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,
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.