Error function

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a function defined as:^[1] $${\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t.}$$

Error function
Plot of the error function
General information
General definition	${\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t}$
Fields of application	Probability, thermodynamics, digital communications
Domain, codomain and image
Domain	${\displaystyle \mathbb {R} }$
Image	${\displaystyle \left(-1,1\right)}$
Basic features
Parity	Odd
Specific features
Root	0
Derivative	${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {erf} z={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}}$
Antiderivative	${\displaystyle \int \operatorname {erf} z\,dz=z\operatorname {erf} z+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}+C}$
Series definition
Taylor series	${\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {z}{2n+1}}\prod _{k=1}^{n}{\frac {-z^{2}}{k}}}$

In some old texts,^[2] the error function is defined 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 ${\displaystyle 1/{\sqrt {2}}}$, 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.