Dictionary

gamma function en · NOUN

Etymology

The function itself was initially defined as an integral (in modern representation, Γ(x)=∫₀ ᪲e⁻ᵗtˣ⁻¹dt) for positive real x by Swiss mathematician Leonhard Euler in 1730. The name derives from the notation, Γ(x), which was introduced by Adrien-Marie Legendre (1752—1833) (he referred to it, however, as the Eulerian integral of the second kind). Both Euler's integral and Legendre's notation shift the argument with respect to the factorial, so that for integer n>0, Γ(n) = (n−1)!. Carl Friedrich Gauss preferred π(x), with no shift, but Legendre's notation prevailed. Generalisation to non-integer negative and to complex numbers was achieved by analytic continuation.

Meanings

1. A meromorphic function which generalizes the notion of factorial to complex numbers and has singularities at the nonpositive integers; any of certain generalizations or analogues of said function, such as extend the factorial to domains other than the complex numbers.
   • We select one mathematical object, the gamma function, and show how it grew in concept and in content from the time of Euler to the recent mathematical treatise of Bourbaki, and how, in this growth, it partook of the general development of mathematics over the past two and a quarter centuries. 2007, Philip J. Davis, “Leonhard Euler's Integral: A Historical Profile of the Gamma Function”, in William Dunham, editor, The Genius of Euler: Reflections on His Life and Work, American Mathematical Society, page 167:
   • Chapter 3 deals with the p-adic gamma function. 1987, Kit Ming Yeung, Applications of p-adic gamma function to congruences of binomial coefficients, University of California, San Diego, page 3:
   • 2002, M. Aslam Chaudhry, Syed M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman & Hall / CRC Press, page 2, In particular, the exponential, circular, and hyperbolic functions are rational combinations of gamma functions.

Forms

Hypernyms

function

Hyponyms

digamma function · incomplete gamma function · polygamma function · trigamma function

Synonyms

Euler integral of the second kind (function that extends the domain of the factorial)

Translations (15)

it funzione Gamma di Eulero (function which generalizes the notion of a factorial) · es función gamma (function which generalizes the notion of a factorial) · sv gammafunktionen (function which generalizes the notion of a factorial) · cmn Γ函數 /Γ函数 (function which generalizes the notion of a factorial) · fa تابع گاما (function which generalizes the notion of a factorial) · ru га́мма-фу́нкция (function which generalizes the notion of a factorial) · sv Eulers gammafunktion (function which generalizes the notion of a factorial) · pl funkcja gamma (function which generalizes the notion of a factorial) · ja ガンマ関数 (function which generalizes the notion of a factorial) · de Gammafunktion (function which generalizes the notion of a factorial) · tr gama fonksiyonu (function which generalizes the notion of a factorial) · hu gamma-függvény (function which generalizes the notion of a factorial) · it funzione Gamma (function which generalizes the notion of a factorial) · fr fonction gamma (function which generalizes the notion of a factorial) · cmn 伽馬函數 /伽马函数 (function which generalizes the notion of a factorial)

wikipedia: American Mathematical Monthly · wikipedia: Adrien-Marie Legendre · wikipedia: Leonhard Euler · wikipedia: Carl Friedrich Gauss