Complex Error Function Wiki
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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 d complex error function matlab t . {\displaystyle {\begin − 2\operatorname − 1 (x)&={\frac − 0{\sqrt {\pi }}}\int _{-x}^ 9e^{-t^ error function of complex argument 8}\,\mathrm 7 t\\&={\frac 6{\sqrt {\pi }}}\int _ 5^ 4e^{-t^ 3}\,\mathrm 2 t.\end 1}} The complementary
Gamma Function Wiki
error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx
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( 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 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 wikipedia error function ( 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 the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 4 (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^ 2}\operatorname 1 (-iz)=\operatorname 0 (-iz).} Contents 1 The name "error functio
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Error Function Values
updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at imaginary error function WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Calculus and Analysis>Calculus>Integrals>Definite Integrals> More... Interactive normal distribution wiki Entries>webMathematica Examples> History and Terminology>Wolfram Language Commands> Less... Erfc Erfc is the complementary error function, commonly denoted , is an entire function defined by (1) (2) https://en.wikipedia.org/wiki/Error_function 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) where is the incomplete gamma function. The derivative is given by (4) and the indefinite integral by (5) It has the special values http://mathworld.wolfram.com/Erfc.html (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 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,
allUploadSign inJoinBooksAudiobooksComicsSheet Music You're Reading a Free Preview Pages 2 to 8 are not shown in this preview. Buy the Full https://www.scribd.com/document/50267654/Error-function-Wikipedia-the-free-encyclopedia Version AboutBrowse booksSite directoryAbout ScribdMeet the teamOur blogJoin http://www.qa-show.com/wiki/22874 our team!Contact UsPartnersPublishersDevelopers / APILegalTermsPrivacyCopyrightSupportHelpFAQAccessibilityPressPurchase helpAdChoicesMembershipsJoin todayInvite FriendsGiftsCopyright © 2016 Scribd Inc. .Terms of service.Accessibility.Privacy.Mobile Site.Site Language: English中文EspañolالعربيةPortuguês日本語DeutschFrançaisTurkceРусский языкTiếng việtJęzyk polskiBahasa indonesiaError function - Wikipedia, the free encyclopedia by Deepak Kumar Rout253 viewsEmbedRelated error function interestsFunction (Mathematics), Normal Distribution, IntegralDownloadRead on Scribd mobile: iPhone, iPad and Android.Copyright: Attribution Non-Commercial (BY-NC)List price: $0.00Download as PDF, TXT or read online from ScribdFlag for inappropriate contentMore informationShow less Documents similar to Error function - Wikipedia, the free encyclopediaSpecial Functions of complex error function Signal Processingby Saddat ShamsuddinAbbe2012.pdfby nomore891lectr14.pptby Hareem KhanBooks similar to Error function - Wikipedia, the free encyclopediaDover Books on Mathematicsby Heinrich W. GuggenheimerIntroduction to Asymptotics and Special Functionsby F. W. J. OlverInternational Series in Pure and Applied Mathematicsby Zhe-xian WanBooks about Function (Mathematics)College Math: Quiz Questions and Answersby Arshad IqbalMatlab: A Practical Introduction to Programming and Problem Solvingby Stormy AttawayGenerative Modeling for Computer Graphics and Cad: Symbolic Shape Design Using Interval Analysisby John M. Snyder Are you sure?This action might not be possible to undo. Are you sure you want to continue?CANCELOKWe've moved you to where you read on your other device.Get the full title to continueGet the full title to continue reading from where you left off, or restart the preview.Restart previewscribd
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