transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Leonhard Euler solved the problem of finding a function of a continuous variable which for integer values of agrees with the factorial function . The gamma function is a shift by one of the solution to this problem.
For a complex variable , we define by the formula
where and for positive integer ,
is the “Pochhammer symbol” (or rising factorial). It easily follows that for natural numbers .
As a function of a complex variable, the Gamma function is a meromorphic function with simple poles at .
Extending the recursive definition of the ordinary factorial function, the Gamma function satisfies the following translation formula:
away from .
It also satisfies a reflection formula, due to Euler:
Quite remarkably, the Gamma function (this time as a function of a real variable) is uniquely characterized in the following theorem:
(Bohr-Mollerup)
The restriction of the Gamma function to the open interval is the unique function such that
,
,
is convex.
A number of other representations of the Gamma function are known and frequently utilized, e.g.,
Product representation:
where is Euler's constant.
Integral representation:
Emil Artin, Einführung in die Theorie der Gammafunktion, Hamburger Mathematische Einzelschriften
l. Heft, Verlag B. G. Teubner, Leipzig (1931)
English translation by Michael Butler: The Gamma Function, Holt, Rinehart and Winston (1931) [pdf]
George Andrews, Richard Askey, Ranjan Roy, Special Functions. Encyclopedia of Mathematics and Its Applications 71, Cambridge University Press, 1999.
See also:
Wikipedia, Gamma function
Wikipedia, Bohr-Mollerup theorem
Last revised on December 27, 2022 at 08:59:41. See the history of this page for a list of all contributions to it.