# Error Functions¶

The error function is described in Abramowitz & Stegun, Chapter 7.

gsl_sf_erf(x)

This routine computes the error function $$\operatorname{erf}(x)$$, where

$\operatorname{erf}(x) = (2/\sqrt{\pi}) \int_0^x \exp(-t^2) dt.$
gsl_sf_erfc(x)

This routine computes the complementary error function

$\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = (2/\sqrt{\pi}) \int_x^\infty \exp(-t^2) dt.$
gsl_sf_log_erfc(x)

This routine computes the logarithm of the complementary error function $$\log(\operatorname{erfc}(x))$$.

## Probability functions¶

The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.

gsl_sf_erf_Z(x)

This routine computes the Gaussian probability density function $$Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)$$.

gsl_sf_erf_Q(x)

This routine computes the upper tail of the Gaussian probability function $$Q(x) = (1/\sqrt{2\pi}) \int_x^\infty \exp(-t^2/2) dt$$.

The hazard function for the normal distribution, also known as the inverse Mills’ ratio, is defined as,

$h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \operatorname{erfc}(x/\sqrt 2)$

It decreases rapidly as $$x$$ approaches $$-\infty$$ and asymptotes to $$h(x) \sim x$$ as $$x$$ approaches $$+\infty$$.

gsl_sf_hazard(x)

This routine computes the hazard function for the normal distribution.