Virhefunktio

Virhefunktio, known in English as the error function, is a standard special function in mathematics. It is defined by erf(x) = (2/√π) ∫_0^x e^{−t^2} dt. The function is odd and increasing, with erf(0) = 0 and erf(x) → 1 as x → ∞, erf(x) → −1 as x → −∞.

The error function is closely related to Gaussian distributions. The standard normal cumulative distribution function is Φ(x) = 1/2 [1 + erf(x/√2)], and the complementary error function erfc(x) = 1 − erf(x) = (2/√π) ∫_x^∞ e^{−t^2} dt is often used for tail probabilities. These relationships make erf and erfc fundamental in probability theory and statistical computations involving Gaussian integrals.

Numerical representations and analytic properties include a power series: erf(x) = (2/√π) ∑_{n=0}∞ [(-1)^n x^{2n+1}] / [n!(2n+1)], valid for all x, and asymptotic expansions for large |x|, such as erf(x) ≈ 1 − e^{−x^2}/(√π x) [1 − 1/(2x^2) + …]. The imaginary error function erfi(x) = −i erf(i x) is related to erf for imaginary arguments.

Applications of virhefunktio appear in solving the heat equation and diffusion problems, evaluating probability integrals, and error analysis in statistics. It is implemented in most mathematical libraries and programming languages under the names erf and erfc, reflecting its wide use in scientific computation.