# Error Function is Odd

## Theorem

$\map \erf {-x} = -\map \erf x$

where:

$\erf$ denotes the error function
$x$ is a real number.

## Proof

 $\ds \map \erf {-x}$ $=$ $\ds \frac 2 {\sqrt \pi} \int_0^{-x} e^{-u^2} \rd u$ Definition of Error Function $\ds$ $=$ $\ds -\frac 2 {\sqrt \pi} \int_0^{-\paren {-x} } e^{-\paren {-u}^2} \rd u$ substituting $u \mapsto -u$ $\ds$ $=$ $\ds -\frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u$ $\ds$ $=$ $\ds -\map \erf x$ Definition of Error Function

$\blacksquare$