erf - Maple Help

erf

The Error Function

erfc

The Complementary Error Function and its Iterated Integrals

erfi

The Imaginary Error Function

 Calling Sequence erf(x) erfc(x) erfc(n, x) erfi(x)

Parameters

 x - algebraic expression n - algebraic expression, understood to be an integer $\le$ $-1$

Description

 • The error function is defined for all complex x by

$\mathrm{erf}\left(x\right)=\frac{2\left({\int }_{0}^{x}{ⅇ}^{-{t}^{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt\right)}{\sqrt{\mathrm{\pi }}}$

 • The complementary error function is defined by

$\mathrm{erfc}\left(x\right)=1-\mathrm{erf}\left(x\right)=1-\frac{2}{{\pi }^{\frac{1}{2}}}\underset{0}{\overset{x}{\int }}{ⅇ}^{-{t}^{2}}ⅆt$

 • The iterated integrals of the complementary error function are defined by

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

$\mathrm{erfc}\left(n,x\right)=\underset{x}{\overset{\infty }{\int }}\mathrm{erfc}\left(n-1,t\right)ⅆtn\ge 0$

 (Note $\mathrm{erfc}\left(0,x\right)=\mathrm{erfc}\left(x\right)$.)
 • The imaginary error function is defined by

$\mathrm{erfi}\left(x\right)=-I\mathrm{erf}\left(Ix\right)=\frac{2}{\sqrt{\pi }}\underset{0}{\overset{x}{\int }}{ⅇ}^{{t}^{2}}ⅆt$

 • All of these functions are entire.

Examples

 > $\mathrm{erf}\left(\mathrm{\infty }\right)$
 ${1}$ (1)
 > $\mathrm{erf}\left(3\right)$
 ${\mathrm{erf}}{}\left({3}\right)$ (2)
 > $\mathrm{evalf}\left(\right)$
 ${0.9999779095}$ (3)
 > $\mathrm{erfc}\left(3.\right)$
 ${0.00002209049700}$ (4)
 > $\mathrm{erf}\left(1.-1.I\right)$
 ${1.316151282}{-}{0.1904534692}{}{I}$ (5)
 > $\mathrm{erfc}\left(1.5-2.85I\right)$
 ${-62.82064889}{-}{10.56167495}{}{I}$ (6)
 > $\mathrm{diff}\left(\mathrm{erf}\left(x\right),x\right)$
 $\frac{{2}{}{{ⅇ}}^{{-}{{x}}^{{2}}}}{\sqrt{{\mathrm{\pi }}}}$ (7)
 > $\mathrm{diff}\left(\mathrm{erfc}\left(5,x\right),x\right)$
 ${-}{\mathrm{erfc}}{}\left({4}{,}{x}\right)$ (8)
 > $\mathrm{erfi}\left(-x\right)$
 ${-}{\mathrm{erfi}}{}\left({x}\right)$ (9)
 > $\mathrm{series}\left(\mathrm{erfi}\left(x\right),x,4\right)$
 $\frac{{2}}{\sqrt{{\mathrm{\pi }}}}{}{x}{+}\frac{{2}}{{3}}{}\frac{{1}}{\sqrt{{\mathrm{\pi }}}}{}{{x}}^{{3}}{+}{O}{}\left({{x}}^{{5}}\right)$ (10)
 > $\mathrm{expand}\left(\mathrm{erfc}\left(2,x\right),x\right)$
 $\frac{{{x}}^{{2}}}{{2}}{-}\frac{{{x}}^{{2}}{}{\mathrm{erf}}{}\left({x}\right)}{{2}}{-}\frac{{x}{}{{ⅇ}}^{{-}{{x}}^{{2}}}}{{2}{}\sqrt{{\mathrm{\pi }}}}{+}\frac{{1}}{{4}}{-}\frac{{\mathrm{erf}}{}\left({x}\right)}{{4}}$ (11)
 > $\mathrm{convert}\left(,\mathrm{erfc}\right)$
 $\frac{{{x}}^{{2}}}{{2}}{-}\frac{{{x}}^{{2}}{}\left({1}{-}{\mathrm{erfc}}{}\left({x}\right)\right)}{{2}}{-}\frac{{x}{}{{ⅇ}}^{{-}{{x}}^{{2}}}}{{2}{}\sqrt{{\mathrm{\pi }}}}{+}\frac{{\mathrm{erfc}}{}\left({x}\right)}{{4}}$ (12)

References

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