Error Function Complex Argument
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Bessel Function Complex Argument
developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and gamma function complex argument Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the
Delta Function Complex Argument
"error function" 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 complex error function matlab (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) where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. For , error function values (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 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
where it was expressed through the following integral: Later C. Kramp (1799) used this integral for the definition of the complementary error function . P.‐S. Laplace
Complementary Error Function
(1812) derived an asymptotic expansion of the error function. The probability integrals error function calculator were so named because they are widely applied in the theory of probability, in both normal and limit
Inverse Error Function
distributions. To obtain, say, a normal distributed random variable from a uniformly distributed random variable, the inverse of the error function, namely is needed. The inverse was systematically investigated in http://mathworld.wolfram.com/Erf.html the second half of the twentieth century, especially by J. R. Philip (1960) and A. J. Strecok (1968).
Definitions of probability integrals and inverses The probability integral (error function) , the generalized error function , the complementary error function , the imaginary error function , the inverse error function , the inverse of the generalized error function , and http://functions.wolfram.com/GammaBetaErf/Erfi/introductions/ProbabilityIntegrals/ShowAll.html the inverse complementary error function are defined through the following formulas: These seven functions are typically called probability integrals and their inverses. Instead of using definite integrals, the three univariate error functions can be defined through the following infinite series. A quick look at the probability integrals and inversesHere is a quick look at the graphics for the probability integrals and inverses along the real axis. Connections within the group of probability integrals and inverses and with other function groups Representations through more general functions The probability integrals , , , and are the particular cases of two more general functions: hypergeometric and Meijer G functions. For example, they can be represented through the confluent hypergeometric functions and : Representations of the probability integrals , , , and through classical Meijer G functions are rather simple: The factor in the last four formulas can be removed by changing the classical Meijer G functions to the generalized one: The probability integrals , , , and are the particular cases of the incomplete gamma function, regularized incomplete gamma function,Search All Support Resources Support Documentation MathWorks Search MathWorks.com MathWorks Documentation Support Documentation Toggle https://www.mathworks.com/help/symbolic/mupad_ref/erf.html navigation Trial Software Product Updates Documentation Home Symbolic Math Toolbox Examples Functions and Other Reference Release Notes PDF Documentation MuPAD Mathematics Mathematical Constants and Functions Special Functions Error and Exponential Integral Functions Symbolic Math Toolbox MuPAD Functions erf On this page Syntax Description Environment Interactions Examples Example error function 1 Example 2 Example 3 Parameters Return Values Algorithms See Also More About 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 function complex argument 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 functionexpand all in page MuPAD notebooks are not recommended. Use MATLAB live scripts instead.MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.Syntaxerf(x) Descriptionerf(x) represents the error function 2π∫0xe−t2dt.This function is defined for all complex arguments x. For floating-point arguments, erf returns floating-point results. The implemented e
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