CylinderU and CylinderV are the parabolic cylinder functions. They satisfy the first real standard distinct form of the Parabolic Cylinder equation:

CylinderD and CylinderU are related in the following way:

$\mathrm{aa}≔\mathrm{CylinderU}\left(3\,0\right)$

$\mathrm{aa}≔\frac{22^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{\Gamma }\left(\frac{3}{4}\right)}{5\sqrt{\mathrm{\pi }}}$

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

$0.4650946536$

$\mathrm{CylinderU}\left(-\frac{5}{2}\,x\right)$

$\frac{{\ⅇ}^{-\frac{x^2}{4}}\mathrm{HermiteH}\left(2\,\frac{x\sqrt{2}}{2}\right)}{2}$

$\mathrm{CylinderD}\left(3.2\,1\right)$

$\mathrm{-1.819497238}$

$\mathrm{diff}\left(\mathrm{CylinderU}\left(a\,x\right)\,x\right)$

$-\frac{x\mathrm{CylinderU}\left(a\,x\right)}{2}-\left(a+\frac{1}{2}\right)\mathrm{CylinderU}\left(a+1\,x\right)$

$\mathrm{convert}\left(\mathrm{CylinderD}\left(\frac{3}{2}\,x\right)\,\mathrm{CylinderU}\right)$

$\mathrm{CylinderU}\left(\mathrm{-2}\,x\right)$

$\mathrm{convert}\left(\mathrm{CylinderU}\left(a\,x\right)+\mathrm{CylinderD}\left(b\,x\right)\,\mathrm{CylinderV}\right)$

$\frac{\mathrm{\pi }\left(\mathrm{CylinderV}\left(a\,-x\right)-\sin \left(a\mathrm{\pi }\right)\mathrm{CylinderV}\left(a\,x\right)\right)}{{\cos \left(a\mathrm{\pi }\right)}^2\mathrm{\Gamma }\left(a+\frac{1}{2}\right)}+\frac{\mathrm{\pi }\left(\mathrm{CylinderV}\left(-b-\frac{1}{2}\,-x\right)-\sin \left(\left(-b-\frac{1}{2}\right)\mathrm{\pi }\right)\mathrm{CylinderV}\left(-b-\frac{1}{2}\,x\right)\right)}{{\cos \left(\left(-b-\frac{1}{2}\right)\mathrm{\pi }\right)}^2\mathrm{\Gamma }\left(-b\right)}$

$\mathrm{series}\left(\mathrm{CylinderV}\left(0\,x\right)\,x\right)$

$\frac{2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{2\mathrm{\Gamma }\left(\frac{3}{4}\right)}+\frac{1}{2}\frac{2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{\Gamma }\left(\frac{3}{4}\right)}{\mathrm{\pi }}x+\frac{1}{96}\frac{2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}{x}^4+\frac{1}{160}\frac{2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{\Gamma }\left(\frac{3}{4}\right)}{\mathrm{\pi }}{x}^5+O\left(x^6\right)$