Overview

• Complex numbers
• Complex functions
  • Complex derivative
  • Holomorphic functions
  • Harmonic functions
• Elementary functions
  • Polynomial functions
  • Exponential functions
  • Trigonometric functions
  • Logarithmic functions
  • Inverse trigonometric functions
• Residue theory

Related fields

• Number theory
  Number theory
  Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
  
  
• Hydrodynamics
• Thermodynamics
  Thermodynamics
  Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
  
  
• Electrical engineering
  Electrical engineering
  Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
  
  

Local theory

• Holomorphic function
  Holomorphic function
  In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
  
  
• Antiholomorphic function
  Antiholomorphic function
  In mathematics, antiholomorphic functions are a family of functions closely related to but distinct from holomorphic functions....
  
  
• Cauchy-Riemann equations
  Cauchy-Riemann equations
  In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable...
  
  
• Conformal mapping
• Power series
• Radius of convergence
  Radius of convergence
  In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
  
  
• Laurent series
  Laurent series
  In mathematics, the Laurent series of a complex function f 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...
  
  
• Meromorphic function
  Meromorphic function
  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...
  
  
• Entire function
  Entire function
  In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
  
  
• Pole (complex analysis)
• Zero (complex analysis)
  Zero (complex analysis)
  In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...
  
  
• Residue (complex analysis)
  Residue (complex analysis)
  In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...
  
  
• Isolated singularity
• Removable singularity
• Essential singularity
  Essential singularity
  In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.The category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the...
  
  
• Branch point
  Branch point
  In the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point...
  
  
• Principal branch
  Principal branch
  In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function. Most often, this applies to functions defined on the complex plane: see branch cut....
  
  
• Weierstrass-Casorati theorem
• Landau's constants
  Landau's constants
  In complex analysis, a branch of mathematics, Landau's constants are certain mathematical constants that describe the behaviour of holomorphic functions defined on the unit disk...
  
  
• Holomorphic functions are analytic
  Holomorphic functions are analytic
  In complex analysis, a branch of mathematics, a complex-valued function ƒ 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...
  
  
• Schwarzian derivative
  Schwarzian derivative
  In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all linear fractional transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and...
  
  
• Analytic capacity
• Disk algebra
  Disk algebra
  In function theory, the disk algebra A is the set of holomorphic functionswhere D is the open unit disk in the complex plane C, f extends to a continuous function on the closure of D...
  
  

Growth

• Bieberbach conjecture
• Borel-Carathéodory theorem
• Hadamard three-circle theorem
  Hadamard three-circle theorem
  In complex analysis, a branch of mathematics, theHadamard three-circle theorem is a result about the behavior of holomorphic functions.Let f be a holomorphic function on the annulusr_1\leq\left| z\right| \leq r_3....
  
  
• Hardy space
  Hardy space
  In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...
  
  
  • Hardy's theorem
    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 constant function...
    
    
  • Progressive function
  • Corona theorem
    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 ....
    
    
• Maximum modulus principle
  Maximum modulus principle
  In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f| cannot exhibit a true local maximum that is properly within the domain of f....
  
  
• Nevanlinna theory
  Nevanlinna theory
  Nevanlinna theory is a branch of complex analysis developed by Rolf Nevanlinna. It deals with the value distribution theory of holomorphic functions in one variable, usually denoted z....
  
  
• Picard's theorem
• Paley-Wiener theorem
• Value distribution theory of holomorphic functions
  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 assumes a value a, as z grows in size, refining the Picard theorem on behaviour close to an essential singularity...
  
  

Contour integrals

• Line integral
  Line integral
  In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
  
  
• Cauchy integral theorem
• Cauchy's integral formula
  Cauchy's integral formula
  In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all...
  
  
• Residue theorem
  Residue theorem
  The residue theorem, sometimes called Cauchy's 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 real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula...
  
  
• Liouville's theorem (complex analysis)
  Liouville's theorem (complex analysis)
  In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f| ≤ M for all z in C is constant.The theorem is considerably improved by...
  
  
• Examples of contour integration
• Fundamental theorem of algebra
  Fundamental theorem of algebra
  The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
  
  
• Simply connected
• Winding number
  Winding number
  In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
  
  
  • Principle of the argument
  • Rouché's theorem
    Rouché's theorem
    Rouché's theorem, named after , states that if the complex-valued functions f and g are holomorphic inside and on some closed contour K, with |g| ...
    
    
• Bromwich integral
  Bromwich integral
  -Mellin's inverse formula:An integral formula for the inverse Laplace transform, called the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula, is given by the line integral:...
  
  
• Morera's theorem
  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....
  
