Erf

The ``error function'' encountered in integrating the Gaussian Distribution.

$$\mathop{\rm erf}\nolimits (z) \equiv {2\over\sqrt{\pi}} \int^z_0 e^{-t^2}\, dt$$ (1)

$$= 1-\mathop{\rm erfc}\nolimits (z)$$ (2)

$$= \sqrt{\pi}\, \gamma({\textstyle{1\over 2}}, z^2),$$ (3)

where Erfc is the complementary error function and $\gamma(x,a)$ is the incomplete Gamma Function. It can also be defined as a Maclaurin Series

$$\mathop{\rm erf}\nolimits (z)={2\over\sqrt{\pi}}\sum_{n=0}^\infty {(-1)^nz^{2n+1}\over n!(2n+1)}.$$ (4)

Erf has the values

$$\mathop{\rm erf}\nolimits (0) = 0$$ (5)

$$\mathop{\rm erf}\nolimits (\infty) = 1.$$ (6)

It is an Odd Function

$$\mathop{\rm erf}\nolimits (-z)=-\mathop{\rm erf}\nolimits (z),$$ (7)

and satisfies

$$\mathop{\rm erf}\nolimits (z)+\mathop{\rm erfc}\nolimits (z) = 1.$$ (8)

Erf may be expressed in terms of a Confluent Hypergeometric Function of the First Kind $M$ as

$$\mathop{\rm erf}\nolimits (z)= {2z\over \sqrt{\pi}} M({\text... ...2z\over \sqrt{\pi}}e^{-z^2} M(1, {\textstyle{3\over 2}}, z^2).$$ (9)

Erf is bounded by

$${1\over x+\sqrt{x^2+2}} < e^{x^2}\int_x^\infty e^{-t^2}\,dt \leq {1\over x+\sqrt{x^2+{4\over \pi}}}.$$ (10)

Its Derivative is

$${d^n\over dz^n} \,\mathop{\rm erf}\nolimits (z) = (-1)^{n-1} {2\over \sqrt{\pi}} H_n(z)e^{-z^2},$$ (11)

where $H_n$ is a Hermite Polynomial. The first Derivative is

$${d\over dz} \mathop{\rm erf}\nolimits (z)={2\over\sqrt{\pi}} e^{-z^2/2},$$ (12)

and the integral is

$$\int \mathop{\rm erf}\nolimits (z)\,dz = z\mathop{\rm erf}\nolimits (z)+{e^{-z^2}\over\sqrt{\pi}}.$$ (13)

For $x\ll 1$, erf may be computed from

$$\mathop{\rm erf}\nolimits (x) = {2\over \sqrt{\pi}} \int_0^x e^{-t^2}\,dt$$ (14)

$$= {2\over \sqrt{\pi}}\int_0^x \sum_{k=0}^\infty {{(-t^2)}^k\over k!}\,dt$$

$$= {2\over\sqrt{\pi}} \int_0^x \sum_{k=0}^\infty {(-1)^kt^{2k}\over k!}\,dt$$

$$= {2\over\sqrt{\pi}} \sum_{k=0}^\infty {x^{2k+1}(-1)^k\over k!(2k+1)}$$ (15)

$$= {2\over\sqrt{\pi}} (x-{\textstyle{1\over 3}}x^3+{\textstyle{1\over 10}}x^5-{\textstyle{1\over 42}}x^7+{\textstyle{1\over 216}}x^9$$

$$\phantom{=} -{\textstyle{1\over 1320}}x^{11}+\ldots)$$ (16)

$$= {2\over\sqrt{\pi}} e^{-x^2}x \left[{1+{2x^2\over 1\cdot 3}+{(2x)^2\over 1\cdot 3\cdot 5}+\ldots}\right]$$ (17)

(Acton 1990). For $x\gg 1$,

$$\mathop{\rm erf}\nolimits (x) = {2\over \sqrt{\pi}}\left({\int_0^\infty e^{-t^2}\,dt -\int_x^\infty e^{-t^2}\,dt}\right)$$

$$= 1-{2\over \sqrt{\pi}} \int_x^\infty e^{-t^2}\,dt.$$ (18)

Using Integration by Parts gives

$$\int_x^\infty e^{-t^2}\,dt = -{1\over 2}\int_x^\infty {1\over t}d(e^{-t^2})$$

$$= -{1\over 2}\left[{e^{-t^2}\over t}\right]^\infty_x -{1\over 2}\int_x^\infty {e^{-t^2}\,dt\over t^2}$$

$$= {e^{-x^2}\over 2x} +{1\over 4}\int_x^\infty {1\over t^3}d(e^{-t^2})$$

$$= {e^{-x^2}\over 2x} - {e^{-x^2}\over 4x^3}-\ldots,$$ (19)

so

$$\mathop{\rm erf}\nolimits (x) = 1-{e^{-x^2}\over \sqrt{\pi}\,x} \left({1-{1\over 2x^2}-\ldots}\right)$$ (20)

and continuing the procedure gives the Asymptotic Series

$$\mathop{\rm erf}\nolimits (x)=1-{e^{-x^2}\over\sqrt{\pi}}(x^... ...yle{15\over 8}}x^{-7}+{\textstyle{105\over 16}}x^{-9}+\ldots).$$ (21)

A Complex generalization of $\mathop{\rm erf}\nolimits$ is defined as

$$w(z) \equiv e^{-z^2}\mathop{\rm erfc}\nolimits (-iz)$$ (22)

$$= e^{-z^2} \left({1+{2i\over \sqrt{\pi}} + {2i\over \sqrt{\pi}} \int_0^z e^{t^2}\,dt}\right)$$ (23)

$$= {i\over \pi} \int_{-\infty}^\infty {e^{-t^2}\,dt\over z-t} = {2iz\over \pi}\int_0^\infty {e^{-t^2}\,dt\over z^2-t^2}.$$ (24)

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 $\mathop{\rm erf}\nolimits (x)$ and Its Complement $\mathop{\rm erfc}\nolimits (x)$.'' Ch. 40 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 385-393, 1987.