In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:

The complementary error function, denoted erfc, is defined as

which also defines erfcx, the scaled complementary error function (which can be used instead of erfc to avoid arithmetic underflow). Another form of is known as Craig's formula:

The imaginary error function, denoted erfi, is defined as

where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow).

Despite the name "imaginary error function", is real when x is real.

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function:

The name "error function"

The error function is used in measurement theory (using probability and statistics), and its use in other branches of mathematics is typically unrelated to the characterization of measurement errors.