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Gamma

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The gamma function,  is defined as:

 

 

The gamma function can be thought of as the extension of the factorial to the entire real (or complex) number set. For non-negative integers it is merely the familiar factorial function, , but offset by 1, i.e. . Like factorials, it satisfies the following recurrence relationship:

 

 

The Nematrian website approximates the gamma function using a so-called Lanczos approximation, see also Press et al. (2007), Toth (2004) or Wikipedia: Lanczos approximation.

 

 

The particular Lanczos approximation the Nematrian website uses involves:

 

 

For large  there is a risk of overflow, which can be mitigated by using the MnLogGamma function, defined as .

 

The Lanczos approximation is valid for arguments in the right complex half-plane, but can be extended to the entire complex plane (where the function is not singular) using the reflection formula, i.e.

 

 

See also MnCGamma.

 

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