BesselJ and BesselY are the Bessel functions of the first and second kinds, respectively.  They satisfy Bessel's equation:

BesselI and BesselK are the modified Bessel functions of the first and second kinds, respectively.  They satisfy the modified Bessel equation:

HankelH1 and HankelH2 are the Hankel functions, also known as the Bessel functions of the third kind.  They also satisfy Bessel's equation, and are related to BesselJ and BesselY by

$\mathrm{BesselJ}\left(0\,2\right)$

$\mathrm{BesselJ}\left(0\,2\right)$

$\mathrm{evalf}\left(\right)$

$0.2238907791$

$\mathrm{BesselK}\left(1\,-3.\right)$

$\mathrm{-0.04015643113}-12.41987883\mathrm{I}$

$\mathrm{BesselI}\left(0\,0\right)$

$1$

$\mathrm{BesselY}\left(1.5+\mathrm{I}\,3.5-\mathrm{I}\right)$

$0.9566518512-1.465483431\mathrm{I}$

$\mathrm{series}\left(\mathrm{BesselJ}\left(3\,x\right)\,x\right)$

$\frac{1}{48}{x}^3-\frac{1}{768}{x}^5+O\left(x^7\right)$

$\mathrm{diff}\left(\mathrm{BesselJ}\left(v\,x\right)\,x\right)$

$-\mathrm{BesselJ}\left(v+1\,x\right)+\frac{v\mathrm{BesselJ}\left(v\,x\right)}{x}$

$\mathrm{HankelH1}\left(2.5\,3.7+\mathrm{I}\right)$

$0.1809260572-0.08706107529\mathrm{I}$

$\mathrm{diff}\left(\mathrm{HankelH2}\left(v\,x^2\right)\,x\right)$

$2\left(-\mathrm{HankelH2}\left(v+1\,x^2\right)+\frac{v\mathrm{HankelH2}\left(v\,x^2\right)}{x^2}\right)x$

$\mathrm{convert}\left(\mathrm{HankelH2}\left(v\,x\right)\,\mathrm{Bessel}\right)$

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

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

$-\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}$

$\mathrm{convert}\left(\mathrm{KelvinKer}\left(v\,x\right)\,\mathrm{BesselK}\right)$

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