Loterre: Mathematics: error function

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mathematical physics > special function > hypergeometric function > error function

mathematical analysis > functional analysis > special function > hypergeometric function > error function

Preferred term

error function

Definition(s)

In mathematics, the error function (also called the Gauss error function), often denoted by $erf$ , is a complex function of a complex variable defined as:

${\\displaystyle \\operatorname {erf} z={\ rac {2}{\\sqrt {\\pi }}}\\int _{0}^{z}e^{-t^{2}}\\,\\mathrm {d} t.}$

Some authors define ${\\displaystyle \\operatorname {erf} }$ without the factor of ${\\displaystyle 2/{\\sqrt {\\pi }}}$ .
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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/√2$ , $erf x$ is the probability that $Y$ falls in the range $[- x, x]$ .
Two closely related functions are the complementary error function ( $erfc$ ) defined as

${\\displaystyle \\operatorname {erfc} z=1-\\operatorname {erf} z,}$

and the imaginary error function ( $erfi$ ) defined as

${\\displaystyle \\operatorname {erfi} z=-i\\operatorname {erf} iz,}$

where $i$ is the imaginary unit.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Error_function)

Broader concept(s)

Synonym(s)

Gauss error function

In other languages

fonction d'erreur

French
fonction d'erreur de Gauss

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http://data.loterre.fr/ark:/67375/PSR-SDBHRQ52-J

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