Complementary Error Function Asymptotic Expansions
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x error function integral x e − t 2 d t = 2 π ∫ 0
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x e − t 2 d t . {\displaystyle {\begin − 5\operatorname − 4 (x)&={\frac − 3{\sqrt {\pi error function table }}}\int _{-x}^ − 2e^{-t^ − 1}\,\mathrm − 0 t\\&={\frac 9{\sqrt {\pi }}}\int _ 8^ 7e^{-t^ 6}\,\mathrm 5 t.\end 4}} The complementary error function, denoted
Erf(inf)
erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 1\operatorname 0 (x)&=1-\operatorname Φ 9 (x)\\&={\frac Φ 8{\sqrt {\pi }}}\int _ Φ 7^{\infty }e^{-t^ Φ 6}\,\mathrm Φ asymptotic expansion examples 5 t\\&=e^{-x^ Φ 4}\operatorname Φ 3 (x),\end Φ 2}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 1 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname Φ 9 (x|x\geq 0)={\frac Φ 8{\pi }}\int _ Φ 7^{\pi /2}\exp \left(-{\frac Φ 6}{\sin ^ Φ 5\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin − 9\operatorname − 8 (x)&=-i\operatorname − 7 (ix)\\&={\frac − 6{\sqrt {\pi }}}\int _ − 5^ − 4e^ − 3}\,\mathrm − 2 t\\&={\frac − 1{\sqrt {\pi }}}e^ − 0}D
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and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire
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Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error asymptotic expansion of exponential function 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 https://en.wikipedia.org/wiki/Error_function 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 http://mathworld.wolfram.com/Erf.html 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) Min Max Re Im 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.3
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