Vipul: 2 revisions

2008-05-18T19:17:49Z

2 revisions

← Older revision	Revision as of 19:17, 18 May 2008
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Vipul: Functional equation for gamma function moved to Recurrence relation for gamma function

2008-05-01T02:08:46Z

Functional equation for gamma function moved to Recurrence relation for gamma function

← Older revision	Revision as of 02:08, 1 May 2008
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Vipul: New page: ==Statement== Let $\Gamma$ denote the gamma function as defined on the right half-plane by: $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} \, dt$ Then, $\Ga...$

2008-05-01T02:06:05Z

New page: ==Statement== Let <math>\Gamma</math> denote the gamma function as defined on the right half-plane by: <math>\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} \, dt</math> Then, <math>\Ga...

New page

==Statement==

Let <math>\Gamma</math> denote the [[gamma function]] as defined on the [[right half-plane]] by:

<math>\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} \, dt</math>

Then, <math>\Gamma</math> satisfies the following functional equation:

<math>\Gamma(z + 1) = z \Gamma(z)</math>

whenever <math>z</math> is in the [[right half-plane]].

In fact, we use this functional equation to extend <math>\Gamma</math> to a meromorphic function on <math>\mathbb{C}</math>, so the functional equation holds more generally for any <math>z</math> that is not a non-positive integer (non-positive integers are precisely the simple poles).

==Proof==

The proof is using integration by parts.

Recurrence relation for gamma function - Revision history

Vipul: 2 revisions

Vipul: Functional equation for gamma function moved to Recurrence relation for gamma function

Vipul: New page: ==Statement== Let \Gamma denote the gamma function as defined on the right half-plane by: \Gamma(z) = \int_0^\infty t^{z-1}e^{-t} \, dt Then, \Ga...

Vipul: New page: ==Statement== Let $\Gamma$ denote the gamma function as defined on the right half-plane by: $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} \, dt$ Then, $\Ga...$