Error Function Erf Erfc
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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> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive erf vs erfc Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the
Excel Error Function Erf
"error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function
Erf Error Function Ti-89
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
Q Function Erf
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 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 normal distribution erf 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 by Laplace in 1805 and Legendre in 1826 (Olds 1963, p.139), proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp.8-9). Definite integrals involving include Definite integrals involving include (34) (35) (36) (37) (38) The first two of these appear in Prudnikov et al. (1990, p.123, eqns. 2.8.19.8 and 2.8.19.11), with , . A complex generalization of is defined as (39) (40) Integral representations valid only in the upper half-plane are given by (41) (42) SEE ALSO: Dawson's Integral, Erf
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 gaussian erf Functions MATLAB Functions erfc On this page Syntax Description Examples Find Complementary Error Function mathematica erf Find Bit Error Rate of Binary Phase-Shift Keying Avoid Roundoff Errors Using Complementary Error Function Input Arguments x More About wiki erf Complementary 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 http://mathworld.wolfram.com/Erf.html 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 https://www.mathworks.com/help/matlab/ref/erfc.html 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) exampleDescriptionexampleerfc(x
) returns the Complementary Error Function evaluated for each element of x. Use the erfc function to replace 1 - erf(x) for greater accuracy when erf(x) is close to 1.Examplescollapse allFind Complementary Error FunctionOpen ScriptFind the complementary error function of a value.erfc(0.35) ans = 0.6206 Find the complementary error function of the elements of a vector.V = [-0.5 0 1 0.72]; erfc(V) ans = 1.5205 1.0000 0.1573 0.3086 Find the complementary error function of the elements of a matrix.M = [0.29 -0.11; 3.1 -2.9]; erfc(M) ans = 0.6817 1.1236 0.0000 2.0000 Find Bit Error Rate of Binary Phase-Shift KeyingOpen ScriptThe bit error rate (BER) of binary phase-shift keying (BPSK), assuming additive white gaussian noise (AWGN), is Plot the BER for BPSK for values of from 0dB to 10dB.EbN0_dB = 0:0.1:10; EbN0 = 10.^(EbN0_dB/10); BER = 1/2.*erfc(sqrt(EbN0)); semilogy(EbN0_dB,BER) grid on ylabel('BER') xlabel('E_b/N_0 (dB)') title('Bit Error Rate for Binary Phase-Shift Keying') Avoid Roundoff Errors Using Complementary Error FunctionOpen ScriptYou can use the complementary error functi
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/matlab/ref/erf.html 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 error function 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 error function erf 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,