Neumann polynomial
In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case , are a sequence of polynomials in used to expand functions in term of Bessel functions. The first few polynomi
Legendre chi function
In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by As such, it resembles the Dirichlet series for the polylogarithm, and, indeed,
Dirichlet integral
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function o
Step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function i
Exponential integral
In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument.
Lommel polynomial
A Lommel polynomial Rm,ν(z), introduced by Eugen von Lommel, is a polynomial in 1/z giving the recurrence relation where Jν(z) is a Bessel function of the first kind. They are given explicitly by
Painlevé transcendents
In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities a
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 of the modular group, and also
Coshc function
In mathematics, the coshc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For , it is defined as It is a solution of the following differe
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
Holonomic function
In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polyn
Spence's function
In mathematics, Spence's function, or dilogarithm, denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm its
Inverse tangent integral
The inverse tangent integral is a special function, defined by: Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function.
Harish-Chandra's Ξ function
In mathematical harmonic analysis, Harish-Chandra's Ξ function is a special spherical function on a semisimple Lie group, studied by Harish-Chandra . Harish-Chandra used it to define Harish-Chandra's
Böhmer integral
In mathematics, a Böhmer integral is an integral introduced by generalizing the Fresnel integrals. There are two versions, given by Consequently, Fresnel integrals can be expressed in terms of the Böh
Student's t-distribution
In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normal
Einstein function
In mathematics, Einstein function is a name occasionally used for one of the functions
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
Conical function
In mathematics, conical functions or Mehler functions are functions which can be expressed in terms of Legendre functions of the first and second kind, and The functions were introduced by Gustav Ferd
Arithmetic–geometric mean
In mathematics, the arithmetic–geometric mean of two positive real numbers x and y is defined as follows: Call x and y a0 and g0: Then define the two interdependent sequences (an) and (gn) as These tw
Bateman Manuscript Project
The Bateman Manuscript Project was a major effort at collation and encyclopedic compilation of the mathematical theory of special functions. It resulted in the eventual publication of five important r
Transport function
In mathematics and the field of transportation theory, the transport functions J(n,x) are defined by Note that
Jackson integral
In q-analog theory, the Jackson integral series in the theory of special functions that expresses the operation inverse to q-differentiation. The Jackson integral was introduced by Frank Hilton Jackso
Pearcey integral
In mathematics, the Pearcey integral is defined as The Pearcey integral is a class of canonical diffraction integrals, often used in wave propagation and optical diffraction problems The first numeric
Hyperbolic growth
When a quantity grows towards a singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function has a hyperbola as a
Parabolic cylinder function
In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This equation is found when the technique of separation of variables is used on
Barnes G-function
In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin c
Struve function
In mathematics, the Struve functions Hα(x), are solutions y(x) of the non-homogeneous Bessel's differential equation: introduced by Hermann Struve. The complex number α is the order of the Struve func
Dickman function
In analytic number theory, the Dickman function or Dickman–de Bruijn function ρ is a special function used to estimate the proportion of smooth numbers up to a given bound.It was first studied by actu
Dawson function
In mathematics, the Dawson function or Dawson integral(named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function.
Lamé function
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper (Gabriel Lamé ). Lamé
Multiplication theorem
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero fo
Bateman function
In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by Bateman discovered this function,
Scorer's function
In mathematics, the Scorer's functions are special functions studied by and denoted Gi(x) and Hi(x). Hi(x) and -Gi(x) solve the equation and are given by The Scorer's functions can also be defined in
Confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three re
Carr–Madan formula
In financial mathematics, the Carr–Madan formula of Peter Carr and Dilip B. Madan shows that the analytical solution of the European option price can be obtained once the explicit form of the characte
Heine's identity
In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as where is a Legendre function of the second kin
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
Strömgren integral
In mathematics and astrophysics, the Strömgren integral, introduced by Bengt Strömgren while computing the Rosseland mean opacity, is the integral: discussed applications of the Strömgren integral in
Mayer f-function
The Mayer f-function is an auxiliary function that often appears in the series expansion of thermodynamic quantities related to classical many-particle systems. It is named after chemist and physicist
Tau function (integrable systems)
Tau functions are an important ingredient in the modern theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in h
Minkowski's question-mark function
In mathematics, the Minkowski question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rationa
Bickley–Naylor functions
In physics, engineering, and applied mathematics, the Bickley–Naylor functions are a sequence of special functions arising in formulas for thermal radiation intensities in hot enclosures. The solution
Barnes integral
In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by Ernest William Barnes . They are closely related to ge
Baer function
Baer functions and , named after Karl Baer, are solutions of the Baer differential equation which arises when separation of variables is applied to the Laplace equation in paraboloidal coordinates. Th
Q-function
In statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, is the probability that a normal (Gaussian) random variable will obtain a value lar
Whittaker function
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker to make the formulas involving the
Griewank function
In mathematics, the Griewank function is often used in testing of optimization. It is defined as follows: The following paragraphs display the special cases of first, second and third orderGriewank fu
Legendre form
In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because the seco
Oblate spheroidal wave function
In applied mathematics, oblate spheroidal wave functions (like also prolate spheroidal wave functions and other related functions) are involved in the solution of the Helmholtz equation in oblate sphe
Crenel function
In mathematics, the crenel function is a periodic discontinuous function P(x) defined as 1 for x belonging to a given interval and 0 outside of it. It can be presented as a difference between two Heav
Ferrers function
In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions.They are named after Norman Macleod Ferrers.
