convert/Bessel converts Airy wave functions, their derivatives, and Bessel functions in an expression to the Bessel functions.

The option Bessel will give the simplest conversion. The other options will force a conversion to an expression in that function, if it is possible.

$\mathrm{convert}\left(\mathrm{AiryAi}\left(x\right)\,\mathrm{Bessel}\right)$

$-\frac{x\mathrm{BesselI}\left(\frac{1}{3}\,\frac{2\sqrt{x^3}}{3}\right)}{3{\left(x^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}+\frac{{\left(x^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\mathrm{BesselI}\left(-\frac{1}{3}\,\frac{2\sqrt{x^3}}{3}\right)}{3}$

$\mathrm{convert}\left(-\mathrm{AiryBi}\left(\mathrm{sqrt}\left(x\right)\right)\,\mathrm{Bessel}\right)$

$-\frac{\sqrt{3}\left(\sqrt{x}\mathrm{BesselI}\left(\frac{1}{3}\,\frac{2\sqrt{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{3}\right)+\mathrm{BesselI}\left(-\frac{1}{3}\,\frac{2\sqrt{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{3}\right){\left(x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}{3{\left(x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$

$\mathrm{convert}\left(\mathrm{AiryAi}\left(1\,x\right)\,\mathrm{BesselI}\right)$

$\frac{x^2\mathrm{BesselI}\left(\frac{2}{3}\,\frac{2\sqrt{x^3}}{3}\right)}{3{\left(x^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}-\frac{{\left(x^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{BesselI}\left(-\frac{2}{3}\,\frac{2\sqrt{x^3}}{3}\right)}{3}$

$\mathrm{convert}\left(\sin \left(\mathrm{HankelH1}\left(v\,x\right)\right)\,\mathrm{Bessel}\right)$

$\sin \left(\mathrm{BesselJ}\left(v\,x\right)+\mathrm{I}\mathrm{BesselY}\left(v\,x\right)\right)$

$\mathrm{convert}\left(\mathrm{BesselJ}\left(v\,2\right)\,\mathrm{BesselY}\right)$

$\frac{\mathrm{BesselY}\left(-v\,2\right)-\mathrm{BesselY}\left(v\,2\right)\cos \left(v\mathrm{\pi }\right)}{\sin \left(v\mathrm{\pi }\right)}$

$\mathrm{convert}\left(\mathrm{KelvinKei}\left(v\,z^2\right)\,\mathrm{BesselK}\right)$

$\frac{\frac{\mathrm{I}}{2}\left({\left({\ⅇ}^{\frac{\mathrm{I}}{2}v\mathrm{\pi }}\right)}^2\mathrm{BesselK}\left(v\,\left(\frac{1}{2}-\frac{\mathrm{I}}{2}\right)z^2\sqrt{2}\right)-\mathrm{BesselK}\left(v\,\left(\frac{1}{2}+\frac{\mathrm{I}}{2}\right)z^2\sqrt{2}\right)\right)}{{\ⅇ}^{\frac{\mathrm{I}}{2}v\mathrm{\pi }}}$