gamma function

Meaning

Noun

• A meromorphic function which generalises the notion of factorial to complex numbers and has singularities at the nonpositive integers; any of certain generalisations or analogues of said function, such as extend the factorial to domains other than the complex numbers.

Related

Similar words

• Euler integral of the second kind

Narrower meaning words

• digamma function
• incomplete gamma function
• polygamma function
• trigamma function

Broader meaning words

• function

Origin

• The function itself was initially defined as an integral (in modern representation, $\backslash textstyle\backslash Gamma(x)=\backslash int\_0^\backslash infty\; e^\{-t\}t^\{x-1\}dt$) 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.