Error Function Derivation
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Derivative Of Error Function Complement
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Derivative Complementary Error Function
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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 How to prove that integration of exp(-x^2) is error function? [closed] up vote -1 down vote favorite Prove that $$\int e^{-x^2} dx=\frac{\sqrt{\pi}}{2}\rm erf (x).$$ integration share|cite|improve this question derivative gamma function edited Apr 17 '13 at 5:13 Mhenni Benghorbal 39.7k52966 asked Apr 17 '13 at 5:10 Litun John 399215 closed as unclear what you're asking by Micah, Adriano, Dominic Michaelis, Danny Cheuk, draks ... Jul 24 '13 at 5:30 Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question. 4 isn't the error function defined over the integral ? –Dominic Michaelis Apr 17 '13 at 5:12 2 By definition, $$\text{erf}(x)=\frac2{\sqrt\pi}\int_0^x\exp(-w^2)\,dw.$$ Is this a different definition than you've been given? Also, since your integral is indefinite, don't forget your constant of integration. –Cameron Buie Apr 17 '13 at 5:17 add a comment| 1 Answer 1 active oldest votes up vote 3 down vote accepted There's nothing here to prove, the definition of the error function is th
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 derivative normal distribution 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> derivative gaussian Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> differentiation error function Less... 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 http://math.stackexchange.com/questions/364112/how-to-prove-that-integration-of-exp-x2-is-error-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, 2
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