The ``error function'' encountered in integrating the Gaussian Distribution.
where Erfc is the complementary error function and
is the incomplete Gamma Function. It can
also be defined as a Maclaurin Series
![\begin{displaymath}
\mathop{\rm erf}\nolimits (z)={2\over\sqrt{\pi}}\sum_{n=0}^\infty {(-1)^nz^{2n+1}\over n!(2n+1)}.
\end{displaymath}](e_1670.gif) |
(4) |
Erf has the values
It is an Odd Function
![\begin{displaymath}
\mathop{\rm erf}\nolimits (-z)=-\mathop{\rm erf}\nolimits (z),
\end{displaymath}](e_1673.gif) |
(7) |
and satisfies
![\begin{displaymath}
\mathop{\rm erf}\nolimits (z)+\mathop{\rm erfc}\nolimits (z) = 1.
\end{displaymath}](e_1674.gif) |
(8) |
Erf may be expressed in terms of a Confluent Hypergeometric Function of the First Kind
as
![\begin{displaymath}
\mathop{\rm erf}\nolimits (z)= {2z\over \sqrt{\pi}} M({\text...
...2z\over \sqrt{\pi}}e^{-z^2} M(1, {\textstyle{3\over 2}}, z^2).
\end{displaymath}](e_1675.gif) |
(9) |
Erf is bounded by
![\begin{displaymath}
{1\over x+\sqrt{x^2+2}} < e^{x^2}\int_x^\infty e^{-t^2}\,dt \leq {1\over x+\sqrt{x^2+{4\over \pi}}}.
\end{displaymath}](e_1676.gif) |
(10) |
Its Derivative is
![\begin{displaymath}
{d^n\over dz^n} \,\mathop{\rm erf}\nolimits (z) = (-1)^{n-1} {2\over \sqrt{\pi}} H_n(z)e^{-z^2},
\end{displaymath}](e_1677.gif) |
(11) |
where
is a Hermite Polynomial. The first Derivative is
![\begin{displaymath}
{d\over dz} \mathop{\rm erf}\nolimits (z)={2\over\sqrt{\pi}} e^{-z^2/2},
\end{displaymath}](e_1679.gif) |
(12) |
and the integral is
![\begin{displaymath}
\int \mathop{\rm erf}\nolimits (z)\,dz = z\mathop{\rm erf}\nolimits (z)+{e^{-z^2}\over\sqrt{\pi}}.
\end{displaymath}](e_1680.gif) |
(13) |
For
, erf may be computed from
(Acton 1990). For
,
Using Integration by Parts gives
so
![\begin{displaymath}
\mathop{\rm erf}\nolimits (x) = 1-{e^{-x^2}\over \sqrt{\pi}\,x} \left({1-{1\over 2x^2}-\ldots}\right)
\end{displaymath}](e_1698.gif) |
(20) |
and continuing the procedure gives the Asymptotic Series
![\begin{displaymath}
\mathop{\rm erf}\nolimits (x)=1-{e^{-x^2}\over\sqrt{\pi}}(x^...
...yle{15\over 8}}x^{-7}+{\textstyle{105\over 16}}x^{-9}+\ldots).
\end{displaymath}](e_1699.gif) |
(21) |
A Complex generalization of
is defined as
See also Dawson's Integral,
Erfc, Erfi, Gaussian Integral, Normal Distribution Function, Probability Integral
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Error Function'' and ``Repeated Integrals of the Error Function.''
§7.1-7.2 in Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 297-300, 1972.
Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 16, 1990.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 568-569, 1985.
Spanier, J. and Oldham, K. B. ``The Error Function
and Its Complement
.''
Ch. 40 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 385-393, 1987.
© 1996-9 Eric W. Weisstein
1999-05-25