Antiderivative 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 e − t integral complementary error function 2 d t = 2 π ∫ 0 x e − t integral of error function with gaussian density function 2 d t . {\displaystyle {\begin − 2\operatorname − 1 (x)&={\frac − 0{\sqrt {\pi }}}\int _{-x}^ 9e^{-t^ 8}\,\mathrm error function values 7 t\\&={\frac 6{\sqrt {\pi }}}\int _ 5^ 4e^{-t^ 3}\,\mathrm 2 t.\end 1}} The complementary error function, denoted erfc, is defined as erfc ( error function integral table x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin Φ 8\operatorname Φ 7 (x)&=1-\operatorname Φ 6 (x)\\&={\frac Φ 5{\sqrt {\pi }}}\int _ Φ 4^{\infty }e^{-t^ Φ 3}\,\mathrm Φ 2 t\\&=e^{-x^ Φ 1}\operatorname Φ 0 (x),\end 9}} which
Error Function Integral Calculation
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 Φ 8 (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 Φ 6 (x|x\geq 0)={\frac Φ 5{\pi }}\int _ Φ 4^{\pi /2}\exp \left(-{\frac Φ 3}{\sin ^ Φ 2\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 − 6\operatorname − 5 (x)&=-i\operatorname − 4 (ix)\\&={\frac − 3{\sqrt {\pi }}}\int _ − 2^ − 1e^ − 0}\,\mathrm − 9 t\\&={\frac − 8{\sqrt {\pi }}}e^ − 7}D(x),\end − 6}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite t
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workings and policies of this site About Us Learn more about erf function table Stack Overflow the company Business Learn more about hiring developers or posting ads with us Mathematics Questions Tags inverse error function 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; https://en.wikipedia.org/wiki/Error_function 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 What is the antiderivative of $e^{-x^2}$ up vote 35 down vote favorite 13 I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it http://math.stackexchange.com/questions/523824/what-is-the-antiderivative-of-e-x2 I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space \text{erf} (x) + C$$ So, I of course didn't know what this $\text{erf}$ was and I looked it up on wikipedia, where it was defined as: $$ \text{erf}(x) = \dfrac{2}{\sqrt{\pi}} \displaystyle \int_0^x e^{-t^2} \textrm{d}t $$ To my mathematically illiterate mind, this is a bit too circular to understand. Why can't we express $\int e^{-x^2} \textrm{d}x$ as a 'normal function'? Also, what is the use of the error function? error-function share|cite|improve this question edited Oct 14 '13 at 9:48 in_wolframAlpha_we_trust 2,175419 asked Oct 12 '13 at 19:34 Phaptitude 9071023 1 You should read this: en.wikipedia.org/wiki/Elementary_function –M Turgeon Oct 12 '13 at 19:37 9 This pretty much says that an antiderivative of $e^{-x^2}$ is an antiderivative of $e^{-x^2}$, and that's the best you can do. –1015 Oct 12 '13 at 19:38 You can read this paper on impossibility theorems for elementary integration: claymath.org/programs/outreach/academy/LectureNotes05/… –Kaa1el Oct 12 '13 at 21:30 add a comment| 6 Answers 6 active oldest votes up vote
ei pi SubscribeSubscribedUnsubscribe226226 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 Need to report the video? Sign https://www.youtube.com/watch?v=CcFUQhorgdc in to report inappropriate content. Sign in Transcript Statistics 16,332 views 43 Like this video? Sign in to make your opinion count. Sign in 44 5 Don't like this video? Sign in to make your opinion count. Sign in 6 Loading... Loading... Transcript The interactive transcript could not be loaded. Loading... Loading... Rating is available when the video has been rented. This feature error function 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 video will automatically play next. Up next Error Function and Complimentary Error Function - Duration: 5:01. StudyYaar.com 11,719 error function integral 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 201,476 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 46,916 views 5:22 Fick's Law of Diffusion - Duration: 12:21. khanacademymedicine 132,903 views 12:21 Video 1690 - ERF Function - Duration: 5:46. Chau Tu 566 views 5:46 The Gaussian Distribution - Duration: 9:49. Steve Grambow 22,354 views 9:49 Hyperbolic Sine and Cosine Functions (Tanton Mathematics) - Duration: 13:45. DrJamesTanton 13,165 views 13:45 Gaussian - Duration: 4:28. Paul Francis 15,579 views 4:28 Diffusion - Coefficients and Non Steady State - Duration: 23:29. Engineering and Design Solutions 10,724 views 23:29 B15 Example problem with a linear equation using the error function - Duration: 5:08. Juan Klopper 802 views 5:08 INTEGRAL DE GAUSS-CAMPANA DE GAUSS - Duration: 7:56. RUBIÑOS 6,307 views 7:56 The Laplace transform of the error function erf(t) (MathsCasts) - Duration: 5:04. Swinburne Commons 4,107 views 5:04 Calculus Surprise The Gaussian Integral - Duration
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