Erf 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 erf q function ∫ − x x e − t 2 d t =
Erf Normal Distribution
2 π ∫ 0 x e − t 2 d t . {\displaystyle {\begin{aligned}\operatorname {erf} (x)&={\frac {1}{\sqrt erf erfc {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,\mathrm {d} t\\&={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,\mathrm {d} t.\end{aligned}}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf
Erf Error Function Ti-89
( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin{aligned}\operatorname {erfc} (x)&=1-\operatorname {erf} (x)\\&={\frac {2}{\sqrt {\pi }}}\int _{x}^{\infty }e^{-t^{2}}\,\mathrm {d} t\\&=e^{-x^{2}}\operatorname {erfcx} (x),\end{aligned}}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc error function erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname {erfc} (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 {erfc} (x|x\geq 0)={\frac {2}{\pi }}\int _{0}^{\pi /2}\exp \left(-{\frac {x^{2}}{\sin ^{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{aligned}\operatorname {erfi} (x)&=-i\operatorname {erf} (ix)\\&={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{t^{2}}\,\mathrm {d} t\\&={\frac {2}{\sqrt {\pi }}}e^{x^{2}}D(x),\end{aligned}}} 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 {erfi} (x)} is real when x i
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Mathematica Erf
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Wiki Erf
nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... complementary error function History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of https://en.wikipedia.org/wiki/Error_function the Gaussian function). It is an entire function defined by (1) Note that some authors (e.g., Whittaker and 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 http://mathworld.wolfram.com/Erf.html 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 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
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/erf.html Documentation Home Symbolic Math Toolbox Examples Functions and Other Reference Release Notes PDF Documentation Mathematics Mathematical Functions Symbolic Math Toolbox Functions erf On this page Syntax Description Examples Error Function for Floating-Point and Symbolic Numbers Error Function for Variables and Expressions Error Function for Vectors and Matrices Special Values of Error Function Handling error function Expressions That Contain Error Function Plot Error Function Input Arguments X More About 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 erf error function 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 erfError functioncollapse all in page Syntaxerf(X) exampleDescriptionexampleerf(X
) represents the error function of X. If X is a vector or a matrix, erf(X) computes the error function of each element of X.ExamplesError Function for Floating-Point and Symbolic Numbers Depending on its arguments, erf can return floating-point or exact symbolic results. Compute the error function for these numbers. Because these numbers are not symbolic objects, you get the floating-point results:A = [erf(1/2), erf(1.41), erf(sqr
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