Vector spherical harmonics
In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the
Bring radical
In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial The Bring radical of a complex number a is either any of the five roots of the above polynomi
Goodwin–Staton integral
In mathematics the Goodwin–Staton integral is defined as : It satisfies the following third-order nonlinear differential equation：
Sudan function
In the theory of computation, the Sudan function is an example of a function that is recursive, but not primitive recursive. This is also true of the better-known Ackermann function. The Sudan functio
Kummer's function
In mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm. Bo
Lerche–Newberger sum rule
The Lerche–Newberger, or Newberger, sum rule, discovered by B. S. Newberger in 1982, finds the sum of certain infinite series involving Bessel functions Jα of the first kind. It states that if μ is an
Boxcar function
In mathematics, a boxcar function is any function which is zero over the entirereal line except for a single interval where it is equal to a constant, A. The boxcar function can be expressed in terms
Lommel function
The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation: Solutions are given by the Lommel functions sμ,ν(z) and Sμ,ν(z), introduce
Real analytic Eisenstein series
In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely rel
Al-Salam–Carlitz polynomials
In mathematics, Al-Salam–Carlitz polynomials U(a)n(x;q) and V(a)n(x;q) are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Waleed Al-Salam and Leon
Jack function
In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal
Bailey pair
In mathematics, a Bailey pair is a pair of sequences satisfying certain relations, and a Bailey chain is a sequence of Bailey pairs. Bailey pairs were introduced by W. N. Bailey while studying the sec
Jacobi–Anger expansion
In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example,
Walsh function
In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric function
Spin-weighted spherical harmonics
In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphe
Triangular function
A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle. Often this is an isosceles triangle of height 1 and b
Hough function
In applied mathematics, the Hough functions are the eigenfunctions of Laplace's tidal equations which govern fluid motion on a rotating sphere. As such, they are relevant in geophysics and meteorology
Quantum dilogarithm
In mathematics, the quantum dilogarithm is a special function defined by the formula It is the same as the q-exponential function . Let be "q-commuting variables", that is elements of a suitable nonco
Thomae's function
Thomae's function is a real-valued function of a real variable that can be defined as: It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, th
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
Logit
In statistics, the logit (/ˈloʊdʒɪt/ LOH-jit) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially i
Incomplete polylogarithm
In mathematics, the Incomplete Polylogarithm function is related to the polylogarithm function. It is sometimes known as the incomplete Fermi–Dirac integral or the incomplete Bose–Einstein integral. I
Abramowitz and Stegun
Abramowitz and Stegun (AS) is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the Nati
Ackermann function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive.
Hahn–Exton q-Bessel function
In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, and satisfies the Hahn-Exton q-difference equation (Swarttouw). This funct
Whipple formulae
In the theory of special functions, Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression, concerning associated Legendre functions.
Askey–Gasper inequality
In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper and used in the proof of the Bieberbach conjecture.
Mathieu function
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation where and are parameters. They were first introduced by Émile Léonard Ma
Tracy–Widom distribution
The Tracy–Widom distribution is a probability distribution from random matrix theory introduced by Craig Tracy and Harold Widom . It is the distribution of the normalized largest eigenvalue of a rando
Heun function
In mathematics, the local Heun function Hℓ(a,q;α,β,γ,δ;z) (Karl L. W. Heun ) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point z = 0. The local Heun func
Floor and ceiling functions
In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted floor(x) or ⌊x⌋.
Clausen's formula
In mathematics, Clausen's formula, found by Thomas Clausen, expresses the square of a Gaussian hypergeometric series as a generalized hypergeometric series. It states In particular it gives conditions
Tanc function
In mathematics, the tanc function is defined for as
Chapman function
A Chapman function describes the integration of atmospheric absorption along a slant path on a spherical earth, relative to the vertical case. It applies to any quantity with a concentration decreasin
Special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics,
Sinhc function
In mathematics, the sinhc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For , it is defined as The sinhc function is the hyperbolic anal
Hyperbolastic functions
The hyperbolastic functions, also known as hyperbolastic growth models, are mathematical functions that are used in medical statistical modeling. These models were originally developed to capture the
Carotid–Kundalini function
The Carotid–Kundalini function is closely associated with Carotid-Kundalini fractals coined by popular science columnist Clifford A. Pickover and it is defined as follows:
Prolate spheroidal wave function
The prolate spheroidal wave functions are eigenfunctions of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions on certain ellipsoids of revolution (an ellipse rotated arou
Polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s does the polylogarithm
Jacobi elliptic functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the
Closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − ×
Spheroidal wave function
Spheroidal wave functions are solutions of the Helmholtz equation that are found by writing the equation in spheroidal coordinates and applying the technique of separation of variables, just like the
Ramp function
The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous , for example "0 for negative inputs, output equals input for non-negative inputs". The t
Sievert integral
The Sievert integral, named after Swedish medical physicist Rolf Sievert, is a special function commonly encountered in radiation transport calculations. It plays a role in the sievert (symbol: Sv) un
F. H. Jackson
The Reverend Frank Hilton Jackson (16 August 1870, Hull, England – 27 April 1960) was an English clergyman and mathematician who worked on basic hypergeometric series. He introduced several q-analogs
Spinor spherical harmonics
In quantum mechanics, the spinor spherical harmonics (also known as spin spherical harmonics, spinor harmonics and Pauli spinors) are special functions defined over the sphere. The spinor spherical ha
Fox H-function
In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles Fox.It is defined by a Mellin–Barnes integral where L is a certai
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
Colog
No description available.
