Stirling's Theorem - Stirling's Formula For The Gamma Function

Stirling's Formula For The Gamma Function

For all positive integers,

where Γ denotes the gamma function.

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 Re(z) > 0 then

Repeated integration by parts gives

where B_n is the n-th 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 |arg(z)| < π−ε, 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 Re(z). See for example the Stirling formula applied in Im(z) = t of the Riemann-Siegel theta function on the straight line 1/4 + it.