- (Graph: see below, red line)

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

*E*_{1}(*x*)=1-e^{-x}, erf(*x*)=*E*_{2}(*x*), *E*_{3}(*x*), *E*_{4}(*x*) and *E*_{5}(*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.