What is "error function"

n. (context mathematics statistics English) A sigmoid function giving the probability that a measurement, under the influence of normal distribution errors with standard deviation, is within a certain distance from the mean value.

In mathematics, the error function (also called the Gauss error function) is a special function (non- elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:

$$\begin{align} \operatorname{erf}(x) & = \frac{1}{\sqrt\pi}\int_{-x}^x e^{-t^2}\,\mathrm dt \\ & = \frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,\mathrm dt. \end{align}$$

The complementary error function, denoted erfc, is defined as

$$\begin{align} \operatorname{erfc}(x) & = 1-\operatorname{erf}(x) \\ & = \frac{2}{\sqrt\pi} \int_x^{\infty} e^{-t^2}\,\mathrm dt \\ & = e^{-x^2} \operatorname{erfcx}(x), \end{align}$$

which also defines erfcx, the scaled complementary error function (which can be used instead of erfc to avoid arithmetic underflow). Another form of erfc(x) is known as Craig's formula:

$$\begin{align} \operatorname{erfc}(x) & = \frac{2}{\pi} \int_0^{\pi/2} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) d\theta. \end{align}$$

The imaginary error function, denoted erfi, is defined as

$$\begin{align} \operatorname{erfi}(x) & = -i\operatorname{erf}(ix) \\ & = \frac{2}{\sqrt\pi} \int_0^x e^{t^2}\,\mathrm dt \\ & = \frac{2}{\sqrt{\pi}} e^{x^2} D(x), \end{align}$$

where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow).

Despite the name "imaginary error function", erfi(x) is real when x is real.

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function:

w(z) = eerfc( − iz) = erfcx( − iz).