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The Lanczos Approximation

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The Lanczos approximation is a method for computing the gamma function numerically, originally derived by Cornelius Lanczos in 1964 and involving the following formula:

 

 

Here  is an arbitrary constant subject to the restriction that  and  and  are as follows, where  is the ’th element of the Chebyshev polynomial coefficient matrix:

 

 

The series  converges. By choosing an appropriate , typically a small positive number, only a few terms are needed to calculate the gamma function to a high degree of accuracy. The series approximation can then be recast into the following form, with the  calculated in advance:

 

 

According to Wikipedia (2015), Lanczos derived the formula by deriving the following integral representation for the gamma function and then deriving a series expansion for the integral within this representation:

 

 


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