Gamma function

English

A hand-drawn graph of the absolute value of the gamma function for complex argument, from 1909, E. Jahnke, F. Emde, Funktionentafeln mit Formeln und Kurven (English title: Tables of Higher Functions)

Etymology

The function itself was initially defined as an integral (in modern representation, $\mathrm{\Gamma }(x)={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{x-1}dt$) for positive real x by Swiss mathematician Template:W in 1730. The name derives from the notation, Γ(x), which was introduced by Template:W (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)!. Template:W 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.

Noun

1. Template:Lb 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. Template:C
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   • 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.

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Synonyms

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Hypernyms

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Hyponyms

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Translations

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Further reading

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