Spouge's approximation

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In mathematics, Spouge's approximation is a formula for the gamma function due to John L. Spouge. The formula is a modification of Stirling's approximation, and has the form

$\Gamma(z+1) = (z+a)^{z+1/2} e^{-(z+a)} \left[ c_0 + \sum_{k=1}^{a-1} \frac{c_k}{z+k} + \epsilon_a(z) \right]$

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

$c_0 = \sqrt{2 \pi}\,$

$c_k = \frac{(-1)^{k-1}}{(k-1)!} (-k+a)^{k-1/2} e^{-k+a} \quad k=1,2,\dots, a-1.$

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

$\epsilon_a(z) \le a^{-1/2} (2 \pi)^{-(a+1/2)}.$

The formula is similar to the Lanczos approximation, but has some distinct features. 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.