Differentiation Of Complementary Error Function
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Complementary Error Function Mathematica
and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire complementary error function ti 89 Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" error function values encountered in integrating the normal 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 http://functions.wolfram.com/GammaBetaErf/Erfc/20/ Watson 1990, p.341) define without the 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 http://mathworld.wolfram.com/Erf.html incomplete gamma function. Erf can also be defined as 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
Search All Support Resources Support Documentation MathWorks Search MathWorks.com MathWorks Documentation Support Documentation Toggle navigation Trial Software Product Updates Documentation Home Symbolic Math Toolbox Examples Functions and https://www.mathworks.com/help/symbolic/erfc.html 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 Complementary Error Function Handling Expressions That Contain Complementary Error Function Plot Complementary Error Function Input Arguments error function 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 the English verison of the page. Back to English × Translate This Page Select Language Bulgarian Catalan Chinese Simplified Chinese Traditional Czech Danish Dutch complementary error function 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 arguments, erfc can return floating-point or exact symbolic results. Compute the complementary error function for these numbers. Because these numbers are not symbolic objects, you get the floating-point results:A = [erfc(1/2), erfc(1.41), erfc(sqrt(2))]A = 0.4795 0.0461 0.0455Compute the complementary error function for the same numbers converted to symbolic objects