Integral representation of the Digamma Function
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\psi(s):=\frac{\Gamma'(s)}{\Gamma(s)}=\int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-st}}{1-e^{-t}}\right)\mathrm{d}t
The digamma Function is the Logarithmic Derivative of the Gamma Function, but it’s used so often in number theory that a proper name and studies are required.
Noteworthy are also its connections to the Riemann Zeta Function: First Formula using the Digamma Function, Second Formula Using the Digamma Function and Third Formula using the Digamma Function.
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