Gamma function: Difference between revisions

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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]]



Latest revision as of 19:13, 18 May 2008

Definition

The gamma function is a meromorphic function on , with simple poles at all the non-positive integers, having residue at equal to:

It is defined in a number of equivalent ways.

Euler's integral formula

Further information: Euler's integral formula for gamma function

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

and extended analytically to by the prescription:

where is an integer chosen such that has positive real part.

Facts

External links