Imaginary Error Function Wikipedia
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around the origin In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson[1]) is either F ( x ) = D + ( x ) = e − x 2 ∫ 0 x e t 2 d t {\displaystyle F(x)=D_{+}(x)=e^{-x^{2}}\int _{0}^{x}e^{t^{2}}\,dt} , also denoted as F(x) complementary error function or D(x), or alternatively D − ( x ) = e x 2 ∫ 0 x e −
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t 2 d t {\displaystyle D_{-}(x)=e^{x^{2}}\int _{0}^{x}e^{-t^{2}}\,dt\!} . The Dawson function is the one-sided Fourier-Laplace sine transform of the Gaussian function, D + ( x ) =
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1 2 ∫ 0 ∞ e − t 2 / 4 sin ( x t ) d t . {\displaystyle D_{+}(x)={\frac {1}{2}}\int _{0}^{\infty }e^{-t^{2}/4}\,\sin {(xt)}\,dt.} It is closely related to the error function erf, as D + ( x ) = π 2
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e − x 2 e r f i ( x ) = − i π 2 e − x 2 e r f ( i x ) {\displaystyle D_{+}(x)={{\sqrt {\pi }} \over 2}e^{-x^{2}}\mathrm {erfi} (x)=-{i{\sqrt {\pi }} \over 2}e^{-x^{2}}\mathrm {erf} (ix)} where erfi is the imaginary error function, erfi(x) = −i erf(ix). Similarly, D − ( x ) = π 2 e x 2 e r f ( x ) {\displaystyle D_{-}(x)={\frac {\sqrt {\pi }}{2}}e^{x^{2}}\mathrm {erf} (x)} in terms of the real error function, erf. In terms of either error function matlab erfi or the Faddeeva function w(z), the Dawson function can be extended to the entire complex plane:[2] F ( z ) = π 2 e − z 2 e r f i ( z ) = i π 2 [ e − z 2 − w ( z ) ] {\displaystyle F(z)={{\sqrt {\pi }} \over 2}e^{-z^{2}}\mathrm {erfi} (z)={\frac {i{\sqrt {\pi }}}{2}}\left[e^{-z^{2}}-w(z)\right]} , which simplifies to D + ( x ) = F ( x ) = π 2 Im [ w ( x ) ] {\displaystyle D_{+}(x)=F(x)={\frac {\sqrt {\pi }}{2}}\operatorname {Im} [w(x)]} D − ( x ) = i F ( − i x ) = − π 2 [ e x 2 − w ( − i x ) ] {\displaystyle D_{-}(x)=iF(-ix)=-{\frac {\sqrt {\pi }}{2}}\left[e^{x^{2}}-w(-ix)\right]} for real x. For |x| near zero, F(x) ≈ x, and for |x| large, F(x) ≈ 1/(2x). More specifically, near the origin it has the series expansion F ( x ) = ∑ k = 0 ∞ ( − 1 ) k 2 k ( 2 k + 1 ) ! ! x 2 k + 1 = x − 2 3 x 3 + 4 15 x 5 − ⋯ {\displaystyle F(x)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\,2^{k}}{(2k+1)!!}}\,x^{2k+1}=x-{\frac {2}{3}}x^{3}+{\frac {4}{15}}x^{5}-\cdots } , while for large x it has the asymptotic expansion F ( x ) = ∑ k = 0 ∞ ( 2 k − 1 ) ! ! 2 k + 1 x 2 k + 1 = 1 2 x + 1 4 x 3 + 3 8 x 5 + ⋯ {\displaystyle F(x)=\sum _{k=0}^{\infty }{\frac {(2k-1)!!}{2^{k+1}x^{2k+1}}}={\frac {1}{2x}}+{\frac {1}
Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Tue Sep 27 error function excel 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> error function python Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... error function properties Erf 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 https://en.wikipedia.org/wiki/Dawson_function (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 http://mathworld.wolfram.com/Erf.html 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) 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). M
where it was expressed through the following integral: Later C. Kramp (1799) used this integral for the definition of the complementary error function . P.‐S. Laplace (1812) derived an asymptotic expansion of the error function. The probability http://functions.wolfram.com/GammaBetaErf/Erfi/introductions/ProbabilityIntegrals/ShowAll.html integrals were so named because they are widely applied in the theory of probability, in both normal and limit distributions. To obtain, say, a normal distributed random variable from a uniformly distributed random variable, the inverse of the error function, namely is needed. The inverse was systematically investigated in the second half of the twentieth century, especially by J. R. Philip (1960) and A. J. Strecok (1968).
Definitions of error function probability integrals and inverses The probability integral (error function) , the generalized error function , the complementary error function , the imaginary error function , the inverse error function , the inverse of the generalized error function , and the inverse complementary error function are defined through the following formulas: These seven functions are typically called probability integrals and their inverses. Instead of using definite integrals, the three univariate error functions imaginary error function can be defined through the following infinite series. A quick look at the probability integrals and inversesHere is a quick look at the graphics for the probability integrals and inverses along the real axis. Connections within the group of probability integrals and inverses and with other function groups Representations through more general functions The probability integrals , , , and are the particular cases of two more general functions: hypergeometric and Meijer G functions. For example, they can be represented through the confluent hypergeometric functions and : Representations of the probability integrals , , , and through classical Meijer G functions are rather simple: The factor in the last four formulas can be removed by changing the classical Meijer G functions to the generalized one: The probability integrals , , , and are the particular cases of the incomplete gamma function, regularized incomplete gamma function, and exponential integral : Representations through related equivalent functions The probability integrals , , and can be represented through Fresnel integrals by the following formulas: Representations through other probability integrals and inverses The probability integrals and their inverses , , , , , , and are interconnected by the following formulas: The best-known properties and formulas for probability integrals and inverses Real values f