In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a function defined as:[1]
Error function
Plot of the error function
General information
General definition
Fields of application
Probability, thermodynamics, digital communications
Domain, codomain and image
Domain
Image
Basic features
Parity
Odd
Specific features
Root
0
Derivative
Antiderivative
Series definition
Taylor series
Some authors define without the factor of .[2]
This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.
In statistics, for non-negative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and standard deviation 1/√2, erf x is the probability that Y falls in the range [−x, x].
Two closely related functions are the complementary error function (erfc) defined as
and the imaginary error function (erfi) defined as
where i is the imaginary unit.
^Andrews, Larry C. (1998). Special functions of mathematics for engineers. SPIE Press. p. 110. ISBN 9780819426161.
^Whittaker, E. T.; Watson, G. N. (1927). A Course of Modern Analysis. Cambridge University Press. p. 341. ISBN 978-0-521-58807-2.
In mathematics, the errorfunction (also called the Gauss errorfunction), often denoted by erf, is a function defined as: erf z = 2 π ∫ 0 z e − t 2...
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example is the errorfunction, which is related to the cumulative distribution function of a normal distribution; another is the arctan function, which is...
Robert (August 7, 2003). "The Q-function". Connexions. Barak, Ohad (April 6, 2006). "Q Function and ErrorFunction" (PDF). Tel Aviv University. Archived...
lack elementary antiderivatives; the integral of the Gaussian function is the errorfunction: ∫ e − x 2 d x = π 2 erf x + C . {\displaystyle \int e^{-x^{2}}\...
applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery...
integrals thereof, such as the errorfunction. There are many interrelations between these functions and the gamma function; notably, the factor π {\displaystyle...
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squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding...
the probit function is available as norm.s.inv(p). In computing environments where numerical implementations of the inverse errorfunction are available...
statistical mechanics, to find its partition function. Although no elementary function exists for the errorfunction, as can be proven by the Risch algorithm...
The Faddeeva function or Kramp function is a scaled complex complementary errorfunction, w ( z ) := e − z 2 erfc ( − i z ) = erfcx ( − i z ) = e...
from interpolation error and give better performance in calculation process. This table gives some values of an unknown function f ( x ) {\displaystyle...
{\text{ for }}a>0,\ a\neq 1} In the following formulas, erf is the errorfunction and Ei is the exponential integral. ∫ e c x ln x d x = 1 c ( e c x...
propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them...
two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the errorfunction (erf). They arise...
on the function code. If there is no error during those steps, server will response PDU = Function code + data response. As there is no error during those...
In statistical hypothesis testing, a type I error, or a false positive, is the rejection of the null hypothesis when it is actually true. For example...
is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What...
logarithmic integral (Li or li) and Fresnel (S and C) integrals. the errorfunction, e r f ( x ) = 2 π ∫ 0 x e − t 2 d t , {\displaystyle \mathrm {erf}...
behavior of the approximation error as a function of the number of integrand evaluations. A method that yields a small error for a small number of evaluations...
\operatorname {erf} } is the errorfunction, and erfc {\displaystyle \operatorname {erfc} } is the complementary errorfunction, erfc ( x ) = 1 − erf ...
(VTA) and substantia nigra (SNc) appear to mimic the errorfunction in the algorithm. The errorfunction reports back the difference between the estimated...
}}'={\hat {H}}\mathbf {{\hat {x}}_{i}} } that minimize the reprojection errorfunction given by ∑ i d ( x i , x i ^ ) 2 + d ( x i ′ , x i ^ ′ ) 2 {\displaystyle...
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