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Fresnel integral

The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the de

Exponential minus 1

No description available.

Wetzel's problem

In mathematics, Wetzel's problem concerns bounds on the cardinality of a set of analytic functions that, for each of their arguments, take on few distinct values. It is named after John Wetzel, a math

Hypertranscendental function

A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in Z

Liouville's theorem (complex analysis)

In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, e

Milne-Thomson method for finding a holomorphic function

In mathematics, the Milne-Thomson method is a method for finding a holomorphic function whose real or imaginary part is given. It is named after Louis Melville Milne-Thomson.

Goniometric function

No description available.

Radius of convergence

In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . Wh

Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynom

Hermite class

The Hermite or Pólya class is a set of entire functions satisfying the requirement that if E(z) is in the class, then: 1.
* E(z) has no zero (root) in the upper half-plane. 2.
* for x and y real and

Faddeeva function

The Faddeeva function or Kramp function is a scaled complex complementary error function, It is related to the Fresnel integral, to Dawson's integral, and to the Voigt function. The function arises in

Transcendental function

In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcen

Exponential function

The exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued functio

Univalent function

In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.

Harmonic map

In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equa

Soboleva modified hyperbolic tangent

The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ([P]SMHTAF), is a special S-shaped function based on the hyperbolic tangen

Schwarz function

The Schwarz function of a curve in the complex plane is an analytic function which maps the points of the curve to their complex conjugates. It can be used to generalize the Schwarz reflection princip

Nevanlinna's criterion

In mathematics, Nevanlinna's criterion in complex analysis, proved in 1920 by the Finnish mathematician Rolf Nevanlinna, characterizes holomorphic univalent functions on the unit disk which are starli

Complex logarithm

In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm

Error function

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: This integral is a special (non-elementary) sig

Holomorphic functional calculus

In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is

Residue theorem

In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to com

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 va

Reciprocal gamma function

In mathematics, the reciprocal gamma function is the function where Γ(z) denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal

Analytic capacity

In the mathematical discipline of complex analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes "how big" a bounded analytic function on C \ K can become.

Removable singularity

In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the r

Algebraic function

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms,

Laguerre–Pólya class

The Laguerre–Pólya class is the class of entire functions consisting of those functions which are locally the limit of a series of polynomials whose roots are all real.Any function of Laguerre–Pólya c

Hadamard's gamma function

In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function. This function, with its argument shifte

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 s

Weierstrass functions

In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass. The relation between

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 fur

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

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 a

Geometric function theory

Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.

Doubly periodic function

In mathematics, a doubly periodic function is a function defined on the complex plane and having two "periods", which are complex numbers u and v that are linearly independent as vectors over the fiel

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

Lacunary function

In analysis, a lacunary function, also known as a lacunary series, is an analytic function that cannot be analytically continued anywhere outside the radius of convergence within which it is defined b

Algebroid function

In mathematics, an algebroid function is a solution of an algebraic equation whose coefficientsare analytic functions. So y(z) is an algebroid function if it satisfies where are analytic. If this equa

Analyticity of holomorphic functions

In complex analysis, a complex-valued function of a complex variable :
* is said to be holomorphic at a point if it is differentiable at every point within some open disk centered at , and
* is said

Harmonic morphism

In mathematics, a harmonic morphism is a (smooth) map between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphism

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

Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with

Mittag-Leffler star

In complex analysis, a branch of mathematics, the Mittag-Leffler star of a complex-analytic function is a set in the complex plane obtained by attempting to extend that function along rays emanating f

Mittag-Leffler function

In mathematics, the Mittag-Leffler function is a special function, a complex function which depends on two complex parameters and . It may be defined by the following series when the real part of is s

Neville theta functions

In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows: where: K(m) is the complete elliptic integral of the first kind, , and is the elliptic nome. Note

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