Cylinder - Maple Help

convert/Cylinder

convert special functions admitting 1F1 or 0F1 hypergeometric representation into Cylinder functions

 Calling Sequence convert(expr, Cylinder)

Parameters

 expr - Maple expression, equation, or set or list of them

Description

 • convert/Cylinder converts, when possible, special functions admitting a 1F1 or 0F1 hypergeometric representation into Cylinder functions. The Cylinder functions are
 The 3 functions in the "Cylinder" class are:
 $\left[{\mathrm{CylinderD}}{,}{\mathrm{CylinderU}}{,}{\mathrm{CylinderV}}\right]$ (1)

Examples

 > $\mathrm{HermiteH}\left(a,z\right)\mathrm{LaguerreL}\left(\frac{1}{2}a,-\frac{1}{2},\frac{1}{2}{z}^{2}\right)$
 ${\mathrm{HermiteH}}{}\left({a}{,}{z}\right){}{\mathrm{LaguerreL}}{}\left(\frac{{a}}{{2}}{,}{-}\frac{{1}}{{2}}{,}\frac{{{z}}^{{2}}}{{2}}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Cylinder}\right)$
 $\frac{{\mathrm{CylinderD}}{}\left({a}{,}{z}{}\sqrt{{2}}\right){}{{ⅇ}}^{\frac{{{z}}^{{2}}}{{2}}}{}\left(\genfrac{}{}{0}{}{\frac{{a}}{{2}}{-}\frac{{1}}{{2}}}{\frac{{a}}{{2}}}\right){}{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{-}\frac{{a}}{{2}}\right){}{\mathrm{CylinderD}}{}\left({a}{,}{z}\right)}{{2}{}\sqrt{{\mathrm{\pi }}}}{+}\frac{{\mathrm{CylinderD}}{}\left({a}{,}{z}{}\sqrt{{2}}\right){}{{ⅇ}}^{\frac{{{z}}^{{2}}}{{2}}}{}\left(\genfrac{}{}{0}{}{\frac{{a}}{{2}}{-}\frac{{1}}{{2}}}{\frac{{a}}{{2}}}\right){}{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{-}\frac{{a}}{{2}}\right){}{\mathrm{CylinderD}}{}\left({a}{,}{-}{z}\right)}{{2}{}\sqrt{{\mathrm{\pi }}}}$ (3)
 > $\mathrm{erfc}\left(a,z\right)$
 ${\mathrm{erfc}}{}\left({a}{,}{z}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{Cylinder}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a::\mathrm{posint}$
 $\frac{{\mathrm{CylinderD}}{}\left({-}{1}{-}{a}{,}{z}{}\sqrt{{2}}\right){}\sqrt{{2}}}{{{ⅇ}}^{\frac{{{z}}^{{2}}}{{2}}}{}\sqrt{{{2}}^{{a}}{}{\mathrm{\pi }}}}$ (5)
 > $\frac{2{\mathrm{\pi }}^{\frac{1}{2}}z}{{2}^{\frac{1}{2}a-\frac{1}{4}}\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{1}{4}\right)}\mathrm{hypergeom}\left(\left[\frac{1}{2}a+\frac{3}{4}\right],\left[\frac{3}{2}\right],\frac{1}{2}{z}^{2}\right)+\frac{{\mathrm{\pi }}^{\frac{1}{2}}\mathrm{hypergeom}\left(\left[\frac{1}{2}a+\frac{1}{4}\right],\left[\frac{1}{2}\right],\frac{1}{2}{z}^{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{3}{4}\right){2}^{\frac{1}{2}a-\frac{3}{4}}}$
 $\frac{{2}{}\sqrt{{\mathrm{\pi }}}{}{z}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{a}}{{2}}{+}\frac{{3}}{{4}}\right]{,}\left[\frac{{3}}{{2}}\right]{,}\frac{{{z}}^{{2}}}{{2}}\right)}{{{2}}^{\frac{{a}}{{2}}{-}\frac{{1}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{a}}{{2}}{+}\frac{{1}}{{4}}\right)}{+}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{a}}{{2}}{+}\frac{{1}}{{4}}\right]{,}\left[\frac{{1}}{{2}}\right]{,}\frac{{{z}}^{{2}}}{{2}}\right)}{{\mathrm{\Gamma }}{}\left(\frac{{a}}{{2}}{+}\frac{{3}}{{4}}\right){}{{2}}^{\frac{{a}}{{2}}{-}\frac{{3}}{{4}}}}$ (6)
 > $\mathrm{convert}\left(,\mathrm{Cylinder}\right)$
 $\frac{{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}\left({{2}}^{\frac{{a}}{{2}}{+}\frac{{1}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{-}\frac{{1}}{{4}}}{-}{{2}}^{\frac{{a}}{{2}}{+}\frac{{3}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{-}\frac{{3}}{{4}}}\right){}{\mathrm{CylinderD}}{}\left({-}{a}{-}\frac{{1}}{{2}}{,}{z}\right)}{{2}{}{{2}}^{\frac{{a}}{{2}}{-}\frac{{1}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{-}\frac{{3}}{{4}}}}{+}\frac{{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}\left({{2}}^{\frac{{a}}{{2}}{+}\frac{{1}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{-}\frac{{1}}{{4}}}{+}{{2}}^{\frac{{a}}{{2}}{+}\frac{{3}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{-}\frac{{3}}{{4}}}\right){}{\mathrm{CylinderD}}{}\left({-}{a}{-}\frac{{1}}{{2}}{,}{-}{z}\right)}{{2}{}{{2}}^{\frac{{a}}{{2}}{-}\frac{{1}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{-}\frac{{3}}{{4}}}}$ (7)
 > $\mathrm{collect}\left(,\mathrm{CylinderD},\mathrm{simplify}\right)$
 ${2}{}{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}{\mathrm{CylinderD}}{}\left({-}{a}{-}\frac{{1}}{{2}}{,}{-}{z}\right)$ (8)

