Laplace Transform Csch(x) (1/Sinh(x))

Question

I need to find the Laplace Transform of $Csch(x)=\frac{1}{\sinh(x)}$. Wolfram Alpha and Mathematica say $-H(\frac{s-1}{2})$, where H(n) is the $n$-th Harmonic Number. I hope someone have a nice hint to proof this.

Thank You

Answer 1

The integral for the Laplace transform for this function does not converge: for positive $s$, $$ \int_\delta^1\frac{2e^{-sx}}{e^x-e^{-x}}\,dx\geq e^{-s}\int_\delta^1\frac1{x}\,dx=e^{-s}\log\frac1\delta. $$ So as $\delta\to0$, we see the integral diverge.

I'm assuming that Mathematica uses some relation with another transform, where the other transform does converge.