The error function is described in Abramowitz & Stegun, Chapter 7.  The functions in this section are declared in the header file gsl_sf_erf.h.

Error Function
--------------

 -- Function: double gsl_sf_erf (double X)
 -- Function: int gsl_sf_erf_e (double X, gsl_sf_result * RESULT)
     These routines compute the error function $\erf(x)$ erf(x), where $\erf(x) = (2/\sqrt\pi) \int_0^x dt \exp(-t^2)$ erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).

Complementary Error Function
----------------------------

 -- Function: double gsl_sf_erfc (double X)
 -- Function: int gsl_sf_erfc_e (double X, gsl_sf_result * RESULT)
     These routines compute the complementary error function $\erfc(x) = 1 - \erf(x) = (2/\sqrt\pi) \int_x^\infty \exp(-t^2)$ erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).

Log Complementary Error Function
--------------------------------

 -- Function: double gsl_sf_log_erfc (double X)
 -- Function: int gsl_sf_log_erfc_e (double X, gsl_sf_result * RESULT)
     These routines compute the logarithm of the complementary error function \log(\erfc(x)).

Probability functions
---------------------

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

 -- Function: double gsl_sf_erf_Z (double X)
 -- Function: int gsl_sf_erf_Z_e (double X, gsl_sf_result * RESULT)
     These routines compute the Gaussian probability density function $Z(x) = (1/\sqrt2\pi) \exp(-x^2/2)$ Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2).

 -- Function: double gsl_sf_erf_Q (double X)
 -- Function: int gsl_sf_erf_Q_e (double X, gsl_sf_result * RESULT)
     These routines compute the upper tail of the Gaussian probability function $Q(x) = (1/\sqrt2\pi) \int_x^\infty dt \exp(-t^2/2)$ Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2).

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) / \erfc(x/\sqrt 2)

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

 -- Function: double gsl_sf_hazard (double X)
 -- Function: int gsl_sf_hazard_e (double X, gsl_sf_result * RESULT)
     These routines compute the hazard function for the normal distribution.