 1F1 - Maple Help

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:
 $\left[{\mathrm{CoulombF}}{,}{\mathrm{CylinderD}}{,}{\mathrm{CylinderU}}{,}{\mathrm{CylinderV}}{,}{\mathrm{Ei}}{,}{\mathrm{FresnelC}}{,}{\mathrm{FresnelS}}{,}{\mathrm{Fresnelf}}{,}{\mathrm{Fresnelg}}{,}{\mathrm{\Gamma }}{,}{\mathrm{HermiteH}}{,}{\mathrm{KummerM}}{,}{\mathrm{KummerU}}{,}{\mathrm{LaguerreL}}{,}{\mathrm{WhittakerM}}{,}{\mathrm{WhittakerW}}{,}{\mathrm{dawson}}{,}{\mathrm{erf}}{,}{\mathrm{erfc}}{,}{\mathrm{erfi}}\right]$ (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)
 > $\mathrm{convert}\left(,\mathrm{1F1}\right)$
 $\frac{{\left({I}{}{z}\right)}^{{a}}{}{\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{,}{2}{}{I}{}{z}\right)}{{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{{2}}^{{a}}{}\left(\genfrac{}{}{0}{}{{-}\frac{{1}}{{2}}{+}{a}}{{-}\frac{{1}}{{2}}{-}{a}}\right){}{{ⅇ}}^{{I}{}{z}}}$ (3)
 > $\mathrm{hypergeom}\left(\left[\right],\left[c\right],z\right)$
 ${\mathrm{hypergeom}}{}\left(\left[\right]{,}\left[{c}\right]{,}{z}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{1F1}\right)$
 $\frac{{\mathrm{LaguerreL}}{}\left(\frac{{1}}{{2}}{-}{c}{,}{-}{2}{+}{2}{}{c}{,}{4}{}\sqrt{{z}}\right)}{\left(\genfrac{}{}{0}{}{{c}{-}\frac{{3}}{{2}}}{\frac{{1}}{{2}}{-}{c}}\right){}{{ⅇ}}^{{2}{}\sqrt{{z}}}}$ (5)
 > $\mathrm{MeijerG}\left(\left[\left[\right],\left[\right]\right],\left[\left[\frac{1}{2}a,-\frac{1}{2}a\right],\left[\right]\right],\frac{1}{4}{z}^{2}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[\frac{{a}}{{2}}{,}{-}\frac{{a}}{{2}}\right]{,}\left[\right]\right]{,}\frac{{{z}}^{{2}}}{{4}}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{1F1}\right)$
 $\frac{{\mathrm{\Gamma }}{}\left({a}\right){}{\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{+}{a}{,}{-}{2}{}{a}{,}{2}{}{z}\right)}{{\left(\frac{{{z}}^{{2}}}{{4}}\right)}^{\frac{{a}}{{2}}}{}\left(\genfrac{}{}{0}{}{{-}\frac{{1}}{{2}}{-}{a}}{{-}\frac{{1}}{{2}}{+}{a}}\right){}{{ⅇ}}^{{z}}}{+}\frac{{\mathrm{\Gamma }}{}\left({-}{a}\right){}{\left(\frac{{{z}}^{{2}}}{{4}}\right)}^{\frac{{a}}{{2}}}{}{\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{,}{2}{}{z}\right)}{\left(\genfrac{}{}{0}{}{{-}\frac{{1}}{{2}}{+}{a}}{{-}\frac{{1}}{{2}}{-}{a}}\right){}{{ⅇ}}^{{z}}}$ (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)
 > $\mathrm{convert}\left(,\mathrm{1F1}\right)$
 $\frac{{{z}}^{{a}}{}{\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{,}{2}{}{I}{}{z}\right)}{{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{{2}}^{{a}}{}\left(\genfrac{}{}{0}{}{{-}\frac{{1}}{{2}}{+}{a}}{{-}\frac{{1}}{{2}}{-}{a}}\right){}{{ⅇ}}^{{I}{}{z}}}{-}\frac{{I}{}{{ⅇ}}^{\left(\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}}{}{\left(\left({-}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}\right)}^{{a}{-}{1}}{}{\mathrm{LaguerreL}}{}\left(\frac{{1}}{{2}}{-}{a}{,}{-}{2}{+}{2}{}{a}{,}\left({-1}{-}{I}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}\right)}{{\mathrm{\Gamma }}{}\left({a}\right){}\left(\genfrac{}{}{0}{}{{-}\frac{{3}}{{2}}{+}{a}}{\frac{{1}}{{2}}{-}{a}}\right){}{{2}}^{{a}}}{+}\frac{{I}{}{\left(\left({-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}\right)}^{{a}{-}{1}}{}{\mathrm{LaguerreL}}{}\left(\frac{{1}}{{2}}{-}{a}{,}{-}{2}{+}{2}{}{a}{,}\left({-1}{+}{I}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}\right){}{{ⅇ}}^{\left(\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}\left({z}{-}{1}\right){}\sqrt{{2}}}}{{\mathrm{\Gamma }}{}\left({a}\right){}\left(\genfrac{}{}{0}{}{{-}\frac{{3}}{{2}}{+}{a}}{\frac{{1}}{{2}}{-}{a}}\right){}{{2}}^{{a}}}$ (9)