Integral using the Hyperbolic Cosecant
Click on the image for a detailed proof
\zeta(s)=\frac{2^{s-1}}{\Gamma(s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^x}\cdot \mbox{csch}(x)dx
This integral connection between the Riemann Zeta function and the hyperbolic cosecant may seem unusual initially. What one discovers by looking at the proof is that this relation is simple to prove, only requiring a few passages. These straightforward equations are excellent tools for anyone looking to understand the intricacies of the Riemann zeta function.
Mind that the Relation to the Gamma Function 1 is an essential component of the demonstration.
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