Gauss Error Function
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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 error function calculator e − t 2 d t = 2 π ∫ 0 x e
Error Function Table
− t 2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^
Inverse Error Function
− 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined
Error Function Matlab
as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 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 error function excel 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 ) = 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 Da
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 error function python WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> complementary error function table More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal erf(1) distribution (which is a normalized form of 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 https://en.wikipedia.org/wiki/Error_function 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 function of the first kind. For , (5) where is the incomplete gamma function. Erf can also be defined as http://mathworld.wolfram.com/Erf.html 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, 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 (Watso
ei pi SubscribeSubscribedUnsubscribe233233 Loading... Loading... Working... Add to Want to watch this again later? Sign in to add this video to a playlist. Sign in Share More Report https://www.youtube.com/watch?v=CcFUQhorgdc Need to report the video? Sign in to report inappropriate content. Sign in Transcript Statistics 16,913 views 45 Like this video? Sign in to make your opinion count. Sign in 46 6 Don't like this video? Sign in to make your opinion count. Sign in 7 Loading... Loading... Transcript The interactive transcript could not be loaded. Loading... error function Loading... Rating is available when the video has been rented. This feature is not available right now. Please try again later. Published on Nov 8, 2013This is a special function related to the Gaussian. In this video I derive it. Category Education License Standard YouTube License Show more Show less Loading... Autoplay When autoplay is enabled, a suggested error function table video will automatically play next. Up next Error Function and Complimentary Error Function - Duration: 5:01. StudyYaar.com 11,854 views 5:01 Evaluating the Error Function - Duration: 6:36. lesnyk255 1,783 views 6:36 Integral of exp(-x^2) | MIT 18.02SC Multivariable Calculus, Fall 2010 - Duration: 9:34. MIT OpenCourseWare 204,132 views 9:34 erf(x) function - Duration: 9:59. Calculus Society -ROCKS!! 946 views 9:59 Evaluation of the Gaussian Integral exp(-x^2) - Cool Math Trick - Duration: 5:22. TouchHax 47,737 views 5:22 Fick's Law of Diffusion - Duration: 12:21. khanacademymedicine 136,701 views 12:21 Approximation of Error in Hindi - Duration: 42:24. Bhagwan Singh Vishwakarma 4,270 views 42:24 Video 1690 - ERF Function - Duration: 5:46. Chau Tu 629 views 5:46 The Gaussian Distribution - Duration: 9:49. Steve Grambow 22,999 views 9:49 Gaussian - Duration: 4:28. Paul Francis 15,941 views 4:28 MSE101 Data Analysis - L4.2 Integrating the Gaussian between limits - the erf function - Duration: 19:19. David Dye 834 views 19:19 Hyperbolic Sine and Cosine Functions (Tanton Mathematics) - Duration: 13:45. DrJamesTanton 13,324 views 13:45 Diffusi