Question Number 220652 by SdC355 last updated on 17/May/25
$$\mathrm{Show}\:\mathrm{that} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\frac{{e}^{−{st}} }{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}\:\mathrm{d}{t}=\frac{\mathrm{1}}{\mathrm{2}}\pi\left(\boldsymbol{\mathrm{H}}_{\mathrm{0}} ^{\:} \left({s}\right)−{Y}_{\mathrm{0}} \left({s}\right)\right)\:,\:{s}\in\mathbb{R}\backslash\left\{\mathrm{0}\right\} \\ $$