TITLE: Understanding the gamma function in the context of Jitsuro Nagura's Proof
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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.