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.

erf(infinity);

$1$

erf(3);

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

evalf((2));

$0.9999779095$

erfc(3.);

$0.00002209049700$

erf(1.-1.*I);

$1.316151282-0.1904534692\mathrm{I}$

erfc(1.5-2.85*I);

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

diff(erf(x),x);

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

diff(erfc(5,x), x);

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

erfi(-x);

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

series(erfi(x),x,4);

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

expand(erfc(2,x),x);

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

convert((11),erfc);

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

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