Whittaker - Maple Help

convert/Whittaker

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

 Calling Sequence convert(expr, Whittaker)

Parameters

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

Description

 • convert/Whittaker converts, when possible, special functions admitting a 1F1 or 0F1 hypergeometric representation into Whittaker functions. The Whittaker functions are
 The 2 functions in the "Whittaker" class are:
 $\left[{\mathrm{WhittakerM}}{,}{\mathrm{WhittakerW}}\right]$ (1)

Examples

 > $\mathrm{AiryAi}\left(z\right)$
 ${\mathrm{AiryAi}}{}\left({z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Whittaker}\right)$
 $\frac{\sqrt{{3}}{}{\mathrm{WhittakerM}}{}\left({0}{,}{-}\frac{{1}}{{3}}{,}\frac{{4}{}{{z}}^{{3}}{{2}}}}{{3}}\right){}{{4}}^{{5}}{{6}}}}{{12}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right){}{\left({{z}}^{{3}}{{2}}}\right)}^{{1}}{{6}}}}{-}\frac{{3}{}{z}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right){}{\mathrm{WhittakerM}}{}\left({0}{,}\frac{{1}}{{3}}{,}\frac{{4}{}{{z}}^{{3}}{{2}}}}{{3}}\right){}{{4}}^{{1}}{{6}}}}{{8}{}{\mathrm{\pi }}{}{\left({{z}}^{{3}}{{2}}}\right)}^{{5}}{{6}}}}$ (3)
 > $\mathrm{HermiteH}\left(a,z\right)\mathrm{LaguerreL}\left(2,\mathrm{exp}\left(z\right)\right)$
 ${\mathrm{HermiteH}}{}\left({a}{,}{z}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{{ⅇ}}^{{z}}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{Whittaker}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}0<\mathrm{\Re }\left(z\right)$
 $\frac{{{2}}^{{a}}{}\sqrt{{\mathrm{\pi }}}{}\left(\frac{{\mathrm{WhittakerM}}{}\left(\frac{{a}}{{2}}{+}\frac{{1}}{{4}}{,}{-}\frac{{1}}{{4}}{,}{{z}}^{{2}}\right){}{{ⅇ}}^{\frac{{{z}}^{{2}}}{{2}}}}{{\left({{z}}^{{2}}\right)}^{{1}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{-}\frac{{a}}{{2}}\right)}{-}\frac{{2}{}{z}{}{\mathrm{WhittakerM}}{}\left(\frac{{a}}{{2}}{+}\frac{{1}}{{4}}{,}\frac{{1}}{{4}}{,}{{z}}^{{2}}\right){}{{ⅇ}}^{\frac{{{z}}^{{2}}}{{2}}}}{{\left({{z}}^{{2}}\right)}^{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left({-}\frac{{a}}{{2}}\right)}\right){}{\mathrm{WhittakerM}}{}\left(\frac{{5}}{{2}}{,}{0}{,}\frac{{\mathrm{WhittakerM}}{}\left({-1}{,}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{z}}{{2}}}}{{z}}\right){}{{ⅇ}}^{\frac{{\mathrm{WhittakerM}}{}\left({-1}{,}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{z}}{{2}}}}{{2}{}{z}}}}{\sqrt{\frac{{\mathrm{WhittakerM}}{}\left({-1}{,}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{z}}{{2}}}}{{z}}}}$ (5)
 > $\frac{\left(\mathrm{exp}\left(z\right)\mathrm{erf}\left({z}^{2}\right)+\mathrm{KummerU}\left(-1,\frac{1}{2},z\right)\mathrm{exp}\left(\frac{1}{2}z\right)\right)\mathrm{MeijerG}\left(\left[\left[1-a\right],\left[\right]\right],\left[\left[0,1-b\right],\left[\right]\right],\frac{1}{z}\right)}{\mathrm{BesselK}\left(-3,1-z\right)}$
 $\frac{\left({{ⅇ}}^{{z}}{}{\mathrm{erf}}{}\left({{z}}^{{2}}\right){+}{\mathrm{KummerU}}{}\left({-1}{,}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{z}}{{2}}}\right){}{\mathrm{MeijerG}}{}\left(\left[\left[{1}{-}{a}\right]{,}\left[\right]\right]{,}\left[\left[{0}{,}{1}{-}{b}\right]{,}\left[\right]\right]{,}\frac{{1}}{{z}}\right)}{{\mathrm{BesselK}}{}\left({3}{,}{1}{-}{z}\right)}$ (6)
 > $\mathrm{convert}\left(,\mathrm{Whittaker}\right)$
 $\frac{\left(\frac{{2}{}{\mathrm{WhittakerM}}{}\left({-1}{,}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{z}}{{2}}}{}{z}{}{\mathrm{WhittakerM}}{}\left(\frac{{1}}{{4}}{,}\frac{{1}}{{4}}{,}{-}{{z}}^{{4}}\right)}{\sqrt{{\mathrm{\pi }}}{}{{ⅇ}}^{\frac{{{z}}^{{4}}}{{2}}}{}{\left({-}{{z}}^{{4}}\right)}^{{3}}{{4}}}}{+}\frac{{2}{}{\mathrm{WhittakerW}}{}\left(\frac{{5}}{{4}}{,}{-}\frac{{1}}{{4}}{,}{z}\right){}{{ⅇ}}^{\frac{{z}}{{2}}}{}{\mathrm{WhittakerM}}{}\left({-1}{,}\frac{{1}}{{2}}{,}\frac{{z}}{{2}}\right){}{{ⅇ}}^{\frac{{z}}{{4}}}}{{{z}}^{{5}}{{4}}}}\right){}{{ⅇ}}^{\frac{{1}}{{2}{}{z}}}{}\left({\mathrm{\Gamma }}{}\left({a}\right){}{\mathrm{\Gamma }}{}\left({1}{-}{b}\right){}{\mathrm{WhittakerM}}{}\left({-}{a}{+}\frac{{b}}{{2}}{,}\frac{{b}}{{2}}{-}\frac{{1}}{{2}}{,}\frac{{1}}{{z}}\right){+}{\mathrm{WhittakerM}}{}\left({-}{a}{+}\frac{{b}}{{2}}{,}\frac{{1}}{{2}}{-}\frac{{b}}{{2}}{,}\frac{{1}}{{z}}\right){}{\mathrm{\Gamma }}{}\left({1}{+}{a}{-}{b}\right){}{\mathrm{\Gamma }}{}\left({-}{1}{+}{b}\right)\right){}\sqrt{{2}{-}{2}{}{z}}}{{\left(\frac{{1}}{{z}}\right)}^{\frac{{b}}{{2}}}{}\sqrt{{\mathrm{\pi }}}{}{\mathrm{WhittakerW}}{}\left({0}{,}{3}{,}{2}{-}{2}{}{z}\right)}$ (7)