The Gaussian Error Function erf(x) in 2026 Mathematics
The error function $\operatorname{erf}(x)$, first introduced by Carl Friedrich Gauss, is a non-elementary special function representing the area under a Gaussian bell curve. Because the antiderivative of $e^{-t^2}$ cannot be expressed using elementary algebra, $\operatorname{erf}(x)$ is computed via numerical approximations or series expansions.
\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} \, dt
\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} \, dt