Gamma function

In mathematics,Gamma function isfunction that extendsconceptfactorial tocomplex numbers. The notationdueAdrien-Marie Legendre. Ifreal part ofcomplex number zpositive, thenintegral

converges absolutely. Using integration by parts, one can show that

BecauseΓ(1) = 1, this relation implies

for all natural numbers n. It can further be usedextend Γ(z) tomeromorphic function definedall complex numbers z except z = 0,  − 1,− 2, − 3, ... by analytic continuation. Itthis extended version thatcommonly referredasGamma function. An alternative notation whichsomtimes used isPi function, whichterms ofGamma function is

We also sometimes find

whichan entire function, definedevery complex number. That entire entailshas no poles, so has no zeros.

Perhapsmost well-known value ofGamma function atnon-integer is

The Gamma function haspoleorder 1 at z = − nevery natural number n;residue theregiven by

The following multiplicative form ofGamma functionvalidall complex numbers z whichnot non-positive integers:

Here γ isEuler-Mascheroni constant.

External links

• Gamma function at MathWorld.