every point within some open disk centered at a, and
* is said to be analytic at a if in some open disk ...
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Schwarzian derivative
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Analytic capacity
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Disk algebra In mathematics, specifically in functional and complex analysis, the disk algebra ''A''(D) (also spelled disc algebra) is the set of holomorphic functions
:''ƒ'' : D → \mathbb,
(where D is the open unit disk in the complex plane \mathbb) th ...
Growth and distribution of values
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Ahlfors theory Ahlfors theory is a mathematical theory invented by Lars Ahlfors as a geometric counterpart of the Nevanlinna theory. Ahlfors was awarded one of the two very first Fields Medals for this theory in 1936.
It can be considered as a generalization of ...
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Bieberbach conjecture In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was ...
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Borel–Carathéodory theorem In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory.
...
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Corona theorem In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by and proved by .
The commutative Banach algebra and Hardy space ''H''∞ consists of the bounded ...
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Hadamard three-circle theorem
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Hardy space
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Hardy's theorem In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions.
Let f be a holomorphic function on the open ball centered at zero and radius R in the complex plane, and assume that f is not a const ...
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Maximum modulus principle
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Nevanlinna theory
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Paley–Wiener theorem In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (189 ...
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Progressive function
In mathematics, a progressive function ''ƒ'' ∈ ''L''2(R) is a function whose Fourier transform is supported by positive frequencies only:
:\mathop\hat \subseteq \mathbb_+.
It is called super regressive if and only if the time ...
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Value distribution theory of holomorphic functions
In mathematics, the value distribution theory of holomorphic functions is a division of mathematical analysis. It tries to get quantitative measures of the number of times a function ''f''(''z'') assumes a value ''a'', as ''z'' grows in size, refi ...
Contour integrals
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Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
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Cauchy's integral theorem
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in t ...
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Cauchy's integral formula
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Residue theorem
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Liouville's theorem (complex analysis)
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Examples of contour integration
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Fundamental theorem of algebra
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Simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
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Winding number
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Principle of the argument
In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.
Specifically, if ...
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Rouché's theorem
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Bromwich integral
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Morera's theorem
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.
Morera's theorem states that a continuous, complex-valued function ''f'' defined ...
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Mellin transform
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Kramers–Kronig relation, a. k. a.
Hilbert transform
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Sokhotski–Plemelj theorem
Special functions
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Exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
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Beta function
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Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
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Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
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Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
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Generalized Riemann hypothesis
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Elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
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Half-period ratio
In mathematics, the half-period ratio τ of an elliptic function is the ratio
:\tau = \frac
of the two half-periods \frac and \frac of the elliptic function, where the elliptic function is defined in such a way that
:\Im(\tau) > 0
is in the ...
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Jacobi's elliptic functions
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Weierstrass's elliptic functions
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by th ...
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Theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
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Elliptic modular function
In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holom ...
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J-function
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Modular function
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Modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
Riemann surfaces
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Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
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Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
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Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
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Riemann mapping theorem
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Carathéodory's theorem (conformal mapping)
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Riemann–Roch theorem
Other
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Amplitwist In mathematics, the amplitwist is a concept created by Tristan Needham in the book ''Visual Complex Analysis'' (1997) to represent the derivative of a complex function visually.
Definition
The ''amplitwist'' associated with a given function is its ...
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Antiderivative (complex analysis)
In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function ''g'' is a function whose complex derivative is ''g''. More precisely, given an open set U in the complex plane and a function g:U\to \ ...
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Bôcher's theorem
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Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is ...
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Harmonic conjugate
In mathematics, a real-valued function u(x,y) defined on a connected open set \Omega \subset \R^2 is said to have a conjugate (function) v(x,y) if and only if they are respectively the real and imaginary parts of a holomorphic function f(z) of th ...
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Hilbert's inequality In analysis, a branch of mathematics, Hilbert's inequality states that
:
\left, \sum_\dfrac\\le\pi\displaystyle\sum_, u_, ^2.
for any sequence of complex numbers. It was first demonstrated by David Hilbert with the constant instead of ; the sh ...
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Method of steepest descent
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Montel's theorem
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic fu ...
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Periodic points of complex quadratic mappings This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers o ...
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Pick matrix
Pick may refer to:
Places
* Pick City, North Dakota, a town in the United States
* Pick Lake (Cochrane District, Ontario), a lake in Canada
* Pick Lake (Thunder Bay District), a lake in Canada
* Pick Mere, a lake in Pickmere, England
People w ...
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Runge approximation theorem
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Schwarz lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping ...
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Weierstrass factorization theorem In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an e ...
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Mittag-Leffler's theorem
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Sendov's conjecture
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Infinite compositions of analytic functions
Several complex variables
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Biholomorphy
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.
Formal definiti ...
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Cartan's theorems A and B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold . They are significant both as applied to several complex variables, and in the general development of ...
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Cousin problems
In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in special cases by Pierre Cousin in 1895. They ...
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Edge-of-the-wedge theorem
In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is use ...
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Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
History
People
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Augustin Louis Cauchy
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Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
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Carl Friedrich Gauss
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Jacques Hadamard
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Kiyoshi Oka
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Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
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Karl Weierstrass
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Pierre Alphonse Laurent
Pierre Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series, an expansion of a function into an infinite power series, generalizing the Tayl ...
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Brook Taylor
Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis.
Life and work
Brook Taylor w ...
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Siméon Denis Poisson
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Hermann Schwarz
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Camille Jordan
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Carl Gustav Jacob Jacobi
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Eugène Rouché
Eugène Rouché (18 August 1832 – 19 August 1910) was a French mathematician.
Career
He was an alumnus of the École Polytechnique, which he entered in 1852. He went on to become professor of mathematics at the Charlemagne lyceum then at the ...
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Gerardus Mercator
Gerardus Mercator (; 5 March 1512 – 2 December 1594) was a 16th-century geographer, cosmographer and Cartography, cartographer from the County of Flanders. He is most renowned for creating the Mercator 1569 world map, 1569 world map based on ...
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Joseph Liouville
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Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
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August Ferdinand Möbius
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William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...