# Error functions

## Definitions

Symbol: Erf $\operatorname{erf}\!\left(z\right)$ Error function
Symbol: Erfc $\operatorname{erfc}\!\left(z\right)$ Complementary error function
Symbol: Erfi $\operatorname{erfi}\!\left(z\right)$ Imaginary error function

## Illustrations

Image: X-ray of $\operatorname{erf}\!\left(z\right)$ on $z \in \left[-4, 4\right] + \left[-4, 4\right] i$

## Integral representations

$\operatorname{erf}\!\left(z\right) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} {e}^{-{t}^{2}} \, dt$
$\operatorname{erfc}\!\left(z\right) = \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} {e}^{-{t}^{2}} \, dt$
$\operatorname{erfi}\!\left(z\right) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} {e}^{{t}^{2}} \, dt$

## Connection formulas

$\operatorname{erf}\!\left(z\right) + \operatorname{erfc}\!\left(z\right) = 1$
$\operatorname{erfc}\!\left(z\right) = 1 - \operatorname{erf}\!\left(z\right)$
$\operatorname{erfi}\!\left(z\right) = -i \operatorname{erf}\!\left(i z\right)$

## Functional equations

$\operatorname{erf}\!\left(-z\right) = -\operatorname{erf}\!\left(z\right)$
$\operatorname{erfi}\!\left(-z\right) = -\operatorname{erfi}\!\left(z\right)$
$\operatorname{erfc}\!\left(-z\right) = 2 - \operatorname{erfc}\!\left(z\right)$

## Hypergeometric representations

$\operatorname{erf}\!\left(z\right) = \frac{2 z}{\sqrt{\pi}} \,{}_1F_1\!\left(\frac{1}{2}, \frac{3}{2}, -{z}^{2}\right)$
$\operatorname{erf}\!\left(z\right) = \frac{2 z {e}^{-{z}^{2}}}{\sqrt{\pi}} \,{}_1F_1\!\left(1, \frac{3}{2}, {z}^{2}\right)$
$\operatorname{erf}\!\left(z\right) = \frac{z}{\sqrt{{z}^{2}}} - \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)$
$\operatorname{erfc}\!\left(z\right) = \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)$

## Derivatives

$\operatorname{erf}'(z) = \frac{2}{\sqrt{\pi}} {e}^{-{z}^{2}}$
${\operatorname{erf}}^{(n)}(z) = \frac{2}{\sqrt{\pi}} {\left(-1\right)}^{n + 1} H_{n - 1}\!\left(z\right) {e}^{-{z}^{2}}$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC