Derivative Of 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 Functions>Erf> error function values Calculus and Analysis>Complex Analysis>Entire Functions> Calculus and Analysis>Calculus>Integrals>Definite Integrals> More... Interactive Entries>webMathematica Examples> History error function graph and Terminology>Wolfram Language Commands> Less... Erfc Erfc is the complementary error function, commonly denoted , is an entire function defined erfc function 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) where
Erf(1)
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 erfc differential equation erf(0) (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 and 395-403, 1987. Whittaker, E.T. a
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 erf(2) x e − t 2 d t . {\displaystyle {\begin − 6\operatorname − 5
Erf(n)
(x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^
Erf(inf)
8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ http://mathworld.wolfram.com/Erfc.html 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 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 https://en.wikipedia.org/wiki/Error_function ( 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 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:
Search All Support Resources Support Documentation MathWorks Search MathWorks.com MathWorks Documentation Support Documentation Toggle navigation Trial Software Product Updates https://www.mathworks.com/help/symbolic/erfc.html Documentation Home Symbolic Math Toolbox Examples Functions and Other Reference Release Notes PDF Documentation Mathematics Mathematical Functions Symbolic Math Toolbox Functions erfc On this page Syntax Description Examples Complementary Error Function for Floating-Point and Symbolic Numbers Error Function for Variables and Expressions Complementary Error Function for Vectors and Matrices Special Values of error function Complementary Error Function Handling Expressions That Contain Complementary Error Function Plot Complementary Error Function Input Arguments X K More About Complementary Error Function Iterated Integral of Complementary Error Function Tips Algorithms References See Also This is machine translation Translated by Mouse over text to see original. Click the button below to return to derivative of complementary the English verison of the page. Back to English × Translate This Page Select Language Bulgarian Catalan Chinese Simplified Chinese Traditional Czech Danish Dutch English Estonian Finnish French German Greek Haitian Creole Hindi Hmong Daw Hungarian Indonesian Italian Japanese Korean Latvian Lithuanian Malay Maltese Norwegian Polish Portuguese Romanian Russian Slovak Slovenian Spanish Swedish Thai Turkish Ukrainian Vietnamese Welsh MathWorks Machine Translation The automated translation of this page is provided by a general purpose third party translator tool. MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation. Translate erfcComplementary error functioncollapse all in page Syntaxerfc(X) exampleerfc(K,X) exampleDescriptionexampleerfc(X
) represents the complementary error function of X, that is,erfc(X) = 1 - erf(X).exampleerfc(K
,X) represents the iterated integral of the complementary error function of X, that is, erfc(K, X) = int(erfc(K - 1, y), y, X, inf).ExamplesComplementary Error Function for Floating-Point and Symbolic Numbers Depending on its a