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# Gaussian Error Function Definition

## Contents

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

## Derivative Of Error Function

2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int erf function calculator _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ ⁡ 9^ ⁡ 8e^{-t^ ⁡ 7}\,\mathrm ⁡ 6 t.\end ⁡

## Error Function Table

5}} The complementary error function, denoted erfc, is defined as erfc ⁡ ( x ) = 1 − erf ⁡ ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − inverse error function 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 ) {\displaystyle \operatorname ⁡ 2 (x)} for non-negative x {\displaystyle x} is error function matlab 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 − z 2 erfc ⁡ ( − i z ) = erfcx ⁡ ( − i z ) . {\displaystyle w(z)=e^{-z^ ⁡ 6}\operato

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## Erf(1)

updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at error function python WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and

## Erf(inf)

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 the Gaussian https://en.wikipedia.org/wiki/Error_function 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 identities (2) (3) (4) http://mathworld.wolfram.com/Erf.html 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 Erf can also be extended to the complex

» Probability and Statistics All times are UTC [ DST ] What is erf(x)? Moderators: mak, helmut, Shadow, outermeasure, Ilaggoodly Page 1 of 1 [ 3 posts ] http://www.sosmath.com/CBB/viewtopic.php?t=52713 Print view Previous topic | Next topic Author Message yoyobarn Post subject: What is erf(x)?Posted: Fri, 7 Jan 2011 14:57:00 UTC Member Joined: Fri, 12 Nov 2010 07:00:12 UTCPosts: 48 What is erf(x)? (in layman https://www.mathworks.com/help/matlab/ref/erf.html terms) The official wolframalpha definition is here: http://www.wolframalpha.com/input/?i=erf%28x%29 And, can we calculate erf(x) from the Normal Distribution Table? Top Ilaggoodly Post subject: Posted: Fri, 7 Jan 2011 18:16:02 UTC Site Admin Joined: error function Thu, 15 Feb 2007 06:35:15 UTCPosts: 755 its the Error Function, a function in probability theory that is related to the integral cumulative density function of the normal distribution, (the pdf doesn't yield a pretty integral) And because of this, it can be approximated with a normal distribution table, read the wiki page http://en.wikipedia.org/wiki/Normal_distribution Top aswoods Post subject: Posted: Sat, 8 Jan 2011 08:11:28 UTC Member of gaussian error function the 'S.O.S. Math' Hall of Fame Joined: Mon, 23 Feb 2009 23:20:33 UTCPosts: 1049Location: Adelaide, Australia If the same measurement is taken hundreds of times, then in many situations the results tend to look like a bell curve, with most of the reported measurements clustered around the correct value. The standard normal distribution is an idealized version of this curve, centred on 0. If a new measurement is taken and mapped onto this idealized version (as m), and you know that it is over the correct value of "0", then erf(x) tells you the probability that your measurement is less than x. Given 0

Search All Support Resources Support Documentation MathWorks Search MathWorks.com MathWorks Documentation Support Documentation Toggle navigation Trial Software Product Updates Documentation Home MATLAB Examples Functions Release Notes PDF Documentation Mathematics Elementary Math Special Functions MATLAB Functions erf On this page Syntax Description Examples Find Error Function Find Cumulative Distribution Function of Normal Distribution Calculate Solution of Heat Equation with Initial Condition Input Arguments x More About Error Function Tall Array Support Tips 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 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) returns the Error Function evaluated for each element of x.Examplescollapse allFind Error FunctionOpen ScriptFind the error function of a value.erf(0.76) ans = 0.7175 Find the error function of the elements of a vector.V = [-0.5 0 1 0.72]; erf(V) ans = -0.5205 0 0.8427 0.6914 Find the error function of the elements of a matrix.M = [0.29 -0.11; 3.1 -2.9]; erf(M) ans = 0.3183 -0.1236 1.0000 -1.0000 Find Cumulative Distribution Function of Normal DistributionOpen ScriptThe cumulative distribution function (CDF) of the normal, or Gaussian, distribution with standard deviation and mean is Note that for increased computational accuracy, you can rew

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