Gauss 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 how to integrate e^2x d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ error function calculator − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The error function table complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 inverse error function erfcx ( x ) , {\displaystyle {\begin 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
Error Function Matlab
Craig's formula:[5] erfc ( x | x ≥ 0 ) = 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
The blue line is the polynomial y ( x ) = 7 x 3 − 8 x 2 − 3 x + 3 {\displaystyle y(x)=7x^ ω 2-8x^ ω 1-3x+3} , whose integral
Error Function Excel
in [-1, 1] is 2/3. The trapezoidal rule returns the integral of error function python the orange dashed line, equal to y ( − 1 ) + y ( 1 ) = − 10 error function properties {\displaystyle y(-1)+y(1)=-10} . The 2-point Gaussian quadrature rule returns the integral of the black dashed curve, equal to y ( − 1 / 3 ) + y ( 1 / 3 https://en.wikipedia.org/wiki/Error_function ) = 2 / 3 {\displaystyle y({-{\sqrt {\scriptstyle 1/3}}})+y({\sqrt {\scriptstyle 1/3}})=2/3} . Such a result is exact since the green region has the same area as the red regions. In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See https://en.wikipedia.org/wiki/Gaussian_quadrature numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi and weights wi for i = 1, ..., n. The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as ∫ − 1 1 f ( x ) d x = ∑ i = 1 n w i f ( x i ) . {\displaystyle \int _{-1}^ − 8f(x)\,dx=\sum _ − 7^ − 6w_ − 5f(x_ − 4).} Gaussian quadrature as above will only produce good results if the function f(x) is well approximated by a polynomial function within the range [−1, 1]. The method is not, for example, suitable for functions with singularities. However, if the integrated function can be written as f ( x ) = ω ( x ) g ( x ) {\displaystyle f(x)=\omega (x)g(x)\,} , where g(x) is approximately polynomial and ω(x) is known, then alternative weights w i
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: error function 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 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 gauss error integral complex. For 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 involv