The error function $\mathrm{erf}$ is defined by $$\mathrm{erf}(x)=\dfrac{2}{\sqrt{\pi}}\displaystyle\int_0^x e^{-\tau^2} \mathrm{d}\tau,$$ where $\pi$ denotes pi and $e^{-\tau^2}$ denotes the exponential function.

Graph of $\mathrm{erf}$.
Domain coloring of $\mathrm{erf}$.

Properties

Taylor series for error function
Series for erf with exponential factored out
Error function is odd
Complex conjugate of argument of error function
Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x
Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4

Theorem: The following formula holds: $\dfrac{1}{2} \left( 1 + \mathrm{erf} \left( \dfrac{x-\mu}{\sqrt{2}\sigma} \right) \right)=\dfrac{1}{\sigma \sqrt{2 \pi}} \displaystyle\int_{-\infty}^x \exp \left( -\dfrac{(t-\mu)^2}{2\sigma^2} \right)\mathrm{d}t.$

Proof: █