Error function

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

Error function
Plot of the error function over real numbers
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	${\displaystyle \mathbb {C} }$
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	${\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}{\frac {z^{2n+1}}{n!}}}$

The integral here is a complex contour integral which is path-independent because ${\displaystyle \exp(-t^{2})}$ is holomorphic on the whole complex plane ${\displaystyle \mathbb {C} }$. In many applications, the function argument is a real number, in which case the function value is also real.

In some old texts, the error function is defined without the factor of ${\displaystyle {\frac {2}{\sqrt {\pi }}}}$. This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations.

In statistics, for non-negative real values of x, the error function has the following interpretation: for a real random variable Y that is normally distributed with mean 0 and standard deviation ${\displaystyle {\frac {1}{\sqrt {2}}}}$, erf x is the probability that Y falls in the range [−x, x].

Two closely related functions are the complementary error function ${\displaystyle \mathrm {erfc}$ :\mathbb {C} \to \mathbb {C} } is defined as

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

and the imaginary error function ${\displaystyle \mathrm {erfi}$ :\mathbb {C} \to \mathbb {C} } is defined as

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

where i is the imaginary unit.