$ℍ\left(s\right) \= \mathrm{convert}\left(\mathrm{hankel}\left(f\left(t\right)\,t\,s\,v\right)\,\mathrm{int}\right)$

$ℍ\left(s\right)={\int }_0^{\mathrm{\infty }}f\left(t\right)t{J}_v\left(st\right)\ⅆt$

$f\left(t\right)={\int }_0^{\mathrm{\infty }}ℍ\left(s\right)s J_v\left(st\right)\ⅆs$

$f\left(t\right)={\int }_0^{\mathrm{\infty }}ℍ\left(s\right)s{J}_v\left(st\right)\ⅆs$

$\mathrm{inttrans}:-\mathrm{setup}\left(\mathrm{alternativehankeldefinition}\right)$

$\mathrm{alternativehankeldefinition}=\mathrm{false}$

$\mathrm{inttrans}:-\mathrm{setup}\left(\mathrm{alternativehankeldefinition} \= \mathrm{true}\right)$

$\mathrm{alternativehankeldefinition}=\mathrm{true}$

$\mathrm{convert}\left(\mathrm{hankel}\left(g\left(t\right)\,t\,s\,v\right)\,\mathrm{int}\right)$

${\int }_0^{\mathrm{\infty }}g\left(t\right)\sqrt{st}{J}_v\left(st\right)\ⅆt$

$\mathrm{inttrans}:-\mathrm{setup}\left(\mathrm{alternativehankeldefinition} \= \mathrm{false}\right)$

$\mathrm{alternativehankeldefinition}=\mathrm{false}$

$\mathrm{hankel}\left(\frac{1}{\mathrm{\α}\+t}\,t\,s\,0\right)$

$\frac{\mathrm{\pi }{I}_0\left(\mathrm{\alpha }s\right)}{2\mathrm{\alpha }}$

$\mathrm{hankel}\left(\exp \left(-\frac{a^2\cdot t^2}{2}\right)\,t\,s\,0\right)\;$

$\frac{{\ⅇ}^{-\frac{s^2}{2a^2}}}{a^2}$

$\mathrm{hankel}\left(\frac{\log \left(t\right)}{t}\,t\,s\,0\right)\;$

$-\frac{\ln \left(2\mathrm{\gamma }s\right)}{s}$

$\mathrm{hankel}\left(\frac{\sin \left(a\cdot t\right)}{t}\,t\,s\,1\right)\;$

$\frac{\mathrm{\theta }\left(s-a\right)a}{s\sqrt{-a^2+s^2}}$

$\mathrm{hankel}\left(\frac{\cos \left(a\cdot t\right)}{t}\,t\,s\,0\right)\;$

$\frac{\mathrm{\theta }\left(s-a\right)}{\sqrt{-a^2+s^2}}$

$\mathrm{hankel}\left(\frac{\arctan \left(t^2\right)}{t}\,t\,s\,1\right)\;$

$-2{\mathrm{kei}}_0\left(s\right)$

$\mathrm{hankel}\left(\frac{\exp \left(-a\cdot t\right)\cdot \sinh \left(b\cdot t\right)}{t}\,t\,s\,0\right) assuming a\ge b$

$\frac{\sqrt{abs}\sqrt{\sqrt{a^2-2ab+b^2+s^2}-\sqrt{a^2+2ab+b^2+s^2}}}{\sqrt{s}\sqrt{a^2-2ab+b^2+s^2}\sqrt{a^2+2ab+b^2+s^2}\sqrt{\sqrt{a^2-2ab+b^2+s^2}+\sqrt{a^2+2ab+b^2+s^2}}}$

$\mathrm{hankel}\left(\mathrm{Ci}\left(\mathrm{\α}{t}^2\right)\,t\,s\,0\right)$

$-\frac{2\left(-1+\cos \left(\frac{s^2}{4\mathrm{\alpha }}\right)\right)}{s^2}$

$\mathrm{hankel}\left(\mathrm{Ssi}\left(\mathrm{\α}{t}^2\right)\,t\,s\,0\right)$

$-\frac{2\sin \left(\frac{s^2}{4\mathrm{\alpha }}\right)}{s^2}$

$\mathrm{hankel}\left(\mathrm{Ei}\left(\mathrm{\β}{t}^2\right)\,t\,s\,0\right)$

$\frac{2\left(-1+{\ⅇ}^{\frac{s^2}{4\mathrm{\beta }}}\right)}{s^2}$

$\mathrm{hankel}\left(\frac{\mathrm{erf}\left(\mathrm{\α}t\right)}{t}\,t\,s\,0\right)$

$\frac{\mathrm{erfc}\left(\frac{s}{2\mathrm{\alpha }}\right)}{s}$

$\mathrm{hankel}\left(\frac{1\mathrm{HankelH1}\left(\mathrm{\&mu;}\&comma;\frac{\mathrm{\&alpha;}}{t}\right)\mathrm{HankelH2}\left(\mathrm{\&mu;}\&comma;\frac{\mathrm{\&alpha;}}{t}\right)}{t}\&comma;t\&comma;s\&comma;0\right) assuming -\frac{1}{4}<\mathrm{\&mu;}<\frac{1}{4}$

