Bessel related - Maple Help

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convert/Bessel_related

convert special functions admitting 1F1 or 0F1 hypergeometric representation into Bessel related functions

 Calling Sequence convert(expr, Bessel_related)

Parameters

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

Description

 • convert/Bessel_related converts, when possible, special functions admitting a 1F1 or 0F1 hypergeometric representation into Bessel related functions. The Bessel related functions are
 > FunctionAdvisor( Bessel_related );
 The 14 functions in the "Bessel_related" class are:
 $\left[{\mathrm{AiryAi}}{,}{\mathrm{AiryBi}}{,}{\mathrm{BesselI}}{,}{\mathrm{BesselJ}}{,}{\mathrm{BesselK}}{,}{\mathrm{BesselY}}{,}{\mathrm{HankelH1}}{,}{\mathrm{HankelH2}}{,}{\mathrm{KelvinBei}}{,}{\mathrm{KelvinBer}}{,}{\mathrm{KelvinHei}}{,}{\mathrm{KelvinHer}}{,}{\mathrm{KelvinKei}}{,}{\mathrm{KelvinKer}}\right]$ (1)

Examples

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

 See Also