In 1952, Jitsuro Nagura published a classic proof that shows that for $n \ge 25$, there is always a prime between $n$ and $\frac{6n}{5}$.
For those interested the paper itself can be found here.
I'm going through this short proof and I am immediately unclear on the first point. I would greatly appreciate it if someone can explain the details of the equation below:
$\frac{\Gamma'}{\Gamma}(s) = \int_{0}^∞(\frac{e^{-t}}{t} - \frac{e^{-st}}{1-e^{-t}})dt$ when $s > 0$
From the Wikipedia article on the gamma function is:
$\Gamma(s) = \int_{0}^∞t^{s-1}e^{-t}dt$
and also from Wikipedia, the polygamma function is:
$\frac{\Gamma'}{\Gamma}(s) = \psi(s) = \int_0^∞\frac{te^{-st}}{1-e^{-t}}dt$
I am interested in understanding how Nagura arrived at this first expression in the proof.
The identity in Nagura's paper is in section 6.3.21 of the treatise by Abramowitz and Stegun, and the derivative of a formula in section 6.1.50.