Error function

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: Some authors define without the factor of . This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real. In statistics, for non-negative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and standard deviation 1/, erf x is the probability that Y falls in the range . Two closely related functions are the complementary error function (erfc) defined as and the imaginary error function (erfi) defined as where i is the imaginary unit. The name "error function" and its abbreviation erf were proposed by J. W. L. Glaisher in 1871 on account of its connection with "the theory of Probability, and notably the theory of Errors." The error function complement was also discussed by Glaisher in a separate publication in the same year. For the "law of facility" of errors whose density is given by (the normal distribution), Glaisher calculates the probability of an error lying between p and q as: When the results of a series of measurements are described by a normal distribution with standard deviation σ and expected value 0, then erf (a/σ ) is the probability that the error of a single measurement lies between −a and +a, for positive a. This is useful, for example, in determining the bit error rate of a digital communication system. 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. The error function and its approximations can be used to estimate results that hold with high probability or with low probability.