Editing Gamma function (section)

=== Properties ===
The [[Bohr–Mollerup theorem]] states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is [[log-convex]], that is, its [[natural logarithm]] is [[convex function|convex]] on the positive real axis. Another characterisation is given by the [[Wielandt theorem]].

The gamma function is the unique function that simultaneously satisfies
# <math>\Gamma(1) = 1</math>,
# <math>\Gamma(z+1) = z \Gamma(z)</math> for all complex numbers <math>z</math> except the non-positive integers, and,
# for integer {{mvar|n}}, <math display="inline">\lim_{n \to \infty} \frac{\Gamma(n+z)}{\Gamma(n)\;n^z} = 1</math> for all complex numbers <math>z</math>.<ref name="Davis" />

In a certain sense, the log-gamma function is the more natural form; it makes some intrinsic attributes of the function clearer. A striking example is the [[Taylor series]] of {{math|logΓ}} around 1:
<math display="block">\operatorname{log\Gamma}(z+1)= -\gamma z +\sum_{k=2}^\infty \frac{\zeta(k)}{k} \, (-z)^k \qquad \forall\; |z| < 1</math>
with {{math|''ζ''(''k'')}} denoting the [[Riemann zeta function]] at {{mvar|k}}.

So, using the following property:
<math display="block">\zeta(s) \Gamma(s) = \int_0^\infty \frac{t^s}{e^t-1} \, \frac{dt}{t}</math>
an integral representation for the log-gamma function is:
<math display="block">\operatorname{log\Gamma}(z+1)= -\gamma z + \int_0^\infty \frac{e^{-zt} - 1 + z t}{t \left(e^t - 1\right)} \, dt </math>
or, setting {{math|1=''z'' = 1}} to obtain an integral for {{math|''γ''}}, we can replace the {{math|''γ''}} term with its integral and incorporate that into the above formula, to get:
<math display="block">\operatorname{log\Gamma}(z+1)= \int_0^\infty \frac{e^{-zt} - ze^{-t} - 1 + z}{t \left(e^t -1\right)} \, dt\,. </math>

There also exist special formulas for the logarithm of the gamma function for rational {{mvar|z}}. 
For instance, if <math>k</math> and <math>n</math> are integers with <math>k<n</math> and <math>k\neq n/2 \,,</math> then<ref name="iaroslav_07">{{cite journal |last=Blagouchine |first=Iaroslav V. |year=2015 |title=A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations |journal=Journal of Number Theory |volume=148 |pages=537–592 |arxiv=1401.3724 |doi=10.1016/j.jnt.2014.08.009}}</ref>
<math display="block">
\begin{align}
\operatorname{log\Gamma} \left(\frac{k}{n}\right) = {} & \frac{\,(n-2k)\log2\pi\,}{2n} + \frac{1}{2}\left\{\,\log\pi-\log\sin\frac{\pi k}{n} \,\right\} + \frac{1}{\pi}\!\sum_{r=1}^{n-1}\frac{\,\gamma+\log r\,}{r}\cdot\sin\frac{\,2\pi r k\,}{n} \\
& {} - \frac{1}{2\pi}\sin\frac{2\pi k}{n}\cdot\!\int_0^\infty \!\!\frac{\,e^{-nx}\!\cdot\log x\,}{\,\cosh x -\cos( 2\pi k/n )\,}\,{\mathrm d}x.
\end{align}
</math>This formula is sometimes used for numerical computation, since the integrand decreases very quickly.