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gamma function

Extension of the factorial function, with its argument shifted down by 1, to real and complex numbers

Entitymeromorphic function special functionQ190573

Also known as complete gamma function · Euler integral of the second kind

Instance of: meromorphic function
, special function

Overview

From Wikipedia, the free encyclopedia · Read full article · Text CC BY-SA 4.0

In mathematics, the gamma function, denoted by ⁠⁠, is the most common extension of the factorial function to complex numbers. First studied by Daniel Bernoulli, the gamma function is defined for all complex numbers except non-positive integers, and satisfies for every positive integer ⁠⁠. The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles.

Overview

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36 statements · 14% with external sources · numbers in brackets are citations

Defining formula: \operatorname{\Gamma}\left(z\right) = \int\limits_{0}^{\infty} t^{z-1} \mathrm{e}^{-t} \, \mathrm{d}t
2
Sources
1.dlmf.nist.gov/5.2.E1
2.Stated in ISO 80000-2:2019 Quantities and units — Part 2: Mathematics
View statement on Wikidata ↗
Discovered by: Leonhard Euler discovered: 1729
1
Sources
1.scholarlycommons.pacific.edu/euler-works/19/
View statement on Wikidata ↗
Different from: factorial
In defining formula: \operatorname{\Gamma}(z) symbol represents: gamma function
\int_a^b f(x) \, \mathrm{d}x symbol represents: definite integral
\mathrm{e} symbol represents: Euler's number
Mathematical inverse: reciprocal gamma function relative to: multiplication

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25 identifiers linking gamma function across the web's reference databases

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Source last edited September 8, 2026. AIDB refreshes this page daily.

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AIDB. (2026). gamma function (Q190573). Retrieved from https://aidb.si/e/gamma-function-Q190573