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: