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Error function

mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. It is defined as:

<math>\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt = \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infin\frac{(-1)^n x^{2n+1}}{(2n+1)n!} = \frac{2}{\sqrt{\pi}} \left(x-\frac{x^3}{3}+\frac{x^5}{10}-\frac{x^7}{42}+\frac{x^9}{216}-\ \cdots\right)<math>

for every real number x.

The complementary error function, denoted erfc, is defined in terms of the error function.

<math>\mbox{erfc}(x) = 1-\mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt<math>

General theory

The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, as they differ only by scaling and translation. Indeed,

<math>\Phi(x) = \frac{1}{2}\left(1+\mbox{erf}\,\frac{x}{\sqrt{2}}\right)<math>

The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.

When the results of a series of measurements are described by a normal distribution with standard deviation σ, then erf(a/(σ√2)) is the probability that the error of a single measurement lies between -a and +a.

By analytic continuation, the error function can be defined for complex arguments as well. It occurs, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.

The integral defining the error function cannot be evaluated in closed form in terms of elementary functions, but the integrand may be expanded in a power series and integrated term by term. Values of the integral, as functions of x, have been tabulated.

Sometimes the more general functions

<math>E_n(x) = \frac{n!}{\sqrt{\pi}} \int_0^x e^{-t^n}dt

=\frac{n!}{\sqrt{\pi}}\sum_{p=0}^\infin(-1)^p\frac{x^{np+1}}{(np+1)p!}<math>

are discussed; E₂(x) is the error function.

Graph of generalized error functions E_n(x). Grey curve: E₁(x)=1-e^-x, red curve: erf(x)=E₂(x), green curve: E₃(x), blue curve: E₄(x), and yellow curve: E₅(x). (The yellow curve is quite close to the y-axis and may not be visible.) After division by n!, all the E_n for odd n look similar (but not identical) to each other. Similarly, the E_n for even n look similar (but not identical) to each other after division by n!. The E_n with odd and even n look similar on the positive x side of the graph.

Asymptotic expansion

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large x is

<math>\mbox{erfc}(x) = \frac{e^{-x^2}}{\sqrt{\pi}}\sum_{n=0}^\infty (-1)^{n+1} \frac{(2n)!}{n!4^n}\frac{1}{x^{n+1}}\,<math>

This series diverges for every finite x. However, in practice only the first few terms of this expansion are needed to obtain a good approximation of erfc(x), whereas the Taylor series given above converges very slowly.

See also

• Fresnel integral.

References

• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 7.)