Product Of Error Functions
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at any level and professionals in related fields. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Integral of product of two error functions (erf) up vote 10 down vote error function properties favorite 3 In the course of my research I came across the following integral: $$\int\nolimits_{-\infty}^{\infty}\operatorname{erf}(a+x)\operatorname{erf}(a-x)dx$$ where $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^2}dt$$ is the familiar error function. Does anyone know if this is solvable? If so, a suggestion on how to do this would be very much appreciated. I can tolerate a solution in terms of $\operatorname{erf}$ functions of $a$. I have tried my usual sources in these matters (G&R, the tables on the Wolfram website, Abramowitz's Handbook of Mathematical Formulas, the web) but couldn't find anything of use. Thank you! integration special-functions share|cite|improve this question edited Nov 25 '15 at 18:00 Michael Medvinsky 5,2003930 asked Sep 9 '11 at 5:03 M.B.M. 2,0111827 6 Perhaps you are interested in $$\int_{-\infty}^\infty \left( \text{erf}(a+x) \text{erf}(a-x)+1 \right)\ dx = \frac{2 \sqrt{2}}{\sqrt{\pi}} e^{-2 a^2} + 4 a\ \text{erf}(\sqrt{2} a)$$ –Robert Israel Sep 9 '11 at 5:45 This came from trying to find an expression the differential entropy of a random variable that is d
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Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of error function python 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 http://math.stackexchange.com/questions/63026/integral-of-product-of-two-error-functions-erf 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 http://mathworld.wolfram.com/Erf.html , 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.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. (19
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