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
- Euler's integral formula for gamma function: This states that on the right half-plane:
- Recurrence relation for gamma function: This states that if is not a non-positive integer, we have:
- Reflection principle for gamma function: If is not an integer, we have:
- Integral formula for reciprocal of gamma function
- 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: