Derivative 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 2 d t = 2 π ∫ 0 x differentiation error function e − t 2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac derivative q function − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ derivative gamma function 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − derivative normal distribution 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 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 )
Derivative Gaussian
{\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 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
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
Error Function Values
Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Calculus and Analysis>Calculus>Integrals>Definite Integrals> More... Interactive Entries>webMathematica Examples> how to solve error function History and Terminology>Wolfram Language Commands> Less... Erfc Erfc is the complementary error function, commonly denoted , is an entire function error function e 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 , (3) https://en.wikipedia.org/wiki/Error_function 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 http://mathworld.wolfram.com/Erfc.html 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
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and http://math.stackexchange.com/questions/1755149/derivative-of-error-function 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 https://www.youtube.com/watch?v=CcFUQhorgdc 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 derivative of error function up vote 0 down vote favorite How can I calculate the derivatives $$\frac{\partial \mbox{erf}\left(\frac{\ln(t)-\mu}{\sqrt{2}\sigma}\right)}{\partial \mu}$$ and $$\frac{\partial \mbox{erf}\left(\frac{\ln(t)-\mu}{\sqrt{2}\sigma}\right)}{\partial \sigma}$$ where $\mbox{erf}$ denotes the error function can be given by $$\mbox{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\exp(-t^2)\,dt$$ I derivative error function have tried it using WA derivative calculator but I am not able to understand the steps. derivatives error-function share|cite|improve this question edited Apr 23 at 9:02 kamil09875 4,3592729 asked Apr 23 at 7:44 Rakesh 11 The error function erf($x$) is just $\frac{2}{\sqrt\pi}\int_0^xe^{-t^2}\ dt$, so its derivative is just $\frac{2}{\sqrt\pi}e^{-x^2}$. All you have to do for your examples is use the chain rule. –almagest Apr 23 at 7:58 add a comment| 1 Answer 1 active oldest votes up vote 0 down vote You have error in your definition of error function :-). The definition of error function is $$\operatorname{erf}(x) = \frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,\mathrm dt = \int_0^x \frac{2}{\sqrt\pi}e^{-t^2}\,\mathrm dt.$$ Derivative of this integral with variable is it's integrand applied to upper boundary and multiplicated by boundary's derivative. ($\frac{\partial x}{\partial x}=1$) $$\frac{\partial \operatorname{erf}(x) }{\partial x}=1\cdot\frac{2}{\sqrt\pi}e^{-x^2}$$ The next step is calculating derivative of a composite function. I hope you can do it yourself. ==Added== You should treat $t$ and $\mu$ as a parameters. For example: $$\frac{\partial \frac{\ln(t)-\mu}{\sqrt{2}\sigma}}{\partial \sigma}=\frac{\ln(t)-\mu}{\sqrt{2}}{\ln|\sigma|}$$ Continue it. share|cite|improve this answer edited Apr 23 at 8:20 answered Apr 23 at 7:5
Google. Het beschrijft hoe wij gegevens gebruiken en welke opties je hebt. Je moet dit vandaag nog doen. Navigatie overslaan NLUploadenInloggenZoeken Laden... Kies je taal. Sluiten Meer informatie View this message in English Je gebruikt YouTube in het Nederlands. Je kunt deze voorkeur hieronder wijzigen. Learn more You're viewing YouTube in Dutch. You can change this preference below. Sluiten Ja, nieuwe versie behouden Ongedaan maken Sluiten Deze video is niet beschikbaar. WeergavewachtrijWachtrijWeergavewachtrijWachtrij Alles verwijderenOntkoppelen Laden... Weergavewachtrij Wachtrij __count__/__total__ The Error Function ei pi AbonnerenGeabonneerdAfmelden229229 Laden... Laden... Bezig... Toevoegen aan Wil je hier later nog een keer naar kijken? Log in om deze video toe te voegen aan een afspeellijst. Inloggen Delen Meer Rapporteren Wil je een melding indienen over de video? Log in om ongepaste content te melden. Inloggen Transcript Statistieken 16.598 weergaven 44 Vind je dit een leuke video? Log in om je mening te geven. Inloggen 45 6 Vind je dit geen leuke video? Log in om je mening te geven. Inloggen 7 Laden... Laden... Transcript Het interactieve transcript kan niet worden geladen. Laden... Laden... Beoordelingen zijn beschikbaar wanneer de video is verhuurd. Deze functie is momenteel niet beschikbaar. Probeer het later opnieuw. Gepubliceerd op 8 nov. 2013This is a special function related to the Gaussian. In this video I derive it. Categorie Onderwijs Licentie Standaard YouTube-licentie Meer weergeven Minder weergeven Laden... Autoplay Wanneer autoplay is ingeschakeld, wordt een aanbevolen video automatisch als volgende afgespeeld. Volgende Error Function and Complimentary Error Function - Duur: 5:01. StudyYaar.com 11.854 weergaven 5:01 Evaluating the Error Function - Duur: 6:36. lesnyk255 1.783 weergaven 6:36 Integral of exp(-x^2) | MIT 18.02SC Multivariable Calculus, Fall 2010 - Duur: 9:34. MIT OpenCourseWare 202.717 weergaven 9:34 erf(x) function - Duur: 9:59. Calculus Society -ROCKS!! 946 weergaven 9:59 Evaluation of the Gaussian Integral exp(-x^2) - Cool Math Trick - Duur: