# Category: Complex analysis

Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name
Principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case ar
Dirichlet space
In mathematics, the Dirichlet space on the domain (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space , for
Schur class
In mathematics, the Schur class consists of the Schur functions: the holomorphic functions from the open unit disk to the closed unit disk. These functions were studied by Schur. The Schur parameters
Lacunary value
In complex analysis, a subfield of mathematics, a lacunary value or gap of a complex-valued function defined on a subset of the complex plane is a complex number which is not in the image of the funct
Pseudoanalytic function
In mathematics, pseudoanalytic functions are functions introduced by Lipman Bers that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations.
Lie algebra bundle
In mathematics, a weak Lie algebra bundle is a vector bundle over a base space X together with a morphism which induces a Lie algebra structure on each fibre . A Lie algebra bundle is a vector bundle
Quasicircle
In mathematics, a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by and ,
Edmund Schuster
Edmund Schuster (7 September 1851 – 5 July 1932) was a German engineer and mathematician who contributed to the field of special functions and complex analysis being a pioneer in the field of harmonic
Wirtinger derivatives
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of
Loewner differential equation
In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Or
Complex polytope
In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A co
Nørlund–Rice integral
In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the nth forward difference of a function to a line integral on the complex plane. It commonly appears in the theory o
Hessian polyhedron
In geometry, the Hessian polyhedron is a regular complex polyhedron 3{3}3{3}3, , in . It has 27 vertices, 72 3{} edges, and 27 3{3}3 faces. It is self-dual. Coxeter named it after Ludwig Otto Hesse fo
Bispectrum
In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions.
Bounded type (mathematics)
In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a fu
In mathematics, the conformal radius is a way to measure the size of a simply connected planar domain D viewed from a point z in it. As opposed to notions using Euclidean distance (say, the radius of
Ramification (mathematics)
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is also used from the opposite
Fuchs relation
In mathematics, the Fuchs relation is a relation between the starting exponents of formal series solutions of certain linear differential equations, so called Fuchsian equations. It is named after Laz
Berezin transform
In mathematics — specifically, in complex analysis — the Berezin transform is an integral operator acting on functions defined on the open unit disk D of the complex plane C. Formally, for a function
Siegel upper half-space
In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is p
Zeros and poles
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Techni
Contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus
Blaschke product
In complex analysis, the Blaschke product is a bounded analytic function in the open unit disc constructed to have zeros at a (finite or infinite) sequence of prescribed complex numbers a0, a1, ... in
Global analytic function
In the mathematical field of complex analysis, a global analytic function is a generalization of the notion of an analytic function which allows for functions to have multiple branches. Global analyti
Indicator function (complex analysis)
In the field of mathematics known as complex analysis, the indicator function of an entire function indicates the rate of growth of the function in different directions.
Pseudo-zero set
In complex analysis (a branch of mathematical analysis), the pseudo-zero set or root neighborhood of a degree-m polynomial p(z) is the set of all complex numbers that are roots of polynomials whose co
Regular singular point
In mathematics, in the theory of ordinary differential equations in the complex plane , the points of are classified into ordinary points, at which the equation's coefficients are analytic functions,
Regular part
In mathematics, the regular part of a Laurent series consists of the series of terms with positive powers. That is, if then the regular part of this Laurent series is In contrast, the series of terms
Infinite compositions of analytic functions
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolvi
Escaping set
In mathematics, and particularly complex dynamics, the escaping set of an entire function ƒ consists of all points that tend to infinity under the repeated application of ƒ.That is, a complex number b
Kramers–Kronig relations
The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. The relations are often
Singular integral operators on closed curves
In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert
Bifurcation locus
In complex dynamics, the bifurcation locus of a family of holomorphic functions informally is a locus of those maps for which the dynamical behavior changes drastically under a small perturbation of t
Almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-pe
Iterated monodromy group
In geometric group theory and dynamical systems the iterated monodromy group of a covering map is a group describing the monodromy action of the fundamental group on all iterations of the covering. A
Subharmonic function
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subha
Szegő kernel
In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named
Upper half-plane
In mathematics, the upper half-plane, is the set of points (x, y) in the Cartesian plane with y > 0.
Asano contraction
In complex analysis, a discipline in mathematics, and in statistical physics, the Asano contraction or Asano–Ruelle contraction is a transformation on a separately affine multivariate polynomial. It w
Complex line
In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has dimension one over C (hence
Complex analysis
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 bra
Fundamental normality test
In complex analysis, a mathematical discipline, the fundamental normality test gives sufficient conditions to test the normality of a family of analytic functions. It is another name for the stronger
Riemann–Hilbert problem
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existe
Schwarzian derivative
In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in
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
Goodman's conjecture
Goodman's conjecture on the coefficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.
General Dirichlet series
In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of where , are complex numbers and is a strictly increasing sequence of nonnegative real num
Positive harmonic function
In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the Herglotz-Ries
Right half-plane
In complex analysis, the (open) right half-plane is the set of all points in the complex plane whose real part is strictly positive, that is, the set .
Partial fractions in complex analysis
In complex analysis, a partial fraction expansion is a way of writing a meromorphic function as an infinite sum of rational functions and polynomials. When is a rational function, this reduces to the
Binomial series
In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like for a nonnegative integer . Specifically, the binomial series is the Taylor
Series multisection
In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series then its
Witting polytope
In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3{3}3{3}3{3}3, and Coxeter diagram . It has 240 vertices, 2160 3{} edges, 2160 3{3}3 faces, and 240 3{3
Neumann–Poincaré operator
In mathematics, the Neumann–Poincaré operator or Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary
Positive-real function
Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis. They are complex functions, Z(s), of a complex v
Beltrami equation
In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation for w a complex distribution of the complex variable z in some open set U, with derivatives th
Grunsky matrix
In complex analysis and geometric function theory, the Grunsky matrices, or Grunsky operators, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single ho
Cauchy product
In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy.
Nevanlinna function
In mathematics, in the field of complex analysis, a Nevanlinna function is a complex function which is an analytic function on the open upper half-plane and has non-negative imaginary part. A Nevanlin
Laurent series
In mathematics, the Laurent series of a complex function 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
Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that for all x i
Residue at infinity
In complex analysis, a branch of mathematics, the residue at infinity is a residue of a holomorphic function on an annulus having an infinite external radius. The infinity is a point added to the loca
Complex convexity
Complex convexity is a general term in complex geometry.
Voorhoeve index
In mathematics, the Voorhoeve index is a non-negative real number associated with certain functions on the complex numbers, named after Marc Voorhoeve. It may be used to extend Rolle's theorem from re
Normal family
In mathematics, with special application to complex analysis, a normal family is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are
Continuous functions on a compact Hausdorff space
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers.
Quasiconformal mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded
Inverse Laplace transform
In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property: where denotes the Laplace transform
Möbius–Kantor polygon
In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in . 3{3}3 has 8 vertices, and 8 edges. It is self-dual. Every vertex is shared by 3 triangular edges. Coxeter named it a M
Bergman space
In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are suff
Antiholomorphic function
In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. A function of the complex variable
Logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where is the derivative of f. Intuitively, this is the infinitesimal
Hilbert's inequality
In analysis, a branch of mathematics, Hilbert's inequality states that for any sequence u1,u2,... of complex numbers. It was first demonstrated by David Hilbert with the constant 2π instead of π; the
Schwarz triangle function
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs fo
Holomorphic separability
In mathematics in complex analysis, the concept of holomorphic separability is a measure of the richness of the set of holomorphic functions on a complex manifold or complex-analytic space.
Hankel contour
In mathematics, a Hankel contour is a path in the complex plane which extends from (+∞,δ), around the origin counter clockwise and back to(+∞,−δ), where δ is an arbitrarily small positive number. The
Essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category essential singularity is a "left-over" or default grou
Swiss cheese (mathematics)
In mathematics, a Swiss cheese is a compact subset of the complex plane obtained by removing from a closed disc some countable union of open discs, usually with some restriction on the centres and rad
Sendov's conjecture
In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex varia
Positive-definite function
In mathematics, a positive-definite function is, depending on the context, either of two types of function.
Argument (complex analysis)
In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as
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 → , (where D is the open unit disk in the c
Algebraic analysis
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of func
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 funct
Line integral convolution
In scientific visualization, line integral convolution (LIC) is a method to visualize a vector field, such as fluid motion.
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. This integral transform is closely connected to the
Planar Riemann surface
In mathematics, a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are character
Hyperfunction
In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order
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.
Bloch space
In the mathematical field of complex analysis, the Bloch space, named after French mathematician André Bloch and denoted or ℬ, is the space of holomorphic functions f defined on the open unit disc D i
Complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-ax
Stokes phenomenon
In complex analysis the Stokes phenomenon, discovered by G. G. Stokes , is that the asymptotic behavior of functions can differ in different regions of the complex plane, and that these differences ca
Pi
The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formula
List of complex analysis topics
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of ma
Calderón projector
In applied mathematics, the Calderón projector is a pseudo-differential operator used widely in boundary element methods. It is named after Alberto Calderón.
Exponential type
In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function eC|z| for some real-valued constant C as |
In mathematics, the Douady–Earle extension, named after Adrien Douady and Clifford Earle, is a way of extending homeomorphisms of the unit circle in the complex plane to homeomorphisms of the closed u
Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz, who named them afte
Distortion (mathematics)
In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it
Dixon elliptic functions
In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map from each regular hexagon in a hexagonal tiling
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
Domain coloring
In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane. By assigning points on the complex
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
Rothe–Hagen identity
In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers except where its denominators vanish: It is a generalization of Vandermonde's identity, and is named a
Connectedness locus
In one-dimensional complex dynamics, the connectedness locus is a subset of the parameter space of rational functions, which consists of those parameters for which the corresponding Julia set is conne
Principal part
In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function.
Cartan's lemma (potential theory)
In potential theory, a branch of mathematics, Cartan's lemma, named after Henri Cartan, is a bound on the measure and complexity of the set on which a logarithmic Newtonian potential is small.
Quasiperiodic function
In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. A function is quasiperiodic with quasiperiod if , where is a "simpler" function than . What
Asymptotic expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite
Bicoherence
In mathematics and statistical analysis, bicoherence (also known as bispectral coherency) is a squared normalised version of the bispectrum. The bicoherence takes values bounded between 0 and 1, which
Koenigs function
In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representatio
Bombieri norm
In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in or (there is also a version for non homogeneous univariate polynomials). This n
Movable singularity
In the theory of ordinary differential equations, a movable singularity is a point where the solution of the equation behaves badly and which is "movable" in the sense that its location depends on the
Motor variable
In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. Will
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
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.
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, togeth
Schramm–Loewner evolution
In probability theory, the Schramm–Loewner evolution with parameter κ, also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling li
Power series
In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical anal
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions,
A Course of Modern Analysis
A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions (colloquially known as Whi
Winding number
In mathematics, the winding number or winding index 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
Dessin d'enfant
In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational number
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 in the com
Logarithmic form
In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.