Error function
Sigmoid shape special function / From Wikipedia, the free encyclopedia
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In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a function defined as:[1]
Error function | |
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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.