Stirling's Formula For The Gamma Function
For all positive integers,
However, the Pi function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied. If then
Repeated integration by parts gives
where Bn is the nth Bernoulli number (note that the infinite sum is not convergent, so this formula is just an asymptotic expansion). The formula is valid for z large enough in absolute value when, where ε is positive, with an error term of when the first m terms are used. The corresponding approximation may now be written:
A further application of this asymptotic expansion is for complex argument z with constant . See for example the Stirling formula applied in of the Riemann-Siegel theta function on the straight line .
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