Error function
= Error function
{title2=$\operatorname{erf}$}
{wiki}
= erf
{c}
{synonym}
The error function records the signed area under a <Gaussian function> between zero and a variable endpoint:
$$
\boxed{\operatorname{erf}(x)=\frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,dt.}
$$
For a negative <real number> $x$, reversing the <integration limits> makes this <integral> negative. The normalization uses the <Gaussian integral>: the full area of $e^{-t^2}$ is $\sqrt\pi$, so its positive half contributes one after multiplication by $2/\sqrt\pi$. The same definition extends to a <complex number> $z$ along any path from zero to $z$, giving an <entire function>.<NIST Digital Library of Mathematical Functions/7.2 Definitions>[§7.2(i), equation 7.2.1, entire-function statement and real-axis limits in 7.2.4]{ref}
For example, $\operatorname{erf}(0)=0$, $\operatorname{erf}(1)\approx0.842701$, and $\operatorname{erf}(-1)\approx-0.842701$. A finite positive endpoint captures only part of the positive <Gaussian integral>; $\operatorname{erf}(x)$ approaches one as $x\to+\infty$ but never reaches it at a finite real $x$.
\Image[/error-function-signed-area-and-accumulation.png]
{title=The <error function> turns signed Gaussian area into a bounded increasing curve}
{description=Original computed real-axis curves. Blue shading represents integration from zero to $1$; orange shading represents the reversed integral from zero to $-1$. The right panel displays their signed values after normalization, not a probability density.}
The useful idea is to name a recurring <integral> once instead of carrying it through every calculation. The <error function and the normal distribution> express <probabilities> and <quantiles> through this area, while an <error-function diffusion front> describes how <diffusion> smooths a sharp jump. The <complementary error function> retains the small area left in the tail.
The <fundamental theorem of calculus> gives
$$
\operatorname{erf}'(x)=\frac{2}{\sqrt\pi}e^{-x^2}>0.
$$
Thus the real-axis <error function> is <strictly increasing>, with values in $(-1,1)$. Its <integrand> is an <even function>, so the <error function> is an <odd function>, $\operatorname{erf}(-x)=-\operatorname{erf}(x)$. The derivative also gives $\operatorname{erf}''(x)=-4xe^{-x^2}/\sqrt\pi$: the curve is <concave> for $x>0$ and <convex> for $x<0$. These real-axis properties do not imply boundedness throughout the <complex plane>; the existing <quadrant asymptotics of the error function> describe a different behavior in other directions.