Ln ⁡ Γ ⁡ ( z ) = ( z − 1 2 ) ⁢ ln ⁡ z − z + 1 2 ⁢ ln ⁡ ( 2 ⁢ π ) + 2 ⁢ ∫ 0 ∞ arctan ⁡ ( t / z ) e 2 ⁢ π ⁢ t − 1 d ⁣ t \operatorname {Ln} \Gamma \left(z\right)=\left(z-{\tfrac {1}{2}}\right)\ln z-z+{\tfrac {1}{2}}\ln \left(2\pi \right)+2\int _{0}^{\infty }{\frac {\operatorname {arctan} \left(t/z\right)}{e^{2\pi t}-1}}\mathrm {d} t