The Error Function $y = \operatorname{erf} x$

The error function $\operatorname{erf} x$ is a special function defined by integrating the Gaussian:

The leading factor $\dfrac{2}{\sqrt{\pi}}$ is a normalizing constant chosen so that the value approaches exactly $1$ as $x \to \infty$.

Domain and range It is defined for all real $x$. Because the integrand $e^{-t^2}$ is always positive, the function increases monotonically, and its range is the open interval $(-1, 1)$.

Symmetry Since the integrand is even, the integral is odd: $\operatorname{erf}(-x) = -\operatorname{erf}(x)$. The graph is therefore point-symmetric about the origin, through which it passes at $(0, 0)$.

Monotonicity and slope By the fundamental theorem of calculus, $\dfrac{d}{dx}\operatorname{erf} x = \dfrac{2}{\sqrt{\pi}}\,e^{-x^2}$, which is always positive, so the function is strictly increasing. Its slope is steepest at the origin, where it equals $\dfrac{2}{\sqrt{\pi}} \approx 1.128$.

Asymptotes and limits As $x \to +\infty$, $y \to 1$, and as $x \to -\infty$, $y \to -1$, giving two horizontal asymptotes $y = \pm 1$. Convergence is rapid: $\operatorname{erf}(2) \approx 0.995$ and $\operatorname{erf}(3) \approx 0.99998$.

Inflection point The second derivative $-\dfrac{4x}{\sqrt{\pi}}\,e^{-x^2}$ changes sign at $x = 0$, the unique inflection point, where the curve turns from concave up to concave down.

Relation to other functions It is tied to the standard normal cumulative distribution $\Phi(x)$ by $\Phi(x) = \dfrac{1}{2}\left(1 + \operatorname{erf}\dfrac{x}{\sqrt{2}}\right)$. The quantity $1 - \operatorname{erf} x$ is the complementary error function $\operatorname{erfc} x$.

Applications and history As its name suggests, it originated in the theory of observational errors. Today it appears wherever the Gaussian integral arises: probability calculations for the normal distribution, solutions of the heat and diffusion equations, and bit-error-rate analysis in communications. Because it cannot be expressed with elementary functions, it is treated as a special function in its own right.