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 .

Read more about this topic: Stirling's Approximation

Famous quotes containing the words version, stirling and/or formula:

“Remember that you were a slave in the land of Egypt, and the LORD your God brought you out from there with a mighty hand and an outstretched arm; therefore the LORD your God commanded you to keep the sabbath day.”
—Bible: Hebrew, Deuteronomy 5:15.

See Exodus 22:8 for a different version of this fourth commandment.

“Oh, if thy pride did not our joys control,
What world of loving wonders shouldst thou see!
For if I saw thee once transformed in me,
Then in thy bosom I would pour my soul;”
—William Alexander, Earl O Stirling (1580?–1640)

“I cannot give you the formula for success, but I can give you the formula for failure—which is: Try to please everybody.”
—Herbert B. Swope (1882–1958)