quality:corruption, structural:reformatting, structural:scholar_edition
Integration over log-gamma

The integral ∫ 0 z l o g Γ (x) d x {\displaystyle \int _{0}^{z}\operatorname {log\Gamma } (x)\,dx} can be expressed in terms of the Barnes G -function (see Barnes G -function for a proof): ∫ 0 z l o g Γ (x) d x = z 2 log (2 π) + z (1 − z) 2 + z l o g Γ (z) − log G (z + 1) {\displaystyle \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)} where Re(z) > −1.

It can also be written in terms of the Hurwitz zeta function: ∫ 0 z l o g Γ (x) d x = z 2 log (2 π) + z (1 − z) 2 − ζ ′ (− 1) + ζ ′ (− 1, z). {\displaystyle \int _{0}^{z}\operatorname {log\Gamma } (x)\,dx={\frac {z}{2}}\log(2\pi)+{\frac {z(1-z)}{2}}-\zeta '(-1)+\zeta '(-1,z).}

When z = 1 {\displaystyle z=1} it follows that ∫ 0 1 l o g Γ (x) d x = 1 2 log (2 π), {\displaystyle \int _{0}^{1}\operatorname {log\Gamma } (x)\,dx={\frac {1}{2}}\log(2\pi),} 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 l o g Γ {\displaystyle \operatorname {log\Gamma } }: ∫ 0 1 log 2 Γ (x) d x = γ 2 12 + π 2 48 + 1 3 γ L 1 + 4 3 L 1 2 − (γ + 2 L 1) ζ ′ (2) π 2 + ζ ′ ′ (2) 2 π 2, {\displaystyle \int _{0}^{1}\log ^{2}\Gamma (x)dx={\frac {\gamma ^{2}}{12}}+{\frac {\pi ^{2}}{48}}+{\frac {1}{3}}\gamma L_{1}+{\frac {4}{3}}L_{1}^{2}-\left(\gamma +2L_{1}\right){\frac {\zeta ^{\prime }(2)}{\pi ^{2}}}+{\frac {\zeta ^{\prime \prime }(2)}{2\pi ^{2}}},} where L 1 {\displaystyle L_{1}} is 1 2 log (2 π) {\displaystyle {\frac {1}{2}}\log(2\pi)}.

D. H. Bailey and his co-authors gave an evaluation for L n:= ∫ 0 1 log n Γ (x) d x {\displaystyle L_{n}:=\int _{0}^{1}\log ^{n}\Gamma (x)\,dx} when n = 1, 2 {\displaystyle n=1,2} in terms of the Tornheim–Witten zeta function and its derivatives.

In addition, it is also known that lim n → ∞ L n n ! = 1. {\displaystyle \lim _{n\to \infty }{\frac {L_{n}}{n!}}=1.}