  
• Mellin transform
  Mellin transform
  In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
  
  
• Kramers–Kronig relation

Special functions

• Exponential function
  Exponential function
  In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
  
  
• Beta function
• Gamma function
  Gamma function
  In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
  
  
• Riemann zeta function
  • Riemann hypothesis
    Riemann hypothesis
    In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
    
    
  • Generalized Riemann hypothesis
    Generalized Riemann hypothesis
    The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
    
    
• Elliptic function
  Elliptic function
  In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
  
  
  • Half-period ratio
  • Jacobi's elliptic functions
    Jacobi's elliptic functions
    In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications...
    
    
  • Weierstrass's elliptic functions
    Weierstrass's elliptic functions
    In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass...
    
    
  • Theta function
• Elliptic modular function
• J-function
  J-function
  The term J-function may refer to* The Klein j-invariant in mathematics* Leverett J-function in petroleum engineering...
  
  
• Modular function
• Modular form
  Modular form
  In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
  
  

Riemann surfaces

• Analytic continuation
  Analytic continuation
  In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
  
  
• Riemann sphere
  Riemann sphere
  In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
  
  
• Riemann surface
  Riemann surface
  In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
  
  
• Riemann mapping theorem
• Carathéodory's theorem (conformal mapping)
  Carathéodory's theorem (conformal mapping)
  In mathematical complex analysis, Carathéodory's theorem, proved by , states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann map...
  
  
• Riemann-Roch theorem

Other

• Antiderivative (complex analysis)
  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...
  
  
• Bôcher's theorem
  Bôcher's theorem
  In mathematics, Bôcher's theorem can refer to one of two theorems proved by the American mathematician Maxime Bôcher.-Bôcher's theorem in complex analysis:...
  
  
• Cayley transform
  Cayley transform
  In mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping in...
  
  
• Complex differential equation
  Complex differential equation
  A complex differential equation is a differential equation whose solutions are functions of a complex variable.Constructing integrals involves choice of what path to take, which means singularities and branch points of the equation need to be studied...
  
  
• Harmonic conjugate
  Harmonic conjugate
  In mathematics, a function u defined on some open domain \Omega\subset\R^2 is said to have as a conjugate a function v if and only if they are respectively real and imaginary part of a holomorphic function f of the complex variable z:=x+iy\in\Omega. That is, v is conjugated to u if f:=u+iv is...
  
  
• Hilbert's inequality
  Hilbert's inequality
  In complex analysis, a branch of mathematics, Hilbert's inequality was first demonstrated by David Hilbert and later improved by Issai Schur, who replaced Hilbert's constant 2\pi with the sharp constant \pi....
  
  
• Method of steepest descent
• Montel's theorem
  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 Paul Montel, and give conditions under which a family of holomorphic functions is normal....
  
  
• Periodic points of complex quadratic mappings
  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 one and two; and a complex map is one in which the...
  
  
• Pick matrix
• Runge approximation theorem
• Schwarz lemma
  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 stronger theorems, such as the Riemann mapping theorem, which it helps to prove...
  
  
• Weierstrass factorization theorem
  Weierstrass factorization theorem
  In mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes...
  
  
• Mittag-Leffler's theorem
  Mittag-Leffler's theorem
  In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. It is sister to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. It is named after Gösta Mittag-Leffler.-Theorem:Let D...
  
  

Several complex variables

• Analytization trick
  Analytization trick
  The analytization trick is a heuristic often applied by physicists.Suppose we have a function f of a complex variable z which is not analytic, but happens to be differentiable with respect to its real and imaginary components separately...
  
  
• Biholomorphy
  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....
  
  
• Cartan's theorems A and B
  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 F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.Theorem A states that F is spanned...
  
  
• Cousin problems
  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 P. Cousin in 1895...
  
  
• Edge-of-the-wedge theorem
  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 used in quantum field theory to construct the...
  
  
• Several complex variables
  Several complex variables
  The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
  
  

People

• Augustin Louis Cauchy
  Augustin Louis Cauchy
  Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
  
  
• Leonhard Euler
  Leonhard Euler
  Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
  
  
• Carl Friedrich Gauss
  Carl Friedrich Gauss
  Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
  
  
• Jacques Hadamard
  Jacques Hadamard
  Jacques Salomon Hadamard FRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...
  
  
• Kiyoshi Oka
  Kiyoshi Oka
  was a Japanese mathematician who did fundamental work in the theory of several complex variables. He was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduating in 1924....
  
  
• Bernhard Riemann
  Bernhard Riemann
  Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
  
  
• Karl Weierstrass
  Karl Weierstrass
  Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....