ERROR FUNCTION

An error function is defined by the integral

(1)

and it occurs frequently in engineering problems; e.g., in heat conduction problems. The error function represents the area under the Gaussian function from t = 0 to t = x, so that erf ∞ = 1. The complementary error function is:

(2)

The error function erf x is a monotonically increasing odd function of x; i.e., erf (–x) = –erf x and erf x₁ ≤ erf x₂ whenever x₁ ≤ x₂. Its Maclaurin series (for small x) is given by:

(3)

and for large values of x, the asymptotic expansion is:

(4)

where erf x may be approximated as

(5)

There exist extensive tabulations of erf x [see Abramowitz and Stegun (1965), for example].