Editing Stirling's approximation (section)

==Stirling's formula for the gamma function==
For all positive integers,
<math display=block>n! = \Gamma(n + 1),</math>
where {{math|Γ}} denotes the [[gamma function]].

However, the gamma 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 {{math|Re(''z'') > 0}}, then
<math display=block>\ln\Gamma (z) = z\ln z - z + \tfrac12\ln\frac{2\pi}{z} + \int_0^\infty\frac{2\arctan\left(\frac{t}{z}\right)}{e^{2\pi t}-1}\,{\rm d}t.</math>

Repeated integration by parts gives
<math display=block>\begin{align}
\ln\Gamma(z) \sim z\ln z - z + \tfrac12\ln\frac{2\pi}{z}
+ \sum_{n=1}^{N-1} \frac{B_{2n}}{2n(2n-1)z^{2n-1}} \\
= z\ln z - z + \tfrac12\ln\frac{2\pi}{z}
+\frac{1}{12z}
-\frac{1}{360z^3}
+\frac{1}{1260z^5}+\dots
,\end{align}</math>

where <math>B_n</math> is the <math>n</math>th [[Bernoulli number]] (note that the limit of the sum as <math>N \to \infty</math> is not convergent, so this formula is just an [[asymptotic expansion]]). The formula is valid for <math>z</math> large enough in absolute value, when {{math|{{abs|arg(''z'')}} < π − ''ε''}}, where {{mvar|ε}} is positive, with an error term of {{math|''O''(''z''<sup>−2''N''+ 1</sup>)}}. The corresponding approximation may now be written:
<math display=block>\Gamma(z) = \sqrt{\frac{2\pi}{z}}\,{\left(\frac{z}{e}\right)}^z \left(1 + O\left(\frac{1}{z}\right)\right).</math>

where the expansion is identical to that of Stirling's series above for <math>n!</math>, except that <math>n</math> is replaced with {{math|''z'' − 1}}.{{r|spiegel}}

A further application of this asymptotic expansion is for complex argument {{mvar|z}} with constant {{math|Re(''z'')}}. See for example the Stirling formula applied in {{math|Im(''z'') {{=}} ''t''}} of the [[Riemann–Siegel theta function]] on the straight line {{math|{{sfrac|1|4}} + ''it''}}.