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# Gamma function

In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. [1]

## Abramowitz and Stegun

Abramowitz and Stegun (AS) is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the National Institute of Standards and Technology (NIST).

## Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.

Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician.

## Algebraic differential equation

In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra.

## Algebraic independence

In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set is algebraically independent over K if and only if α is transcendental over K. In general, all the elements of an algebraically independent set S over K are by necessity transcendental over K, and over all of the field extensions over K generated by the remaining elements of S.

## American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

## Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.

## Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

## Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.

## Android (operating system)

Android is a mobile operating system developed by Google, based on a modified version of the Linux kernel and other open source software and designed primarily for touchscreen mobile devices such as smartphones and tablets.

## Arc length

Determining the length of an irregular arc segment is also called rectification of a curve.

## Argument of a function

In mathematics, an argument of a function is a specific input in the function, also known as an independent variable.

## Arithmetic–geometric mean

In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers and is defined as follows: Call and and: \end Then define the two interdependent sequences and as \end where the square root takes the principal value.

## Astrophysics

Astrophysics is the branch of astronomy that employs the principles of physics and chemistry "to ascertain the nature of the astronomical objects, rather than their positions or motions in space".

## Barnes G-function

In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers.

## Barnes integral

In mathematics, a Barnes integral or Mellin&ndash;Barnes integral is a contour integral involving a product of gamma functions.

## 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.

## Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in number theory.

## Beta function

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by for.

## Binomial coefficient

In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient.

## Bohr–Mollerup theorem

In mathematical analysis, the Bohr–Mollerup theorem is a theorem named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it.

## C (programming language)

C (as in the letter ''c'') is a general-purpose, imperative computer programming language, supporting structured programming, lexical variable scope and recursion, while a static type system prevents many unintended operations.

## C mathematical functions

C mathematical operations are a group of functions in the standard library of the C programming language implementing basic mathematical functions.

## Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

## Carl Johan Malmsten

Carl Johan Malmsten (April 9, 1814 in Uddetorp, Skara County, Sweden – February 11, 1886 in Uppsala, Sweden) was a Swedish mathematician and politician.

## Character (mathematics)

In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers).

## Charles Hermite

Prof Charles Hermite FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.

## Chauvenet Prize

The Chauvenet Prize is the highest award for mathematical expository writing.

## Christian Goldbach

Christian Goldbach (March 18, 1690 &ndash; November 20, 1764) was a German mathematician who also studied law.

## Combinatorics

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

## 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.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Computer algebra system

A computer algebra system (CAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists.

## Contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

## Convex function

In mathematics, a real-valued function defined on an ''n''-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions.

## Cornelius Lanczos

Cornelius (Cornel) Lanczos (Lánczos Kornél,, born as Kornél Lőwy, until 1906: Löwy (Lőwy) Kornél) was a Jewish Hungarian mathematician and physicist, who was born on February 2, 1893, and died on June 25, 1974.

## Daniel Bernoulli

Daniel Bernoulli FRS (8 February 1700 – 17 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family.

## Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

## Digamma function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions.

## Division by zero

In mathematics, division by zero is division where the divisor (denominator) is zero.

## Double factorial

In mathematics, the double factorial or semifactorial of a number (denoted by) is the product of all the integers from 1 up to that have the same parity (odd or even) as.

## Ellipse

In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

## Elliptic gamma function

In mathematics, the elliptic gamma function is a generalization of the q-Gamma function, which is itself the q-analog of the ordinary Gamma function.

## Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.

## Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.

## Ernst Kummer

Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician.

## Error function

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion.

## Eugen Jahnke

Paul Rudolf Eugen Jahnke (born November 30, 1861 in Berlin, died October 18, 1921 in Berlin) was a German mathematician.

## Euler integral

In mathematics, there are two types of Euler integral: For positive integers m and n.

## Euler–Mascheroni constant

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.

## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

## Exponential sum

In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function Therefore, a typical exponential sum may take the form summed over a finite sequence of real numbers xn.

## Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

## Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when n.

## Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

## Finite ring

In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.

## Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids - liquids and gases.

## Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Functional equation

In mathematics, a functional equation is any equation in which the unknown represents a function.

## 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.

## Gamma

Gamma (uppercase, lowercase; gámma) is the third letter of the Greek alphabet.

## Gamma distribution

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions.

## Gamma function

In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

## Gauss sum

In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically where the sum is over elements of some finite commutative ring, is a group homomorphism of the additive group into the unit circle, and is a group homomorphism of the unit group into the unit circle, extended to non-unit where it takes the value 0.

## Gauss's constant

In mathematics, Gauss's constant, denoted by G, is defined as the reciprocal of the arithmetic–geometric mean of 1 and the square root of 2: The constant is named after Carl Friedrich Gauss, who on May 30, 1799 discovered that so that where Β denotes the beta function.

## Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: for arbitrary real constants, and.

## Gaussian integral

The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.

## GNU MPFR

GNU MPFR (GNU Multiple Precision Floating-Point Reliably) is a GNU portable C library for arbitrary-precision binary floating-point computation with correct rounding, based on GNU Multi-Precision Library.

## GNU Octave

GNU Octave is software featuring a high-level programming language, primarily intended for numerical computations.

## GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science.

## Greek alphabet

The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC.

## Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

## Harald Bohr

Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and soccer player.

## Hölder's theorem

In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions.

## Holomorphic function

In mathematics, 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.

## Hurwitz zeta function

In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions.

## Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.

## Hypersphere

In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center.

## Hypertranscendental function

A hypertranscendental function or transcendentally transcendental function is an analytic function which is not the solution of an algebraic differential equation with coefficients in Z (the integers) and with algebraic initial conditions.

