A simple proof of Lerch´s formula

In this blog entry we will present a simple proof for Lerch´s formula


\ln\left(\Gamma\left(x \right) \right)=\zeta^\prime(0,x)-\zeta^\prime(0)(1)


We rely on the previous established result (see proof here)


\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{x + t} dt =\left(x-\frac12\right)\ln x-x+\frac{\ln (2\pi)}{2}-\ln\left( \Gamma (x)\right)


Click here to see the proof of  (1).

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