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In mathematics, the error function, often denoted by erf, is a function defined as: erf ⁡ z = 2 π ∫ 0 z e − t 2 d t. In some old texts,
the error function is defined without the factor of 2 / π.
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 / 2, erf x is the probability that Y falls in the range [−x, x]. Two closely related functions are the complementary error function (erfc) defined as erfc ⁡ z = 1 − erf ⁡ z, and the imaginary error function (erfi) defined as erfi ⁡ z = − i erf ⁡ i z, where i is the imaginary unit.