Erf -- From Wolfram MathWorld

is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by

(1)

Note that some authors (e.g., Whittaker and Watson 1990, p. 341) define without the leading factor of .

Erf is implemented in the Wolfram Language as Erf[z]. A two-argument form giving is also implemented as Erf[z0, z1].

Erf satisfies the identities

			(2)
			(3)
			(4)

where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. For ,

(5)

where is the incomplete gamma function.

Erf can also be defined as a Maclaurin series

			(6)
			(7)

(OEIS A007680). Similarly,

(8)

(OEIS A103979 and A103980).

For , may be computed from

			(9)
			(10)

(OEIS A000079 and A001147; Acton 1990).

For ,

			(11)
			(12)

Using integration by parts gives

			(13)
			(14)
			(15)
			(16)

so

(17)

and continuing the procedure gives the asymptotic series

			(18)
			(19)
			(20)

(OEIS A001147 and A000079).

Erf has the values

			(21)
			(22)

It is an odd function

(23)

and satisfies

(24)

Erf may be expressed in terms of a confluent hypergeometric function of the first kind as

			(25)
			(26)

Its derivative is

(27)

where is a Hermite polynomial. The first derivative is

(28)

and the integral is

(29)

Erf can also be extended to the complex plane, as illustrated above.

A simple integral involving erf that Wolfram Language cannot do is given by

(30)

(M. R. D'Orsogna, pers. comm., May 9, 2004). More complicated integrals include

(31)

(M. R. D'Orsogna, pers. comm., Dec. 15, 2005).

Erf has the continued fraction

			(32)
			(33)

(Wall 1948, p. 357), first stated by Laplace in 1805 and Legendre in 1826 (Olds 1963, p. 139), proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp. 8-9).

Definite integrals involving include Definite integrals involving include

			(34)
			(35)
			(36)
			(37)
			(38)

The first two of these appear in Prudnikov et al. (1990, p. 123, eqns. 2.8.19.8 and 2.8.19.11), with , .

A complex generalization of is defined as

			(39)
			(40)

Integral representations valid only in the upper half-plane are given by

			(41)
			(42)