convert/Cylinder - Maple Programming Help

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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
 > FunctionAdvisor( Cylinder );
 The 3 functions in the "Cylinder" class are:
 $\left[{\mathrm{CylinderD}}{,}{\mathrm{CylinderU}}{,}{\mathrm{CylinderV}}\right]$ (1)

Examples

 > $\mathrm{HermiteH}\left(a,z\right)\mathrm{LaguerreL}\left(\frac{1a}{2},-\frac{1}{2},\frac{1{z}^{2}}{2}\right)$
 ${\mathrm{HermiteH}}{}\left({a}{,}{z}\right){}{\mathrm{LaguerreL}}{}\left(\frac{{1}}{{2}}{}{a}{,}{-}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{}{{z}}^{{2}}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Cylinder}\right)$
 $\frac{{1}}{{2}}{}\frac{{\mathrm{CylinderD}}{}\left({a}{,}{z}{}\sqrt{{2}}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{{z}}^{{2}}}{}{\mathrm{binomial}}{}\left({-}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}{a}{,}\frac{{1}}{{2}}{}{a}\right){}{{ⅇ}}^{\frac{{1}}{{4}}{}{{z}}^{{2}}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{a}\right){}{\mathrm{CylinderD}}{}\left({a}{,}{z}\right)}{\sqrt{{\mathrm{π}}}}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{CylinderD}}{}\left({a}{,}{z}{}\sqrt{{2}}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{{z}}^{{2}}}{}{\mathrm{binomial}}{}\left({-}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}{a}{,}\frac{{1}}{{2}}{}{a}\right){}{{ⅇ}}^{\frac{{1}}{{4}}{}{{z}}^{{2}}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{a}\right){}{\mathrm{CylinderD}}{}\left({a}{,}{-}{z}\right)}{\sqrt{{\mathrm{π}}}}$ (3)
 > $\mathrm{erfc}\left(a,z\right)$
 ${\mathrm{erfc}}{}\left({a}{,}{z}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{Cylinder}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}a::\mathrm{posint}$
 $\frac{{\mathrm{CylinderD}}{}\left({-}{a}{-}{1}{,}{z}{}\sqrt{{2}}\right){}\sqrt{{2}}}{{{ⅇ}}^{\frac{{1}}{{2}}{}{{z}}^{{2}}}{}\sqrt{{{2}}^{{a}}{}{\mathrm{π}}}}$ (5)
 > $\frac{2{\mathrm{π}}^{\frac{1}{2}}z\mathrm{hypergeom}\left(\left[\frac{1a}{2}+\frac{3}{4}\right],\left[\frac{3}{2}\right],\frac{1{z}^{2}}{2}\right)}{{2}^{\frac{1a}{2}-\frac{1}{4}}\mathrm{Γ}\left(\frac{1a}{2}+\frac{1}{4}\right)}+\frac{{\mathrm{π}}^{\frac{1}{2}}\mathrm{hypergeom}\left(\left[\frac{1a}{2}+\frac{1}{4}\right],\left[\frac{1}{2}\right],\frac{1{z}^{2}}{2}\right)}{\mathrm{Γ}\left(\frac{1a}{2}+\frac{3}{4}\right){2}^{\frac{1a}{2}-\frac{3}{4}}}$
 $\frac{{2}{}\sqrt{{\mathrm{π}}}{}{z}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{2}}{}{a}{+}\frac{{3}}{{4}}\right]{,}\left[\frac{{3}}{{2}}\right]{,}\frac{{1}}{{2}}{}{{z}}^{{2}}\right)}{{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{1}}{{4}}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{a}{+}\frac{{1}}{{4}}\right)}{+}\frac{\sqrt{{\mathrm{π}}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{2}}{}{a}{+}\frac{{1}}{{4}}\right]{,}\left[\frac{{1}}{{2}}\right]{,}\frac{{1}}{{2}}{}{{z}}^{{2}}\right)}{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{a}{+}\frac{{3}}{{4}}\right){}{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{3}}{{4}}}}$ (6)
 > $\mathrm{convert}\left(,\mathrm{Cylinder}\right)$
 $\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{\frac{{1}}{{4}}{}{{z}}^{{2}}}{}\left({{2}}^{\frac{{1}}{{2}}{}{a}{+}\frac{{1}}{{4}}}{}{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{1}}{{4}}}{-}{{2}}^{\frac{{1}}{{2}}{}{a}{+}\frac{{3}}{{4}}}{}{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{3}}{{4}}}\right){}{\mathrm{CylinderD}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{z}\right)}{{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{1}}{{4}}}{}{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{3}}{{4}}}}{+}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{\frac{{1}}{{4}}{}{{z}}^{{2}}}{}\left({{2}}^{\frac{{1}}{{2}}{}{a}{+}\frac{{1}}{{4}}}{}{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{1}}{{4}}}{+}{{2}}^{\frac{{1}}{{2}}{}{a}{+}\frac{{3}}{{4}}}{}{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{3}}{{4}}}\right){}{\mathrm{CylinderD}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{-}{z}\right)}{{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{1}}{{4}}}{}{{2}}^{\frac{{1}}{{2}}{}{a}{-}\frac{{3}}{{4}}}}$ (7)
 > $\mathrm{collect}\left(,\mathrm{CylinderD},\mathrm{simplify}\right)$
 ${2}{}{{ⅇ}}^{\frac{{1}}{{4}}{}{{z}}^{{2}}}{}{\mathrm{CylinderD}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{-}{z}\right)$ (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:

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

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