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Function Description

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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