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Author: Tinku Tara

e-st-J-t-dt-s-s-2-1-s-2-1-R-d-ds-e-st-J-t-dt-s-s-2-1-s-s-2-1-s-2-1-3-and-d-ds-e-st-J-t-dt-

Question Number 218043 by SdC355 last updated on 27/Mar/25 $$\int\:\:{e}^{−{st}} {J}_{\nu} \left({t}\right){dt}=\frac{\left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{−\nu} }{\:\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}}.\:\:\nu\in\mathbb{R}^{+} \\ $$$$\frac{\mathrm{d}\:\:}{\mathrm{d}{s}}\int\:\:{e}^{−{st}} {J}_{\nu} \left({t}\right){dt}=\frac{\left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{−\nu} \left({s}+\nu\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)}{\:\sqrt{\left({s}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }} \\…