Error function

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:

  (Graph: see below, red line)

for every real number x. The reason it is conventionally defined that way rather than by an expression with "-x²/2" in the exponent appears to be that it is among experts in the theory of special functions that this convention prevails, and they are relatively ignorant of probability theory. Probabilists and statisticians do not use the notation "erf" nor the name "error function", but incessantly use the "standard normal cumulative probability distribution function" (the "normal cdf"), with "-x²/2" in the exponent.

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, 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

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

E₁(x)=1-e^-x, erf(x)=E₂(x), E₃(x), E₄(x) and E₅(x). After dividing by n!, all the E_n for odd n would look similar to each other (but still different). Similarily, the E_n for even n would look similar to each other (but still different). The E_n with odd and even n look similar on the positive x side of the graph. Despite the horrid appearance of E'_n for n of 4 and above, on the above graph, the functions are still probably useful for something.

See also: Fresnel integral.