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convert/Hankel

convert special functions admitting 1F1 or 0F1 hypergeometric representation into Hankel (Bessel of third kind) functions

 Calling Sequence convert(expr, Hankel)

Parameters

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

Description

 • convert/Hankel converts, when possible, special functions admitting a 1F1 or 0F1 hypergeometric representation into Hankel (Bessel of third kind) functions. The Hankel functions are
 > FunctionAdvisor( Hankel );
 The 2 functions in the "Hankel" class are:
 $\left[{\mathrm{HankelH1}}{,}{\mathrm{HankelH2}}\right]$ (1)

Examples

 > $\mathrm{BesselJ}\left(a,z\right)$
 ${\mathrm{BesselJ}}{}\left({a}{,}{z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Hankel}\right)$
 $\frac{{\mathrm{HankelH1}}{}\left({a}{,}{z}\right)}{{2}}{+}\frac{{\mathrm{HankelH2}}{}\left({a}{,}{z}\right)}{{2}}$ (3)
 > $\mathrm{BesselK}\left(a,z\right)$
 ${\mathrm{BesselK}}{}\left({a}{,}{z}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{Hankel}\right)$
 $\frac{{I}{}\left(\left({-}{\left({{z}}^{{a}}\right)}^{{2}}{}{{ⅇ}}^{{I}{}{a}{}{\mathrm{\pi }}}{+}{\left({\left({I}{}{z}\right)}^{{a}}\right)}^{{2}}{}{{ⅇ}}^{{2}{}{I}{}{a}{}{\mathrm{\pi }}}\right){}{\mathrm{HankelH1}}{}\left({a}{,}{I}{}{z}\right){+}\left({\left({\left({I}{}{z}\right)}^{{a}}\right)}^{{2}}{-}{\left({{z}}^{{a}}\right)}^{{2}}{}{{ⅇ}}^{{I}{}{a}{}{\mathrm{\pi }}}\right){}{\mathrm{HankelH2}}{}\left({a}{,}{I}{}{z}\right)\right){}{\mathrm{\pi }}}{{\left({I}{}{z}\right)}^{{a}}{}{{z}}^{{a}}{}\left({2}{}{{ⅇ}}^{{2}{}{I}{}{a}{}{\mathrm{\pi }}}{-}{2}\right)}$ (5)
 > $\mathrm{KummerU}\left(a+\frac{1}{2},2a+1,z\right)$
 ${\mathrm{KummerU}}{}\left({a}{+}\frac{{1}}{{2}}{,}{2}{}{a}{+}{1}{,}{z}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{Hankel}\right)$
 $\frac{\left(\left({-}\frac{{{ⅇ}}^{{I}{}{a}{}{\mathrm{\pi }}}{}{\left(\frac{{I}}{{2}}{}{z}\right)}^{{a}}{}{{2}}^{{2}{}{a}{+}{1}}}{{8}{}{{z}}^{{2}{}{a}}{}{{2}}^{{a}}}{+}\frac{{{2}}^{{a}}}{{8}{}{\left(\frac{{I}}{{2}}{}{z}\right)}^{{a}}{}{{2}}^{{-}{1}{+}{2}{}{a}}}\right){}{\mathrm{HankelH1}}{}\left({a}{,}\frac{{I}}{{2}}{}{z}\right){+}\left({-}\frac{{\left(\frac{{I}}{{2}}{}{z}\right)}^{{a}}{}{{2}}^{{2}{}{a}{+}{1}}}{{8}{}{{z}}^{{2}{}{a}}{}{{ⅇ}}^{{I}{}{a}{}{\mathrm{\pi }}}{}{{2}}^{{a}}}{+}\frac{{{2}}^{{a}}}{{8}{}{\left(\frac{{I}}{{2}}{}{z}\right)}^{{a}}{}{{2}}^{{-}{1}{+}{2}{}{a}}}\right){}{\mathrm{HankelH2}}{}\left({a}{,}\frac{{I}}{{2}}{}{z}\right)\right){}\sqrt{{\mathrm{\pi }}}{}{{ⅇ}}^{\frac{{z}}{{2}}}}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}\left({a}{+}{1}\right)\right)}$ (7)
 > $\mathrm{LaguerreL}\left(-\frac{1}{2}-a,2a,2Iz\right)-\mathrm{WhittakerM}\left(0,a,2Iz\right)$
 ${\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{,}{2}{}{I}{}{z}\right){-}{\mathrm{WhittakerM}}{}\left({0}{,}{a}{,}{2}{}{I}{}{z}\right)$ (8)
 > $\mathrm{convert}\left(,\mathrm{Hankel}\right)$
 $\frac{\left(\genfrac{}{}{0}{}{{-}\frac{{1}}{{2}}{+}{a}}{{-}\frac{{1}}{{2}}{-}{a}}\right){}{{ⅇ}}^{{I}{}{z}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}\left({\mathrm{HankelH1}}{}\left({a}{,}{-}{z}\right){+}{\mathrm{HankelH2}}{}\left({a}{,}{-}{z}\right)\right){}{{2}}^{{a}}}{{2}{}{\left({-}{z}\right)}^{{a}}}{-}\frac{{\left({2}{}{I}{}{z}\right)}^{{a}{+}\frac{{1}}{{2}}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}\left({\mathrm{HankelH1}}{}\left({a}{,}{-}{z}\right){+}{\mathrm{HankelH2}}{}\left({a}{,}{-}{z}\right)\right){}{{2}}^{{a}}}{{2}{}{\left({-}{z}\right)}^{{a}}}$ (9)
 > $\mathrm{MeijerG}\left(\left[\left[\right],\left[\right]\right],\left[\left[\frac{1a}{2}\right],\left[-\frac{1a}{2}\right]\right],z\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[\frac{{a}}{{2}}\right]{,}\left[{-}\frac{{a}}{{2}}\right]\right]{,}{z}\right)$ (10)
 > $\mathrm{convert}\left(,\mathrm{Hankel}\right)$
 $\frac{{{z}}^{\frac{{a}}{{2}}}{}\left({\mathrm{HankelH1}}{}\left({a}{,}{-}{2}{}\sqrt{{z}}\right){+}{\mathrm{HankelH2}}{}\left({a}{,}{-}{2}{}\sqrt{{z}}\right)\right){}{{2}}^{{a}}}{{2}{}{\left({-}{2}{}\sqrt{{z}}\right)}^{{a}}}$ (11)