erf related - Maple Help

convert/erf_related

convert special functions into erf related functions

 Calling Sequence convert(expr, erf_related)

Parameters

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

Description

 • convert/erf_related converts, when possible, special functions into erf related functions. The erf related functions are
 > FunctionAdvisor( erf_related );
 The 8 functions in the "erf_related" class are:
 $\left[{\mathrm{FresnelC}}{,}{\mathrm{FresnelS}}{,}{\mathrm{Fresnelf}}{,}{\mathrm{Fresnelg}}{,}{\mathrm{dawson}}{,}{\mathrm{erf}}{,}{\mathrm{erfc}}{,}{\mathrm{erfi}}\right]$ (1)

Examples

 > $\mathrm{_C1}\mathrm{KummerM}\left(\frac{1}{2},\frac{3}{2},-{z}^{2}\right)+\mathrm{_C2}z\mathrm{KummerU}\left(1,\frac{3}{2},{z}^{2}\right)$
 ${\mathrm{_C1}}{}{\mathrm{KummerM}}{}\left(\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{,}{-}{{z}}^{{2}}\right){+}{\mathrm{_C2}}{}{z}{}{\mathrm{KummerU}}{}\left({1}{,}\frac{{3}}{{2}}{,}{{z}}^{{2}}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{erf_related}\right)$
 ${-}\frac{\sqrt{{\mathrm{\pi }}}{}\left({2}{}{\mathrm{_C2}}{}{z}{}{{ⅇ}}^{{{z}}^{{2}}}{-}{\mathrm{_C1}}\right){}{\mathrm{erf}}{}\left(\sqrt{{{z}}^{{2}}}\right)}{{2}{}\sqrt{{{z}}^{{2}}}}{+}\frac{{\mathrm{_C2}}{}{z}{}\sqrt{{\mathrm{\pi }}}{}{{ⅇ}}^{{{z}}^{{2}}}}{\sqrt{{{z}}^{{2}}}}$ (3)
 > $\mathrm{_C1}\mathrm{HermiteH}\left(-1,z\right)+\mathrm{_C2}\mathrm{HermiteH}\left(1,-z\right)$
 ${\mathrm{_C1}}{}{\mathrm{HermiteH}}{}\left({-1}{,}{z}\right){+}{\mathrm{_C2}}{}{\mathrm{HermiteH}}{}\left({1}{,}{-}{z}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{erf_related}\right)$
 ${-}\frac{{\mathrm{_C1}}{}\sqrt{{\mathrm{\pi }}}{}{{ⅇ}}^{{{z}}^{{2}}}{}{\mathrm{erf}}{}\left({z}\right)}{{2}}{+}\frac{{\mathrm{_C1}}{}\sqrt{{\mathrm{\pi }}}{}{{ⅇ}}^{{{z}}^{{2}}}}{{2}}{-}{2}{}{\mathrm{_C2}}{}{z}$ (5)
 > $\frac{\frac{2}{{\mathrm{\pi }}^{\frac{1}{2}}}{z}^{\frac{1}{2}}\mathrm{WhittakerM}\left(\frac{1}{4},\frac{1}{4},-z\right)}{\mathrm{exp}\left(\frac{1}{2}z\right){\left(-z\right)}^{\frac{3}{4}}}+\mathrm{sqrt}\left(\mathrm{\pi }\right)z\mathrm{LaguerreL}\left(-\frac{1}{2},\frac{1}{2},{z}^{2}\right)$
 $\frac{{2}{}\sqrt{{z}}{}{\mathrm{WhittakerM}}{}\left(\frac{{1}}{{4}}{,}\frac{{1}}{{4}}{,}{-}{z}\right)}{\sqrt{{\mathrm{\pi }}}{}{{ⅇ}}^{\frac{{z}}{{2}}}{}{\left({-}{z}\right)}^{{3}}{{4}}}}{+}\sqrt{{\mathrm{\pi }}}{}{z}{}{\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{,}{{z}}^{{2}}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{erf_related}\right)$
 ${\mathrm{erf}}{}\left(\sqrt{{z}}\right){+}\frac{{z}{}{\mathrm{erf}}{}\left(\sqrt{{-}{{z}}^{{2}}}\right)}{\sqrt{{-}{{z}}^{{2}}}}$ (7)