Error function
The error function or Gaussian error function is defined in the theory of special functions, the integral
For a real argument x is a real-valued function erf; the generalization to complex arguments, see below.
The error function is a sigmoid function and is used in the statistics and in the theory of partial differential equations and is closely related to the error integral.
In the theory of approximation error function also is the difference between a function and its best approximation.
Designations
The name comes from the error function. The complementary (or conjugate ) error function is given by:
The generalized error function is given by the integral:
Defined.
Properties
It is true.
The error function is odd:
Use
Relationship with the normal distribution
The error function has a certain similarity to the distribution function of the normal distribution. However, it has a target amount, while a distribution function mandatory values from the range must accept.
It applies to the standard normal distribution
Or for the distribution function of any normal distribution with standard deviation and expected value
If the deviations of the individual results of a series of measurements from the common mean by a normal distribution with standard deviation and expected value 0 can be described, then the probability with which is the measurement error of a single measurement between and is ( for positive ).
The error function can be used to generate pseudo-random numbers normally distributed with the aid of the inversion method.
Heat conduction equation
The error function and the complementary error function occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside function.
Numerical
The error function as the distribution function of the normal distribution can not be represented by a closed function and must be determined numerically.
For small real values , the calculation is done with the series expansion
For large real values with the continued fraction expansion
For the complete range of values we have the following approximation with a maximum error of:
With
And
Table of values
Complex error function
The defining equation of the error function can be extended to complex arguments z:
In this case a complex function erf. Applies under complex conjugation
The imaginary error function is given by:
For the numerical calculation, erf, erfi, erfc and other related functions like Faddeeva expressed by the function w ( z). The Faddeeva function is known a scaled complex complementary error function and also as a relativistic plasma dispersion function. It is related to the Dawson integrals and the Voigt profile. A numerical implementation of Steven G. Johnson is available as a C library libcerf available.