## Improper integral

In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, \infty, -\infty, or in some instances as both endpoints approach limits.

## Incomplete gamma function

In mathematics, the upper incomplete gamma function and lower incomplete gamma function are types of special functions, which arise as solutions to various mathematical problems such as certain integrals.

## Infinite product

In mathematics, for a sequence of complex numbers a1, a2, a3,...

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative.

## Interpolation

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

## Jacques Philippe Marie Binet

Jacques Philippe Marie Binet (2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856.

## James Stirling (mathematician)

James Stirling (May 1692, Garden, Stirlingshire – 5 December 1770, Edinburgh) was a Scottish mathematician.

## Johannes Mollerup

Johannes Mollerup (born 3 December 1872 in Nyborg; died 27 June 1937) was a Danish mathematician.

## Jonathan Borwein

Jonathan Michael Borwein (20 May 1951 &ndash; 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia.

## Joseph Ludwig Raabe

Joseph Ludwig Raabe (15 May 1801 in Brody, Galicia – 22 January 1859 in Zürich, Switzerland) was a Swiss mathematician.

## Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

## Lanczos approximation

In mathematics, the Lanczos approximation is a method for computing the gamma function numerically, published by Cornelius Lanczos in 1964.

## Leibniz integral rule

In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form where -\infty, the derivative of this integral is expressible as where the partial derivative indicates that inside the integral, only the variation of f(x, t) with x is considered in taking the derivative.

## Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Limit of a function

Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.

## Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

## Logarithmic derivative

In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where f' is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely f', scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f. This follows directly from the chain rule.

## Logarithmically convex function

In mathematics, a function f defined on a convex subset of a real vector space and taking positive values is said to be logarithmically convex or superconvex if \circ f, the composition of the logarithmic function with f, is a convex function.

## Maple (software)

Maple is a symbolic and numeric computing environment, and is also a multi-paradigm programming language.

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## MATLAB

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks.

## 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.

## Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.

## Michael Berry (physicist)

Sir Michael Victor Berry, (born 14 March 1941), is a mathematical physicist at the University of Bristol, England.

## Multiple gamma function

In mathematics, the multiple gamma function \Gamma_N is a generalization of the Euler Gamma function and the Barnes G-function.

## Multiplication theorem

In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function.

## Multivariate gamma function

In mathematics, the multivariate gamma function Γp is a generalization of the gamma function.

## National Institute of Standards and Technology

The National Institute of Standards and Technology (NIST) is one of the oldest physical science laboratories in the United States.

## Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.

## Normal distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.

## Otto Hölder

Otto Ludwig Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.

In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function.

## PARI/GP

PARI/GP is a computer algebra system with the main aim of facilitating number theory computations.

## Particular values of the gamma function

The gamma function is an important special function in mathematics.

## Philip J. Davis

Philip J. Davis (January 2, 1923 – March 13, 2018) was an American academic applied mathematician.

## Pochhammer k-symbol

In the mathematical theory of special functions, the Pochhammer k-symbol and the k-gamma function, introduced by Rafael Díaz and Eddy Pariguan are generalizations of the Pochhammer symbol and gamma function.

## Polygamma function

In mathematics, the polygamma function of order is a meromorphic function on '''ℂ''' and defined as the th derivative of the logarithm of the gamma function: Thus holds where is the digamma function and is the gamma function.

## 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.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Probability

Probability is the measure of the likelihood that an event will occur.

## Probability theory

Probability theory is the branch of mathematics concerned with probability.

## Q-gamma function

In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary Gamma function closely related to the double gamma function.

## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

## Ramanujan's master theorem

In mathematics, Ramanujan's master theorem (named after Srinivasa Ramanujan) is a technique that provides an analytic expression for the Mellin transform of an analytic function.

## Rational function

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Reciprocal gamma function

In mathematics, the reciprocal gamma function is the function where denotes the gamma function.

## Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

## Reflection formula

In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a − x) and f(x).

## 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.

## Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

## Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.

The Russian Academy of Sciences (RAS; Росси́йская акаде́мия нау́к (РАН) Rossíiskaya akadémiya naúk) consists of the national academy of Russia; a network of scientific research institutes from across the Russian Federation; and additional scientific and social units such as libraries, publishing units, and hospitals.

## Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.

## Sinc function

In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by, has two slightly different definitions.

## Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

## Spouge's approximation

In mathematics, the Spouge's approximation is a formula for computing an approximation of the gamma function.

## Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

## Stirling's approximation

In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials.

## Stretched exponential function

The stretched exponential function is obtained by inserting a fractional power law into the exponential function.

## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## TK Solver

TK Solver (originally TK!Solver) is a mathematical modeling and problem solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc.

## Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.

## TU Dresden

The TU Dresden (abbreviated as TUD and often mistakenly translated from German as Dresden University of Technology) is a public research university, the largest institute of higher education in the city of Dresden, the largest university in Saxony and one of the 10 largest universities in Germany with 37,134 students.

## Volume of an n-ball

In geometry, a ball is a region in space comprising all points within a fixed distance from a given point; that is, it is the region enclosed by a sphere or hypersphere.

## Weierstrass factorization theorem

In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that entire functions can be represented by a product involving their zeroes.

## Well-defined

In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value.

## Windows Calculator

Windows Calculator is a software calculator included in all versions of Windows.

## Wolfram Mathematica

Wolfram Mathematica (usually termed Mathematica) is a modern technical computing system spanning most areas of technical computing — including neural networks, machine learning, image processing, geometry, data science, visualizations, and others.

## Zeros and poles

In mathematics, a zero of a function is a value such that.

## References

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