Spouge's approximation

In mathematics, Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994 paper.[1] The formula is a modification of Stirling's approximation, and has the form

${\displaystyle \Gamma (z+1)=(z+a)^{z+{\frac {1}{2}}}e^{-z-a}\left(c_{0}+\sum _{k=1}^{a-1}{\frac {c_{k}}{z+k}}+\varepsilon _{a}(z)\right)}$

where a is an arbitrary positive integer and the coefficients are given by

${\displaystyle {\begin{aligned}c_{0}&={\sqrt {2\pi }}\\c_{k}&={\frac {(-1)^{k-1}}{(k-1)!}}(-k+a)^{k-{\frac {1}{2}}}e^{-k+a}\qquad k\in \{1,2,\dots ,a-1\}.\end{aligned}}}$

Spouge has proved that, if Re(z) > 0 and a > 2, the relative error in discarding ε_a(z) is bounded by

${\displaystyle a^{-{\frac {1}{2}}}(2\pi )^{-a-{\frac {1}{2}}}.}$

The formula is similar to the Lanczos approximation, but has some distinct features.[2] Whereas the Lanczos formula exhibits faster convergence, Spouge's coefficients are much easier to calculate and the error can be set arbitrarily low. The formula is therefore feasible for arbitrary-precision evaluation of the gamma function. However, special care must be taken to use sufficient precision when computing the sum due to the large size of the coefficients c_k, as well as their alternating sign. For example, for a = 49, one must compute the sum using about 65 decimal digits of precision in order to obtain the promised 40 decimal digits of accuracy.