Definite Integral Complementary Error Function
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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 Calculus and Analysis>Special complementary error function table Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Calculus and Analysis>Calculus>Integrals>Definite Integrals> More... Interactive Entries>webMathematica complementary error function calculator Examples> History and Terminology>Wolfram Language Commands> Less... Erfc Erfc is the complementary error function, commonly denoted , is an entire complementary error function excel function defined by (1) (2) It is implemented in the Wolfram Language as Erfc[z]. Note that some authors (e.g., Whittaker and Watson 1990, p.341) define without the leading factor of . For ,
Inverse Complementary Error Function
(3) where is the incomplete gamma function. The derivative is given by (4) and the indefinite integral by (5) It has the special values (6) (7) (8) It satisfies the identity (9) It has definite integrals (10) (11) (12) For , is bounded by (13) Min Max Re Im Erfc can also be extended to the complex plane, as illustrated above. A generalization is obtained from the complementary error function in matlab erfc differential equation (14) (Abramowitz and Stegun 1972, p.299; Zwillinger 1997, p.122). The general solution is then (15) where is the repeated erfc integral. For integer , (16) (17) (18) (19) (Abramowitz and Stegun 1972, p.299), where is a confluent hypergeometric function of the first kind and is a gamma function. The first few values, extended by the definition for and 0, are given by (20) (21) (22) SEE ALSO: Erf, Erfc Differential Equation, Erfi, Inverse Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfc/ REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). "Repeated Integrals of the Error Function." §7.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp.299-300, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp.568-569, 1985. Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; and Vetterling, W.T. "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp.209-214, 1992. Spanier, J. and Oldham, K.B. "The Error Function and Its Complement " and "The and and Related Functions." Chs.40 and 41 in An Atlas of Functions. Washington, DC: Hemisphere, pp.385-393
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Integral Of Error Function
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Involving exponential function and trigonometric functions Involving exp and sin Involving exp and cos Involving power, exponential and trigonometric functions Involving power, http://mathworld.wolfram.com/Erfc.html exp and sin Involving power, exp and cos Involving hyperbolic functions Involving sinh Involving cosh Involving hyperbolic functions and a power function Involving sinh and power Involving cosh and power Involving exponential function and hyperbolic functions Involving exp and sinh Involving exp and cosh http://functions.wolfram.com/GammaBetaErf/Erfc/21/ShowAll.html Involving power, exponential and hyperbolic functions Involving power, exp and sinh Involving power, exp and cosh Involving logarithm Involving log Involving logarithm and a power function Involving log and power Involving functions of the direct function Involving elementary functions of the direct function Involving powers of the direct function Involving products of the direct function Involving functions of the direct function and elementary functions Involving elementary functions of the direct function and elementary functions Involving powers of the direct function and a power function Involving products of the direct function and a power function Involving power of the direct function and exponential function Involving direct function and Gamma-, Beta-, Erf-type functions Involving erf-type functions Involving erf Involving erf-type functions and a power function Involving erf and power Definite integration For the direct function itself Involving the direct functionhere for a quick overview of the site Help Center Detailed answers http://math.stackexchange.com/questions/1210623/integral-involving-an-error-function to any questions you might have Meta Discuss the workings and policies 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 error function question and answer site for people studying math 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 complementary error function involving an error function up vote 5 down vote favorite 3 For $\sigma>0$, how can we prove that $$\frac{1}{2}\int_{-1}^1 \text{erf}\left(\frac{\sigma}{\sqrt{2}}+\text{erf}^{-1}(x)\right) \, \mathrm{d}x= \text{erf}\left(\frac{\sigma}{2}\right)$$ where erf is the error function, $$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}dt.$$ The result was obtained by tinkering, and I was wondering if there a concise derivation. integration definite-integrals error-function share|cite|improve this question edited Mar 28 '15 at 23:10 Eric Naslund 53.4k10119210 asked Mar 28 '15 at 21:49 Nero 1,492963 1 First thing that comes to mind is to change variable (y = invert of erf). Then look at the integral as a convolution of erf and its derivative, then use convolution theorem. Overall it requires around 5 steps. –user227136 Mar 28 '15 at 22:36 add a comment| 2 Answers 2 active oldest votes up vote 9 down vote accepted Let $u=\text{erf}^{-1}(x)$ so that $x=\text{erf}\left(u\right)$, and $dx=\frac{2}{\sqrt{\pi}}e^{-u^{2}}du.$ Then $$I(\sigma)=\frac{1}{2}\int_{-1}^{1}\text{erf}\left(\frac{\sigma}{\sqrt{2}}+\text{erf}^{-1}(x)\right)dx=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-u^{2}}\text{erf}\left(\frac{\sigma}{\sqrt{2}}+u\right)du.$$ Differentiating with respect to $\sigma$ w