Two Fractional part Integrals related to ln (Γ(z))

In this post we will present proofs for the follwoing two integrals related to the LoGamma function:



\begin{align*} \int_1^\infty \frac{x-\lfloor x\rfloor-1/2}{x}\,dx=\frac{\ln(2 \pi)}{2} -1 \\ \end{align*}


\begin{align*} \int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt=\left(z-\frac12\right)\ln z-z+\frac{\ln (2\pi)}{2}-\ln\left( \Gamma (z)\right) \label{2} \end{align*}


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