Editing Gamma function (section)

=== Integration over log-gamma ===

The integral
<math display="block"> \int_0^z \operatorname{log\Gamma} (x) \, dx</math>
can be expressed in terms of the [[Barnes G-function|Barnes {{math|''G''}}-function]]<ref name="Alexejewsky">{{cite journal|first=W. P. |last=Alexejewsky |title=Über eine Classe von Funktionen, die der Gammafunktion analog sind |trans-title=On a class of functions analogous to the gamma function |journal=Leipzig Weidmannsche Buchhandlung |volume=46 |date=1894 |pages=268–275}}</ref><ref name="Barnes">{{cite journal|first=E. W. |last=Barnes |title=The theory of the ''G''-function |journal=Quart. J. Math. |volume=31 |date=1899 |pages=264–314}}</ref> (see [[Barnes G-function|Barnes {{math|''G''}}-function]] for a proof):
<math display="block">\int_0^z \operatorname{log\Gamma}(x) \, dx = \frac{z}{2} \log (2 \pi) + \frac{z(1-z)}{2} + z \operatorname{log\Gamma}(z) - \log G(z+1)</math>
where {{math|Re(''z'') > −1}}.

It can also be written in terms of the [[Hurwitz zeta function]]:<ref name="Adamchik">{{cite journal|first=Victor S. |last=Adamchik |title=Polygamma functions of negative order |journal=J. Comput. Appl. Math. |volume=100 |issue=2 |date=1998 |pages=191–199 |doi=10.1016/S0377-0427(98)00192-7|doi-access=free }}</ref><ref name="Gosper">{{cite journal|first=R. W. |last=Gosper |title=<math>\textstyle \int_{n/4}^{m/6} \log F(z) \,dz</math> in special functions, ''q''-series and related topics |journal=J. Am. Math. Soc. |volume=14 |date=1997}}</ref>
<math display="block">\int_0^z \operatorname{log\Gamma}(x) \, dx = \frac{z}{2} \log(2 \pi) + \frac{z(1-z)}{2} - \zeta'(-1) + \zeta'(-1,z) .</math>

When <math>z=1</math> it follows that
<math display="block"> \int_0^1 \operatorname{log\Gamma}(x) \, dx = \frac 1 2 \log(2\pi), </math>
and this is a consequence of [[Raabe's formula]] as well. O. Espinosa and V. Moll derived a similar formula for the integral of the square of <math>\operatorname{log\Gamma}</math>:<ref name="EspinosaMoll">{{cite journal|first1=Olivier |last1=Espinosa|first2=Victor H. |last2=Moll|title= On Some Integrals Involving the Hurwitz Zeta Function: Part 1|journal=The Ramanujan Journal |volume=6 |date=2002 |issue=2|pages=159–188 |doi=10.1023/A:1015706300169|s2cid=128246166}}</ref>
<math display="block">\int_{0}^{1} \log ^{2} \Gamma(x) d x=\frac{\gamma^{2}}{12}+\frac{\pi^{2}}{48}+\frac{1}{3} \gamma L_{1}+\frac{4}{3} L_{1}^{2}-\left(\gamma+2 L_{1}\right) \frac{\zeta^{\prime}(2)}{\pi^{2}}+\frac{\zeta^{\prime \prime}(2)}{2 \pi^{2}},</math>
where <math>L_1</math> is <math>\frac12\log(2\pi)</math>.

D. H. Bailey and his co-authors<ref name="Bailey">{{cite journal|first1=David H. |last1=Bailey|first2=David |last2=Borwein|first3=Jonathan M.|last3=Borwein|title= On Eulerian log-gamma integrals and Tornheim-Witten zeta functions|journal=The Ramanujan Journal |volume=36 |date=2015 |issue=1–2|pages=43–68 |doi=10.1007/s11139-012-9427-1|s2cid=7335291}}</ref> gave an evaluation for
<math display="block">L_n:=\int_0^1 \log^n \Gamma(x) \, dx</math>
when <math>n=1,2</math> in terms of the Tornheim–Witten zeta function and its derivatives.

In addition, it is also known that<ref name="ACEKNM">{{cite journal|first1=T. |last1=Amdeberhan|first2=Mark W.|last2=Coffey|first3=Olivier|last3=Espinosa|first4=Christoph|last4=Koutschan|first5=Dante V.|last5=Manna|first6=Victor H.|last6=Moll|title= Integrals of powers of loggamma|journal=Proc. Amer. Math. Soc.|volume=139|issue=2 |date=2011 |pages=535–545 |doi=10.1090/S0002-9939-2010-10589-0|doi-access=free}}</ref>
<math display="block">
\lim_{n\to\infty} \frac{L_n}{n!}=1.
</math>