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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. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
Artin, Emil (2007), Rosen, Michael (ed.), Exposition by Emil Artin: a selection., History of Mathematics, vol. 30, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4172-3, MR 2288274 Reprints Artin's books on the gamma function, Galois theory, the theory of algebraic numbers, and several of his papers.
as the only positive function f , with domain on the interval x > 0, that simultaneously has the following three properties: f (1) = 1, and f (x + 1) = x f (x) for x > 0 and f is logarithmically convex. A treatment of this theorem is in Artin's book The Gamma Function, [4] which has been reprinted by the AMS in a collection of Artin's writings.
In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers for which > by = +,as the only function defined on the half-plane := {: >} such that:
A fair amount of effort has been made to calculate the numerical value of the Fransén–Robinson constant with high accuracy. The value was computed to 36 decimal places by Herman P. Robinson using 11 point Newton–Cotes quadrature, to 65 digits by A. Fransén using Euler–Maclaurin summation, and to 80 digits by Fransén and S. Wrigge using Taylor series and other methods.
In mathematics, Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994 paper. [ 1 ] The formula is a modification of Stirling's approximation , and has the form
where () is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics through Feynman diagrams. [2] A similar result was also obtained by Glaisher. [3]