Error function

From Wikipedia, the free encyclopedia

In 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.</math>

1 Properties
2 Applications
3 Asymptotic expansion
4 Related functions
- 4.1 Generalized error functions
5 See also
6 References
7 External links

[edit] Properties

The error function is evidently odd

<math>\operatorname{erf} (-x) = -\operatorname{erf} (x).</math>

Also, for any complex number x one has

<math>\operatorname{erf} (x^{C}) = \operatorname{erf}(x)^{C} </math>

where x^C is the complex conjugate of x.

The integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand in a Taylor series, one obtains the Taylor series for the error function as follows:

<math>\operatorname{erf}(x)= \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>

which holds for every real number x, and also throughout the complex plane.

The inverse error function has series

<math>\operatorname{erf}^{-1}(x)=\sum_{k=0}^\infin\frac{c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}x\right )^{2k+1}, \,\!</math>

where c₀=1 and

Another series representation is

<math>\operatorname{erf}^{-1}(x)=\frac{1}{2}\sqrt{\pi}\left (x+\frac{\pi x^3}{12}+\frac{7\pi^2 x^5}{480}+\frac{127\pi^3 x^7}{40320}+\frac{4369\pi^4 x^9}{5806080}+\frac{34807\pi^5 x^{11}}{182476800}+\cdots\right ). \,\!</math>[1]

Image:Erfc plot.svg

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>

The complex error function, denoted w(x), is also defined in terms of the error function:

<math>w(x) = e^{-x^2}{\textrm{erfc}}(-ix).\,\!</math>

[edit] Applications

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.

The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.

[edit] Asymptotic expansion

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

<math>\mathrm{erfc}(x) = \frac{e^{-x^2}}{x\sqrt{\pi}}\left [1+\sum_{n=1}^\infty (-1)^n \frac{1\cdot3\cdot5\cdots(2n-1)}{(2x^2)^n}\right ]=\frac{e^{-x^2}}{x\sqrt{\pi}}\left [ 1+\sum_{n=1}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}}\right ].\,</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.

[edit] Related functions

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}\left(\frac{x}{\sqrt{2}}\right)\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.

The error function is a special case of the Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function. It has a simple expression in terms of the Fresnel integral. In terms of the Regularized Gamma function P and the incomplete gamma function,

<math>\operatorname{erf}(x)=\operatorname{signum}(x)*P\left(\frac{1}{2}, x^2\right)={\operatorname{signum}(x)\over \sqrt{\pi}}\gamma\left(\frac{1}{2}, x^2\right).</math>

[edit] Generalized error functions

Some authors discuss 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>

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.

[edit] See also

Gaussian function

[edit] References

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