The error function is defined for all complex x by

The complementary error function is defined by

The iterated integrals of the complementary error function are defined by

(Note $\mathrm{erfc}\left(0\,x\right)=\mathrm{erfc}\left(x\right)$.)

The imaginary error function is defined by

All of these functions are entire.

$\mathrm{erf}\left(\mathrm{\infty }\right)$

$1$

$\mathrm{erf}\left(3\right)$

$\mathrm{erf}\left(3\right)$

$\mathrm{evalf}\left(\right)$

$0.9999779095$

$\mathrm{erfc}\left(3.\right)$

$0.00002209049700$

$\mathrm{erf}\left(1.-1.\mathrm{I}\right)$

$1.316151282-0.1904534692\mathrm{I}$

$\mathrm{erfc}\left(1.5-2.85\mathrm{I}\right)$

$\mathrm{-62.82064889}-10.56167495\mathrm{I}$

$\mathrm{diff}\left(\mathrm{erf}\left(x\right)\,x\right)$

$\frac{2{\ⅇ}^{-x^2}}{\sqrt{\mathrm{\pi }}}$

$\mathrm{diff}\left(\mathrm{erfc}\left(5\,x\right)\,x\right)$

$-\mathrm{erfc}\left(4\,x\right)$

$\mathrm{erfi}\left(-x\right)$

$-\mathrm{erfi}\left(x\right)$

$\mathrm{series}\left(\mathrm{erfi}\left(x\right)\,x\,4\right)$

$\frac{2}{\sqrt{\mathrm{\pi }}}x+\frac{2}{3}\frac{1}{\sqrt{\mathrm{\pi }}}{x}^3+O\left(x^5\right)$

$\mathrm{expand}\left(\mathrm{erfc}\left(2\,x\right)\,x\right)$

$\frac{x^2}{2}-\frac{x^2\mathrm{erf}\left(x\right)}{2}-\frac{x{\ⅇ}^{-x^2}}{2\sqrt{\mathrm{\pi }}}+\frac{1}{4}-\frac{\mathrm{erf}\left(x\right)}{4}$

$\mathrm{convert}\left(\,\mathrm{erfc}\right)$

$\frac{x^2}{2}-\frac{x^2\left(1-\mathrm{erfc}\left(x\right)\right)}{2}-\frac{x{\ⅇ}^{-x^2}}{2\sqrt{\mathrm{\pi }}}+\frac{\mathrm{erfc}\left(x\right)}{4}$

Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953. Vol. 2.