Second Formula using the Digamma Function
\zeta(1+s)=\frac{\sin(\pi s)}{\pi s}\int_{0}^{\infty}\psi'(1+x)\frac{dx}{x^s}
\Downarrow
\zeta(1+s)=\frac{\sin(\pi s)}{\pi s}\int_{0}^{\infty}(\gamma+\psi(1+x))\frac{dx}{x^{s+1}}
This is a second formula connecting the Riemann Zeta to the Digamma Function, it is attributed to N.G. De Brujin, it was first published in the article “Integralen voor de zeta -functie van Riemann” in 1937.
In his demonstration he makes use of some properties of the Gamma Function, the Beta Function, Leibniz’s integral rule Theorem and Lebesgue’s dominated convergence Theorem.
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