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)
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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)
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(3)
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(4)
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where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. For ,
(5)
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where is the incomplete gamma function.
Erf can also be defined as a Maclaurin series
(6)
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(7)
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(OEIS A007680). Similarly,
(8)
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For , may be computed from
(9)
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(10)
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(OEIS A000079 and A001147; Acton 1990).
For ,
(11)
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(12)
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Using integration by parts gives
(13)
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(14)
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(15)
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(16)
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so
(17)
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and continuing the procedure gives the asymptotic series
(18)
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(19)
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(20)
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Erf has the values
(21)
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(22)
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It is an odd function
(23)
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and satisfies
(24)
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Erf may be expressed in terms of a confluent hypergeometric function of the first kind as
(25)
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(26)
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Its derivative is
(27)
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where is a Hermite polynomial. The first derivative is
(28)
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and the integral is
(29)
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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)
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(M. R. D'Orsogna, pers. comm., May 9, 2004). More complicated integrals include
(31)
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(M. R. D'Orsogna, pers. comm., Dec. 15, 2005).
Erf has the continued fraction
(32)
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(33)
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(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)
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(35)
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(36)
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(37)
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(38)
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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)
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(40)
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Integral representations valid only in the upper half-plane are given by
(41)
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(42)
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