In Table of integrals, series and product by Gradshteyn and Ryzhlik I find the following identity, which I should use in my calculation:
$\int\limits_{0}^{\infty}\sin\left(z \cdot\mathrm{ch}x\right)dx=\frac {\pi}{2}J_0(z), \ \ \Re(z)>0$
How to prove it ? I tried check that this integral satisfies differential equation for Bessel function, but there is a problem with convergence of integrals which are obtained by differentiation of given integral.