convert/1F1 - Maple Programming Help

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convert/1F1

convert to special functions admitting a 1F1 hypergeometric representation

 Calling Sequence convert(expr, 1F1)

Parameters

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

Description

 • convert/1F1 converts, when possible, hypergeometric, MeijerG, and special functions admitting a 0F1 hypergeometric representation into special functions admitting a 1F1 hypergeometric representation; that is, into one of
 > FunctionAdvisor( 1F1 );
 The 20 functions in the "1F1" class are particular cases of the hypergeometric function and are given by:
 $\left[{\mathrm{CoulombF}}{,}{\mathrm{CylinderD}}{,}{\mathrm{CylinderU}}{,}{\mathrm{CylinderV}}{,}{\mathrm{Ei}}{,}{\mathrm{FresnelC}}{,}{\mathrm{FresnelS}}{,}{\mathrm{Fresnelf}}{,}{\mathrm{Fresnelg}}{,}{\mathrm{Γ}}{,}{\mathrm{HermiteH}}{,}{\mathrm{KummerM}}{,}{\mathrm{KummerU}}{,}{\mathrm{LaguerreL}}{,}{\mathrm{WhittakerM}}{,}{\mathrm{WhittakerW}}{,}{\mathrm{dawson}}{,}{\mathrm{erf}}{,}{\mathrm{erfc}}{,}{\mathrm{erfi}}\right]$ (1)
 • convert/1F1 accepts as optional arguments all those described in convert/to_special_function.

Examples

 > $\mathrm{BesselI}\left(a,Iz\right)$
 ${\mathrm{BesselI}}{}\left({a}{,}{I}{}{z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{1F1}\right)$
 $\frac{{\left({I}{}{z}\right)}^{{a}}{}{\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{,}{2}{}{I}{}{z}\right)}{{\mathrm{Γ}}{}\left({a}{+}{1}\right){}{{2}}^{{a}}{}{\mathrm{binomial}}{}\left({-}\frac{{1}}{{2}}{+}{a}{,}{-}\frac{{1}}{{2}}{-}{a}\right){}{{ⅇ}}^{{I}{}{z}}}$ (3)
 > $\mathrm{hypergeom}\left(\left[\right],\left[c\right],z\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{}\right]{,}\left[{c}\right]{,}{z}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{1F1}\right)$
 $\frac{{\mathrm{LaguerreL}}{}\left(\frac{{1}}{{2}}{-}{c}{,}{-}{2}{+}{2}{}{c}{,}{4}{}\sqrt{{z}}\right)}{{\mathrm{binomial}}{}\left({c}{-}\frac{{3}}{{2}}{,}\frac{{1}}{{2}}{-}{c}\right){}{{ⅇ}}^{{2}{}\sqrt{{z}}}}$ (5)
 > $\mathrm{MeijerG}\left(\left[\left[\right],\left[\right]\right],\left[\left[\frac{1a}{2},-\frac{1a}{2}\right],\left[\right]\right],\frac{1{z}^{2}}{4}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[{}\right]{,}\left[{}\right]\right]{,}\left[\left[\frac{{1}}{{2}}{}{a}{,}{-}\frac{{1}}{{2}}{}{a}\right]{,}\left[{}\right]\right]{,}\frac{{1}}{{4}}{}{{z}}^{{2}}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{1F1}\right)$
 $\frac{{\mathrm{Γ}}{}\left({a}\right){}{\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{+}{a}{,}{-}{2}{}{a}{,}{2}{}{z}\right)}{{\left(\frac{{1}}{{4}}{}{{z}}^{{2}}\right)}^{\frac{{1}}{{2}}{}{a}}{}{\mathrm{binomial}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{-}\frac{{1}}{{2}}{+}{a}\right){}{{ⅇ}}^{{z}}}{+}\frac{{\mathrm{Γ}}{}\left({-}{a}\right){}{\left(\frac{{1}}{{4}}{}{{z}}^{{2}}\right)}^{\frac{{1}}{{2}}{}{a}}{}{\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{,}{2}{}{z}\right)}{{\mathrm{binomial}}{}\left({-}\frac{{1}}{{2}}{+}{a}{,}{-}\frac{{1}}{{2}}{-}{a}\right){}{{ⅇ}}^{{z}}}$ (7)
 > $\mathrm{BesselJ}\left(a,z\right)+\mathrm{KelvinBei}\left(a-1,z-1\right)$
 ${\mathrm{BesselJ}}{}\left({a}{,}{z}\right){+}{\mathrm{KelvinBei}}{}\left({a}{-}{1}{,}{z}{-}{1}\right)$ (8)
 > $\mathrm{convert}\left(,\mathrm{1F1}\right)$
 $\frac{{{z}}^{{a}}{}{\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{,}{2}{}{I}{}{z}\right)}{{\mathrm{Γ}}{}\left({a}{+}{1}\right){}{{2}}^{{a}}{}{\mathrm{binomial}}{}\left({-}\frac{{1}}{{2}}{+}{a}{,}{-}\frac{{1}}{{2}}{-}{a}\right){}{{ⅇ}}^{{I}{}{z}}}{-}\frac{{I}{}{{ⅇ}}^{\left(\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}{I}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}}{}{\left(\left({-}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}{I}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}\right)}^{{a}{-}{1}}{}{\mathrm{LaguerreL}}{}\left(\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{-}{2}{,}\left({-}{1}{-}{I}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}\right)}{{\mathrm{Γ}}{}\left({a}\right){}{\mathrm{binomial}}{}\left({-}\frac{{3}}{{2}}{+}{a}{,}\frac{{1}}{{2}}{-}{a}\right){}{{2}}^{{a}}}{+}\frac{{I}{}{\left(\left({-}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{I}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}\right)}^{{a}{-}{1}}{}{\mathrm{LaguerreL}}{}\left(\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{-}{2}{,}\left({-}{1}{+}{I}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}\right){}{{ⅇ}}^{\left(\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{I}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}}}{{\mathrm{Γ}}{}\left({a}\right){}{\mathrm{binomial}}{}\left({-}\frac{{3}}{{2}}{+}{a}{,}\frac{{1}}{{2}}{-}{a}\right){}{{2}}^{{a}}}$ (9)

 See Also