convert/Whittaker - Maple Programming Help

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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)
 > $\mathrm{convert}\left(,\mathrm{Whittaker}\right)$
 $\frac{{1}}{{12}}{}\frac{\sqrt{{3}}{}{\mathrm{WhittakerM}}{}\left({0}{,}{-}\frac{{1}}{{3}}{,}\frac{{4}}{{3}}{}{{z}}^{{3}{/}{2}}\right){}{{4}}^{{5}{/}{6}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}{\left({{z}}^{{3}{/}{2}}\right)}^{{1}{/}{6}}}{-}\frac{{3}}{{8}}{}\frac{{z}{}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}{\mathrm{WhittakerM}}{}\left({0}{,}\frac{{1}}{{3}}{,}\frac{{4}}{{3}}{}{{z}}^{{3}{/}{2}}\right){}{{4}}^{{1}{/}{6}}}{{\mathrm{π}}{}{\left({{z}}^{{3}{/}{2}}\right)}^{{5}{/}{6}}}$ (3)
 > $\mathrm{HermiteH}\left(a,z\right)\mathrm{LaguerreL}\left(2,{ⅇ}^{z}\right)$
 ${\mathrm{HermiteH}}{}\left({a}{,}{z}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{{ⅇ}}^{{z}}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{Whittaker}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}0<\mathrm{ℜ}\left(z\right)$
 $\frac{{{2}}^{{a}}{}\sqrt{{\mathrm{π}}}{}\left(\frac{{\mathrm{WhittakerM}}{}\left(\frac{{1}}{{2}}{}{a}{+}\frac{{1}}{{4}}{,}{-}\frac{{1}}{{4}}{,}{{z}}^{{2}}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{{z}}^{{2}}}}{{\left({{z}}^{{2}}\right)}^{{1}{/}{4}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{a}\right)}{-}\frac{{2}{}{z}{}{\mathrm{WhittakerM}}{}\left(\frac{{1}}{{2}}{}{a}{+}\frac{{1}}{{4}}{,}\frac{{1}}{{4}}{,}{{z}}^{{2}}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{{z}}^{{2}}}}{{\left({{z}}^{{2}}\right)}^{{3}{/}{4}}{}{\mathrm{Γ}}{}\left({-}\frac{{1}}{{2}}{}{a}\right)}\right){}{\mathrm{WhittakerM}}{}\left(\frac{{5}}{{2}}{,}{0}{,}{\mathrm{WhittakerM}}{}\left({0}{,}{-}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{\mathrm{WhittakerM}}{}\left({0}{,}{-}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}}}{\sqrt{{\mathrm{WhittakerM}}{}\left({0}{,}{-}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}}}$ (5)
 > $\frac{\left({ⅇ}^{z}\mathrm{erf}\left({z}^{2}\right)+\mathrm{KummerU}\left(-1,\frac{1}{2},z\right){ⅇ}^{\frac{1z}{2}}\right)\mathrm{MeijerG}\left(\left[\left[1-a\right],\left[\right]\right],\left[\left[0,1-b\right],\left[\right]\right],\frac{1}{z}\right)}{\mathrm{BesselK}\left(-3,1-z\right)}$
 $\frac{\left({{ⅇ}}^{{z}}{}{\mathrm{erf}}{}\left({{z}}^{{2}}\right){+}{\mathrm{KummerU}}{}\left({-}{1}{,}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}\right){}{\mathrm{MeijerG}}{}\left(\left[\left[{1}{-}{a}\right]{,}\left[{}\right]\right]{,}\left[\left[{0}{,}{1}{-}{b}\right]{,}\left[{}\right]\right]{,}\frac{{1}}{{z}}\right)}{{\mathrm{BesselK}}{}\left({3}{,}{1}{-}{z}\right)}$ (6)
 > $\mathrm{convert}\left(,\mathrm{Whittaker}\right)$
 $\frac{\left(\frac{{2}{}{\mathrm{WhittakerM}}{}\left({0}{,}{-}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}{}{{z}}^{{2}}{}{\mathrm{WhittakerM}}{}\left(\frac{{1}}{{4}}{,}\frac{{1}}{{4}}{,}{-}{{z}}^{{4}}\right)}{\sqrt{{\mathrm{π}}}{}{{ⅇ}}^{\frac{{1}}{{2}}{}{{z}}^{{4}}}{}{\left({-}{{z}}^{{4}}\right)}^{{3}{/}{4}}}{+}\frac{{\mathrm{WhittakerW}}{}\left(\frac{{5}}{{4}}{,}{-}\frac{{1}}{{4}}{,}{z}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}{}{\mathrm{WhittakerM}}{}\left({0}{,}{-}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{}{z}\right){}{{ⅇ}}^{\frac{{1}}{{4}}{}{z}}}{{{z}}^{{1}{/}{4}}}\right){}{{ⅇ}}^{\frac{{1}}{{2}{}{z}}}{}\left({\mathrm{Γ}}{}\left({a}\right){}{\mathrm{Γ}}{}\left({1}{-}{b}\right){}{\mathrm{WhittakerM}}{}\left({-}{a}{+}\frac{{1}}{{2}}{}{b}{,}\frac{{1}}{{2}}{}{b}{-}\frac{{1}}{{2}}{,}\frac{{1}}{{z}}\right){+}{\mathrm{WhittakerM}}{}\left({-}{a}{+}\frac{{1}}{{2}}{}{b}{,}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{b}{,}\frac{{1}}{{z}}\right){}{\mathrm{Γ}}{}\left({1}{+}{a}{-}{b}\right){}{\mathrm{Γ}}{}\left({-}{1}{+}{b}\right)\right){}\sqrt{{2}{-}{2}{}{z}}}{{\left(\frac{{1}}{{z}}\right)}^{\frac{{1}}{{2}}{}{b}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{WhittakerW}}{}\left({0}{,}{3}{,}{2}{-}{2}{}{z}\right)}$ (7)