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:

$\left[\mathrm{FresnelC}\,\mathrm{FresnelS}\,\mathrm{Fresnelf}\,\mathrm{Fresnelg}\,\mathrm{dawson}\,\mathrm{erf}\,\mathrm{erfc}\,\mathrm{erfi}\right]$

$\mathrm{\_C1}\mathrm{KummerM}\left(\frac{1}{2}\,\frac{3}{2}\,-z^2\right)+\mathrm{\_C2}z\mathrm{KummerU}\left(1\,\frac{3}{2}\,z^2\right)$

$\mathrm{\_C1}\mathrm{KummerM}\left(\frac{1}{2}\,\frac{3}{2}\,-z^2\right)+\mathrm{\_C2}z\mathrm{KummerU}\left(1\,\frac{3}{2}\,z^2\right)$

$\mathrm{convert}\left(\,\mathrm{erf\_related}\right)$

$-\frac{\sqrt{\mathrm{\pi }}\left(2\mathrm{\_C2}z{\ⅇ}^{z^2}-\mathrm{\_C1}\right)\mathrm{erf}\left(\sqrt{z^2}\right)}{2\sqrt{z^2}}+\frac{\mathrm{\_C2}z\sqrt{\mathrm{\pi }}{\ⅇ}^{z^2}}{\sqrt{z^2}}$

$\mathrm{\_C1}\mathrm{HermiteH}\left(-1\,z\right)+\mathrm{\_C2}\mathrm{HermiteH}\left(1\,-z\right)$

$\mathrm{\_C1}\mathrm{HermiteH}\left(\mathrm{-1}\,z\right)+\mathrm{\_C2}\mathrm{HermiteH}\left(1\,-z\right)$

$\mathrm{convert}\left(\,\mathrm{erf\_related}\right)$

$-\frac{\mathrm{\_C1}\sqrt{\mathrm{\pi }}{\ⅇ}^{z^2}\mathrm{erf}\left(z\right)}{2}+\frac{\mathrm{\_C1}\sqrt{\mathrm{\pi }}{\ⅇ}^{z^2}}{2}-2\mathrm{\_C2}z$

$\frac{\frac{2}{{\mathrm{\pi }}^{\frac{1}{2}}}{z}^{\frac{1}{2}}\mathrm{WhittakerM}\left(\frac{1}{4}\,\frac{1}{4}\,-z\right)}{\exp \left(\frac{1}{2}z\right){\left(-z\right)}^{\frac{3}{4}}}+\mathrm{sqrt}\left(\mathrm{\pi }\right)z\mathrm{LaguerreL}\left(-\frac{1}{2}\,\frac{1}{2}\,z^2\right)$

$\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)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}+\sqrt{\mathrm{\pi }}z\mathrm{LaguerreL}\left(-\frac{1}{2}\,\frac{1}{2}\,z^2\right)$

$\mathrm{convert}\left(\,\mathrm{erf\_related}\right)$

$\mathrm{erf}\left(\sqrt{z}\right)+\frac{z\mathrm{erf}\left(\sqrt{-z^2}\right)}{\sqrt{-z^2}}$