When converting to a function class (e.g. Cylinder) it is possible to request additional conversion rules to be performed. Compare for instance these two different outputs:

 > $\mathrm{MeijerG}\left(\left[\left[\frac{1}{4}-\frac{1}{2}a\right],\left[\right]\right],\left[\left[0\right],\left[-\frac{1}{2}\right]\right],-\frac{1}{2}{z}^{2}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[\frac{{1}}{{4}}{-}\frac{{a}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{-}\frac{{1}}{{2}}\right]\right]{,}{-}\frac{{{z}}^{{2}}}{{2}}\right)$ (9)
 > $\mathrm{convert}\left(,\mathrm{Cylinder}\right)$
 ${-}\frac{{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{+}\frac{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{a}\right){}{\mathrm{CylinderD}}{}\left({-}{a}{-}\frac{{1}}{{2}}{,}{z}\right)}{{2}{}\sqrt{{\mathrm{\pi }}}{}{z}{}{{2}}^{{a}{-}\frac{{1}}{{2}}}}{+}\frac{{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}{\mathrm{CylinderD}}{}\left({-}{a}{-}\frac{{1}}{{2}}{,}{-}{z}\right){}{{2}}^{\frac{{a}}{{2}}{+}\frac{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{a}\right)}{{2}{}\sqrt{{\mathrm{\pi }}}{}{z}{}{{2}}^{{a}{-}\frac{{1}}{{2}}}}$ (10)
 > $\mathrm{convert}\left(,\mathrm{Cylinder},"raise a"\right)$
 ${-}\frac{{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{+}\frac{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{a}\right){}{\mathrm{CylinderD}}{}\left(\frac{{3}}{{2}}{-}{a}{,}{z}\right)}{\left({2}{}{a}{-}{1}\right){}\sqrt{{\mathrm{\pi }}}{}{z}{}{{2}}^{{a}{-}\frac{{1}}{{2}}}}{+}\frac{{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{+}\frac{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{a}\right){}{\mathrm{CylinderD}}{}\left(\frac{{3}}{{2}}{-}{a}{,}{-}{z}\right)}{\left({2}{}{a}{-}{1}\right){}\sqrt{{\mathrm{\pi }}}{}{z}{}{{2}}^{{a}{-}\frac{{1}}{{2}}}}{+}\frac{{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}{{2}}^{\frac{{a}}{{2}}{+}\frac{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{a}\right){}{\mathrm{CylinderD}}{}\left({-}{a}{+}\frac{{1}}{{2}}{,}{z}\right)}{\left({2}{}{a}{-}{1}\right){}\sqrt{{\mathrm{\pi }}}{}{{2}}^{{a}{-}\frac{{1}}{{2}}}}{+}\frac{{{ⅇ}}^{\frac{{{z}}^{{2}}}{{4}}}{}{\mathrm{CylinderD}}{}\left({-}{a}{+}\frac{{1}}{{2}}{,}{-}{z}\right){}{{2}}^{\frac{{a}}{{2}}{+}\frac{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{a}\right)}{\left({2}{}{a}{-}{1}\right){}\sqrt{{\mathrm{\pi }}}{}{{2}}^{{a}{-}\frac{{1}}{{2}}}}$ (11)