Gamma function - Revision history

Vipul: 6 revisions

2008-05-18T19:13:00Z

6 revisions

← Older revision	Revision as of 19:13, 18 May 2008
(No difference)

Vipul: /* Facts */

2008-05-01T21:21:38Z

Facts

← Older revision		Revision as of 21:21, 1 May 2008
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	* [[Integral formula for reciprocal of gamma function]]		* [[Integral formula for reciprocal of gamma function]]
	* [[Reciprocal of gamma function is entire]]: In other words, the gamma function is zero-free		* [[Reciprocal of gamma function is entire]]: In other words, the gamma function is zero-free
			* [[Summation formula for logarithmic derivative of gamma function]]: This states that:

			<math>\frac{\Gamma'(z)}{\Gamma(z)} = \sum_{n=0}^\infty \left( \frac{1}{n+1} - \frac{1}{n+z} \right)</math>
			* [[Product formula for reciprocal of gamma function]]
	* [[Gauss's formula for gamma function]]		* [[Gauss's formula for gamma function]]

Vipul: /* Facts */

2008-05-01T20:59:38Z

Facts

← Older revision		Revision as of 20:59, 1 May 2008
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	* [[Integral formula for reciprocal of gamma function]]		* [[Integral formula for reciprocal of gamma function]]
	* [[Reciprocal of gamma function is entire]]: In other words, the gamma function is zero-free		* [[Reciprocal of gamma function is entire]]: In other words, the gamma function is zero-free
			* [[Gauss's formula for gamma function]]

	==External links==		==External links==

	* {{mathworld\|Gamma function}}		* {{mathworld\|Gamma function}}

Vipul: /* Facts */

2008-05-01T19:51:32Z

Facts

← Older revision		Revision as of 19:51, 1 May 2008
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	==Facts==		==Facts==

	* [[~~Functional equation~~ for gamma function]]		* [[Euler's integral formula for gamma function]]: This states that on the right half-plane:
	* [[Reflection principle for gamma function]]
	* [[Reciprocal of gamma function is entire]]		<math>\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt</math>
			* [[Recurrence relation for gamma function]]: This states that if <math>z</math> is not a non-positive integer, we have:

			<math>\Gamma(z + 1) = z\Gamma(z)</math>
			* [[Reflection principle for gamma function]]: If <math>z</math> is not an integer, we have:

			<math>\Gamma(z)\Gamma(1 - z) = \frac{\pi}{\sin (\pi z)}</math>
			* [[Integral formula for reciprocal of gamma function]]
			* [[Reciprocal of gamma function is entire]]: In other words, the gamma function is zero-free

	==External links==		==External links==

	* {{mathworld\|Gamma function}}		* {{mathworld\|Gamma function}}

Vipul: /* Euler integral formula */

2008-05-01T19:46:37Z

Euler integral formula

← Older revision		Revision as of 19:46, 1 May 2008
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	It is defined in a number of equivalent ways.		It is defined in a number of equivalent ways.

	===Euler integral formula===		===Euler's integral formula===

			{{further\|[[Euler's integral formula for gamma function]]}}
	This defines the gamma function in the [[right half-plane]] by the formula:		This defines the gamma function in the [[right half-plane]] by the formula:

Vipul at 01:31, 1 May 2008

2008-05-01T01:31:07Z

← Older revision		Revision as of 01:31, 1 May 2008
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	where <math>n</math> is an integer chosen such that <math>z+n</math> has positive real part.		where <math>n</math> is an integer chosen such that <math>z+n</math> has positive real part.

			==Facts==

			* [[Functional equation for gamma function]]
			* [[Reflection principle for gamma function]]
			* [[Reciprocal of gamma function is entire]]
	==External links==		==External links==

	* {{mathworld\|Gamma function}}		* {{mathworld\|Gamma function}}

Vipul: New page: ==Definition== The '''gamma function''' is a meromorphic function on $\mathbb{C}$ , with simple poles at all the non-positive integers, having residue at $-k$ ...

2008-05-01T01:29:44Z

New page: ==Definition== The '''gamma function''' is a meromorphic function on <math>\mathbb{C}</math>, with simple poles at all the non-positive integers, having residue at <math>-k</math>...

New page

==Definition==

The '''gamma function''' is a [[meromorphic function]] on <math>\mathbb{C}</math>, with [[simple pole]]s at all the non-positive integers, having residue at <math>-k</math> equal to:

<math>\frac{(-1)^k}{k!}</math>

It is defined in a number of equivalent ways.

===Euler integral formula===

This defines the gamma function in the [[right half-plane]] by the formula:

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

and extended analytically to <math>\mathbb{C}</math> by the prescription:

<math>\Gamma(z) = \frac{\Gamma(z+n)}{z(z+1)(z+2),\dots,(z+n-1)}</math>

where <math>n</math> is an integer chosen such that <math>z+n</math> has positive real part.

==External links==

* {{mathworld|Gamma function}}

Gamma function - Revision history

Vipul: 6 revisions

Vipul: /* Facts */

Vipul: /* Facts */

Vipul: /* Facts */

Vipul: /* Euler integral formula */

Vipul at 01:31, 1 May 2008

Vipul: New page: ==Definition== The '''gamma function''' is a meromorphic function on \mathbb{C}, with simple poles at all the non-positive integers, having residue at -k...

Vipul: New page: ==Definition== The '''gamma function''' is a meromorphic function on $\mathbb{C}$ , with simple poles at all the non-positive integers, having residue at $-k$ ...