Formelsammlung Mathematik: Bestimmte Integrale: Form R(x,log,cos,Gamma)

1.1

\int _{0}^{1}\log \Gamma (x)\,\cos(2n\pi x)\,dx={\frac {1}{4n}}\qquad n\in \mathbb {Z} ^{>0}

Beweis

$I:=\int _{0}^{1}\log \Gamma (x)\,\cos(2n\pi x)\,dx=\int _{0}^{1}\log \Gamma (1-x)\,\cos(2n\pi x)\,dx$

$\Rightarrow \,2I=\int _{0}^{1}\log {\Big (}\Gamma (x)\Gamma (1-x){\Big )}\,\cos(2n\pi x)\,dx=\int _{0}^{1}\log \left({\frac {\pi }{\sin \pi x}}\right)\cos(2n\pi x)\,dx$

$\log(2\pi )\underbrace {\int _{0}^{1}\cos(2n\pi x)\,dx} _{=0}+\int _{0}^{1}-\log {\big (}2\sin \pi x{\big )}\,\cos(2n\pi x)\,dx$ , wobei $-\log {\big (}2\sin \pi x{\big )}=\sum _{k=1}^{\infty }{\frac {\cos 2k\pi x}{k}}$ ist.

$\Rightarrow 2I=\sum _{k=1}^{\infty }{\frac {1}{k}}\underbrace {\int _{0}^{1}\cos(2k\pi x)\,\cos(2n\pi x)\,dx} _{={\frac {1}{2}}\,\delta _{nk}}={\frac {1}{2n}}$ .