Stirling's Approximation - A Convergent Version of Stirling's Formula

A Convergent Version of Stirling's Formula

Thomas Bayes showed, in a letter to John Canton published by the Royal Society in 1763, that Stirling's formula did not give a convergent series.

Obtaining a convergent version of Stirling's formula entails evaluating

\int_0^\infty \frac{2\arctan (t/z)}{\exp(2 \pi t)-1}\,{\rm d}t = \ln\Gamma (z) - \left( z-\frac12 \right) \ln z +z - \frac12\ln(2\pi).

One way to do this is by means of a convergent series of inverted rising exponentials. If ; then

\int_0^\infty \frac{2\arctan (t/z)}{\exp(2 \pi t)-1} \,{\rm d}t = \sum_{n=1}^\infty \frac{c_n}{(z+1)^{\bar n}}

where

where s(n, k) denotes the Stirling numbers of the first kind. From this we obtain a version of Stirling's series

\begin{align} \ln \Gamma (z) & = \left( z-\frac12 \right) \ln z -z + \frac{\ln (2 \pi)}{2} \\ & {} + \frac{1}{12(z+1)} + \frac{1}{12(z+1)(z+2)} + \frac{59}{360(z+1)(z+2)(z+3)} \\ & {} + \frac{29}{60(z+1)(z+2)(z+3)(z+4)} + \cdots \end{align}

which converges when .

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