Absolute value
In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if x is a positive number, and if is negative (in which case
Synchrotron function
In mathematics the synchrotron functions are defined as follows (for x ≥ 0):
* First synchrotron function
* Second synchrotron function where Kj is the modified Bessel function of the second kind.
Logistic function
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation where , the value of the sigmoid's midpoint;, the supremum of the values of the function;, the logistic g
Bessel–Maitland function
In mathematics, the Bessel–Maitland function, or Wright generalized Bessel function, is a generalization of the Bessel function, introduced by Edward Maitland Wright. The word "Maitland" in the name o
Dirac comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula for some given period . Here t is a real variable and the sum ext
Rectangular function
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as Alternati
Voigt profile
The Voigt profile (named after Woldemar Voigt) is a probability distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data fro
Chandrasekhar's X- and Y-function
In atmospheric radiation, Chandrasekhar's X- and Y-function appears as the solutions of problems involving diffusive reflection and transmission, introduced by the Indian American astrophysicist Subra
Lambert W function
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function f(w) = wew, where w
Wright omega function
In mathematics, the Wright omega function or Wright function, denoted ω, is defined in terms of the Lambert W function as:
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
Herglotz–Zagier function
In mathematics, the Herglotz–Zagier function, named after Gustav Herglotz and Don Zagier, is the function introduced by who used it to obtain a Kronecker limit formula for real quadratic fields.
Tak (function)
In computer science, the Tak function is a recursive function, named after (竹内郁雄). It is defined as follows: def tak( x, y, z) if y < x tak( tak(x-1, y, z), tak(y-1, z, x), tak(z-1, x, y) ) else z end
Jacobi zeta function
In mathematics, the Jacobi zeta function Z(u) is the logarithmic derivative of the Jacobi theta function Θ(u). It is also commonly denoted as Where E, K, and F are generic Incomplete Elliptical Integr
Sign function
In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign funct
Bernoulli polynomials
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–Mac
Kontorovich–Lebedev transform
In mathematics, the Kontorovich–Lebedev transform is an integral transform which uses a Macdonald function (modified Bessel function of the second kind) with imaginary index as its kernel. Unlike othe
Anger function
In mathematics, the Anger function, introduced by C. T. Anger, is a function defined as and is closely related to Bessel functions. The Weber function (also known as Lommel–Weber function), introduced
Complete Fermi–Dirac integral
In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by This equals where is the polylogarithm. Its derivative is and this derivative r
Trigonometric integral
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.
Selberg integral
In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg.
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,
Neuman–Sándor mean
In mathematics of special functions, the Neuman–Sándor mean M, of two positive and unequal numbers a and b, is defined as: This mean interpolates the inequality of the unweighted arithmetic mean A = (
Airy function
In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(x) and
List of special functions and eponyms
This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory (but not intend
Buchstab function
The Buchstab function (or Buchstab's function) is the unique continuous function defined by the delay differential equation In the second equation, the derivative at u = 2 should be taken as u approac
Debye function
In mathematics, the family of Debye functions is defined by The functions are named in honor of Peter Debye, who came across this function (with n = 3) in 1912 when he analytically computed the heat 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
Cantor function
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions
Jackson q-Bessel function
In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson . The third Jackson q-Bessel function is the same as t
Ruler function
In number theory, the ruler function of an integer can be either of two closely-related functions. One of these functions counts the number of times can be evenly divided by two, which for the numbers
Chandrasekhar's H-function
In atmospheric radiation, Chandrasekhar's H-function appears as the solutions of problems involving scattering, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar. The Chandra
Tanhc function
In mathematics, the tanhc function is defined for as The tanhc function is the hyperbolic analogue of the tanc function.
Pochhammer contour
In mathematics, the Pochhammer contour, introduced by Camille Jordan and Leo Pochhammer, is a contour in the complex plane with two points removed, used for contour integration. If A and B are loops a
Digital Library of Mathematical Functions
The Digital Library of Mathematical Functions (DLMF) is an online project at the National Institute of Standards and Technology (NIST) to develop a database of mathematical reference data for special
Incomplete Fermi–Dirac integral
In mathematics, the incomplete Fermi–Dirac integral for an index j is given by This is an alternate definition of the incomplete polylogarithm.