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The complementary error function describes the Gaussian integral area remaining beyond the endpoint. For a real $x$,It is an entire function for complex arguments when defined by $1-\mathrm{erf}(z)$. On the real line it decreases from two to zero, with $\mathrm{erfc}(0)=1$ and $\mathrm{erfc}(-x)=2-\mathrm{erfc}(x)$.^[ref]

For example, $\mathrm{erfc}(1)\approx 0.157299$ is a sizeable tail, whereas $\mathrm{erfc}(8)\approx 1.12243\times 10^{-29}$ is extremely small. In the tested floating-point arithmetic, a direct evaluation of $1-\mathrm{erf}(8)$ gives zero because the computed error function has already rounded to one. Evaluating the complementary error function directly preserves this tail. It is the useful quantity in an upper-tail normal distribution probability and in the decaying side of an error-function diffusion front.

For large positive real $x$, the leading asymptotic expansion isThe first corrections are $1-1/(2x^2)+3/(4x^4)-\cdots$ multiplying the leading term. This is an asymptotic expansion, not an assertion that summing arbitrarily many terms converges at a fixed $x$.^[ref] The complementary error function tail bound provides finite-$x$ inequalities instead.

The error function records the signed area under a Gaussian function between zero and a variable endpoint: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.^[ref]

For example, $\mathrm{erf}(0)=0$, $\mathrm{erf}(1)\approx 0.842701$, and $\mathrm{erf}(-1)\approx -0.842701$. A finite positive endpoint captures only part of the positive Gaussian integral; $\mathrm{erf}(x)$ approaches one as $x\to +\infty$ but never reaches it at a finite real $x$.

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 givesThus 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, $\mathrm{erf}(-x)=-\mathrm{erf}(x)$. The derivative also gives ${\mathrm{erf}}^{\prime \prime }(x)=-4x{e}^{-x^2}/\sqrt{\pi }$: 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.

For initial data $u_0$ on the real line, Fourier transformation or convolution with the fundamental solution gives

For a bounded function $u_0$, the convolution is well defined at every positive time because the heat kernel has integral one. The error-function diffusion front uses a Heaviside step function initial condition and gives one smoothed interface. The heat equation with interval-indicator initial data instead has two interfaces; subtracting their two error functions produces the finite-interval profile.