Generalised Stirling approximations to N!

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Abstract
Generalised asymptotic approximations to Gamma (x+1), which contain an arbitrary parameter, are derived both from the integral representation of the gamma function without assuming a knowledge of the Stirling series, and through elementary rearrangements of the Stirling series. By optimising the arbitrary parameter according to appropriate criteria, several known Stirling-like approximations are recovered in a unifying way. These are as compact as but numerically superior to the standard Stirling approximation, and are meaningful on intervals that even include parts of the negative x-axis. It is pointed out that these results-arrived at by elementary but generally applicable asymptotic techniques-can be exploited in physics teaching to demonstrate the power and utility of asymptotic methods in the analysis of a variety of physics problems.