$\frac{16\cos \left(\mathrm{\pi }\mathrm{\mu }\right)K_{2\mathrm{\mu }}\left(\left(1+\mathrm{I}\right)\sqrt{\mathrm{\alpha }}\sqrt{s}\sqrt{2}\right)K_{2\mathrm{\mu }}\left(\left(1-\mathrm{I}\right)\sqrt{\mathrm{\alpha }}\sqrt{s}\sqrt{2}\right)}{s{\mathrm{\pi }}^2}$

$\mathrm{hankel}\left(\frac{\mathrm{BesselJ}\left(0\&comma;\mathrm{\&beta;}t\right)}{\mathrm{\&alpha;}\&plus;t^2}\&comma;t\&comma;s\&comma;0\right)$

$\mathrm{\theta }\left(\mathrm{\beta }-s\right)K_0\left(\sqrt{\mathrm{\alpha }}\mathrm{\beta }\right)I_0\left(\sqrt{\mathrm{\alpha }}s\right)+\mathrm{\theta }\left(s-\mathrm{\beta }\right)I_0\left(\sqrt{\mathrm{\alpha }}\mathrm{\beta }\right)K_0\left(\sqrt{\mathrm{\alpha }}s\right)$

$\mathrm{hankel}\left(\frac{\mathrm{BesselJ}\left(v\&comma;\mathrm{\&beta;}t\right)}{\mathrm{\&alpha;}\&plus;t^2}\&comma;t\&comma;s\&comma;v\right)assuming v \&gt; -\frac{1}{2}$

$\mathrm{\theta }\left(\mathrm{\beta }-s\right)K_v\left(\sqrt{\mathrm{\alpha }}\mathrm{\beta }\right)I_v\left(\sqrt{\mathrm{\alpha }}s\right)+\mathrm{\theta }\left(s-\mathrm{\beta }\right)I_v\left(\sqrt{\mathrm{\alpha }}\mathrm{\beta }\right)K_v\left(\sqrt{\mathrm{\alpha }}s\right)$

$\mathrm{hankel}\left(\frac{\mathrm{BesselY}\left(0\&comma;\frac{\mathrm{\&alpha;}}{t}\right)}{t}\&comma;t\&comma;s\&comma;0\right)$

$\frac{\mathrm{\pi }{Y}_0\left(2\sqrt{\mathrm{\alpha }}\sqrt{s}\right)-2K_0\left(2\sqrt{\mathrm{\alpha }}\sqrt{s}\right)}{s\mathrm{\pi }}$

$\mathrm{hankel}\left(\frac{\mathrm{BesselY}\left(\mathrm{\&mu;}\&comma;\frac{\mathrm{\&alpha;}}{t}\right)}{t}\&comma;t\&comma;s\&comma;\mathrm{\&mu;}\right) assuming -\frac{1}{2}<\mathrm{\mu }<\frac{1}{2}$

$\frac{\mathrm{\pi }{Y}_{2\mathrm{\mu }}\left(2\sqrt{\mathrm{\alpha }}\sqrt{s}\right)-2K_{2\mathrm{\mu }}\left(2\sqrt{\mathrm{\alpha }}\sqrt{s}\right)}{s\mathrm{\pi }}$

$\mathrm{hankel}\left(\mathrm{BesselK}\left(0\&comma;\mathrm{\&beta;}t\right)\&comma;t\&comma;s\&comma;0\right)$

$\frac{1}{{\mathrm{\beta }}^2+s^2}$

$\mathrm{hankel}\left(\mathrm{BesselK}\left(v\&comma;\mathrm{\&beta;}t\right)\&comma;t\&comma;s\&comma;v\right)$

$\frac{{\mathrm{\beta }}^{-v}{s}^v}{{\mathrm{\beta }}^2+s^2}$

$\mathrm{hankel}\left({\&ExponentialE;}^{\mathrm{\&alpha;}{t}^2}\mathrm{BesselI}\left(0\&comma;\mathrm{\&beta;}t\right)\&comma;t\&comma;s\&comma;0\right)$

$-\frac{{\&ExponentialE;}^{-\frac{{\mathrm{\beta }}^2-s^2}{4\mathrm{\alpha }}}{J}_0\left(\frac{\mathrm{\beta }s}{2\mathrm{\alpha }}\right)}{2\mathrm{\alpha }}$

$\mathrm{hankel}\left({\&ExponentialE;}^{\mathrm{\&alpha;}{t}^2}\mathrm{BesselI}\left(v\&comma;\mathrm{\&beta;}t\right)\&comma;t\&comma;s\&comma;v\right)$

$\mathrm{hankel}\left({\&ExponentialE;}^{\mathrm{\alpha }{t}^2}{I}_v\left(\mathrm{\beta }t\right)\&comma;t\&comma;s\&comma;v\right)$