erfcx

erfcx, the scaled complementary error function, is a mathematical function defined for real x by erfcx(x) = e^{x^2} erfc(x), where erfc(x) = 1 − erf(x) is the complementary error function. It is an entire function used to stabilize computations involving Gaussian tails, since erfc decays like e^{-x^2} and multiplying by e^{x^2} yields a more well-behaved function for large x.

Basic properties include that erfcx(0) = 1. Its derivative satisfies d/dx erfcx(x) = 2x erfcx(x) − 2/√π. This relationship is useful for analytic work and numerical evaluation. For large positive x, erfcx(x) has the asymptotic expansion erfcx(x) ~ (1/(√π x)) [1 − 1/(2x^2) + 3/(4x^4) − …]. For large negative x, erfcx(x) grows roughly as 2 e^{x^2} because erfc(x) → 2 as x → −∞.

Connections and usage include its role in tail probabilities of the normal distribution, where Q(x) = (1/2) erfc(x/√2). The erfcx form is preferred in numerical work to avoid underflow when x is large, since it removes the rapidly decaying factor e^{-x^2}.

Implementations are widespread in numerical libraries, often available as erfcx in SciPy (scipy.special.erfcx), Boost, Cephes, and other math libraries. It is commonly used in statistics, physics, and engineering for stable evaluation of Gaussian integrals and related expressions.

See also: erf, erfc, erfi, and the Faddeeva function, all of which relate to Gaussian and complex error functions.