Gaussian Error Integral
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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 x e − t 2 d t = 2 π ∫ 0 x e − t 2 d t . {\displaystyle
Gaussian Integral Table
{\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − error function calculator 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is error function table defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin
Inverse Error Function
2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} 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 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) =
Integral Of E^-ax^2
2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\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 Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 8 (x)} is real when x is real. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) = e − z 2 erfc ( − i z ) = erfcx ( − i z ) . {\displaystyle w(z)=e^{-z^ 6}\operatorname 5 (-iz)=\operatorname 4 (-iz).} Contents 1 The name "error function" 2 Properties 2.1 Taylor series 2.2 Derivative and integral 2.3 Bürmann series 2.4 Inverse functions 2.5 Asym
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 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch error function matlab Calculus and Analysis>Special Functions>Erf> Gaussian Integral The Gaussian integral, also called the
Error Function Python
probability integral and closely related to the erf function, is the integral of the one-dimensional Gaussian function over . complementary error function table It can be computed using the trick of combining two one-dimensional Gaussians (1) (2) (3) Here, use has been made of the fact that the variable in the integral is a dummy https://en.wikipedia.org/wiki/Error_function variable that is integrates out in the end and hence can be renamed from to . Switching to polar coordinates then gives (4) (5) (6) There also exists a simple proof of this identity that does not require transformation to polar coordinates (Nicholas and Yates 1950). The integral from 0 to a finite upper limit can be given by the continued fraction (7) http://mathworld.wolfram.com/GaussianIntegral.html (8) where is erf (the error function), as first stated by Laplace, proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp.8-9). The general class of integrals of the form (9) can be solved analytically by setting (10) (11) (12) Then (13) (14) For , this is just the usual Gaussian integral, so (15) For , the integrand is integrable by quadrature, (16) To compute for , use the identity (17) (18) (19) (20) For even, (21) (22) (23) (24) (25) so (26) (27) where is a double factorial. If is odd, then (28) (29) (30) (31) (32) so (33) The solution is therefore (34) The first few values are therefore (35) (36) (37) (38) (39) (40) (41) A related, often useful integral is (42) which is simply given by (43) The more general integral of has the following closed forms, (44) (45) (46) for integer (F.Pilolli, pers. comm.). For (45) and (46), (the punctured plane), , and . Here, is a confluent hypergeometric function of the second kind and is a binomial coefficient. SEE ALSO: Erf, Gauss Integral, Gaussian Function, Leibniz Integral Rule, Norma
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies http://math.stackexchange.com/questions/1735883/definite-integral-of-error-function-times-exponential-and-gaussian of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute: Sign error function 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 Definite integral of error function times exponential and Gaussian up vote 0 down vote favorite I am looking for the solution of the following integral $$\int_{-\infty}^\infty\text{d}x\,\text{erf}(x)e^{-a x^2-bx}$$ where $a$ is real but $b$ is in general complex. For error function table the case of $b$ being real the solution is (Section 4.3, Eq. 13 of http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf) $$\int_{-\infty}^\infty\text{d}x\,\text{erf}(x)e^{-(a x+b)^2}=-\frac{\sqrt{\pi}}{a}\text{erf}\left(\frac{b}{\sqrt{a^2+1}}\right)$$ when $\text{Re}(a^2)>0$. Does anyone know a closed form for the integral when $b\in\mathbb{C}$? integration definite-integrals closed-form complex-integration share|cite|improve this question asked Apr 10 at 11:41 Alex 646211 add a comment| active oldest votes Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook. Your Answer draft saved draft discarded Sign up or log in Sign up using Google Sign up using Facebook Sign up using Email and Password Post as a guest Name Email Post as a guest Name Email discard By posting your answer, you agree to the privacy policy and terms of service. Browse other questions tagged integration definite-integrals closed-form complex-integration or ask your own question. asked 6 months ago viewed 110 times Related 2Integrating an exponential times an error function; expansion needed12Closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) \, dx$0Definite integral involving Error function6Integral involving the error function of log(x)1Integrating a product of to error functions and an exponential3Closed forms for definite integrals involving error functions1Integrals wi
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