erfc

erfc, the complementary error function, is a special function in mathematics related to the error function erf. It is defined for real x by erfc(x) = 1 - erf(x) = (2/√π) ∫_x^∞ e^{-t^2} dt. For complex arguments, erfc(z) = 1 - erf(z) and can be represented by the same integral along appropriate contours. The function is real-valued for real x, with 0 ≤ erfc(x) ≤ 2, and satisfies erfc(-x) = 2 - erfc(x). It decreases monotonically from erfc(-∞) = 2 to erfc(∞) = 0. Its derivative is erfc'(x) = - (2/√π) e^{-x^2}.

Asymptotically, for large x > 0, erfc(x) ~ (e^{-x^2})/(x√π) [1 - 1/(2x^2) + 3/(4x^4) - ...]. The relation to the Gaussian distribution is given by Q-function: Q(x) = (1/2) erfc(x/√2), which describes the upper tail probability of the standard normal distribution.

Erfc is an entire function of complex variable and satisfies the differential equation d/dx erfc(x) = - (2/√π) e^{-x^2}. It is often computed via numerical methods using either its integral definition, its asymptotic expansions, or built-in library routines in scientific computing languages (for example, erfc in C/C++, Python, and MATLAB). A related scaled form erfcx(z) = e^{z^2} erfc(z) is used to improve numerical stability.

Applications include solutions to the heat equation, diffusion processes, probability and statistics, and Gaussian integrals in physics.