Error function

Summary

The error function is $\operatorname{erf} x = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}\, dt$. It is the integral of the Gaussian bell curve, scaled so that $\operatorname{erf} x \to 1$ as $x \to \infty$. It gives the cumulative distribution function of the normal distribution, and it appears in solutions of the heat equation. It has no expression in terms of elementary functions.