.. index::
single: error function
single: erf(x)
single: erfc(x)
The error function is described in Abramowitz & Stegun, Chapter 7. The
functions in this section are declared in the header file
:file:`gsl_sf_erf.h`.
Error Function
--------------
.. function:: double gsl_sf_erf (double x)
int gsl_sf_erf_e (double x, gsl_sf_result * result)
These routines compute the error function :math:`\erf(x)`,
where
:math:`\erf(x) = (2/\sqrt{\pi}) \int_0^x dt \exp(-t^2)`.
.. Exceptional Return Values: none
Complementary Error Function
----------------------------
.. function:: double gsl_sf_erfc (double x)
int gsl_sf_erfc_e (double x, gsl_sf_result * result)
These routines compute the complementary error function
:math:`\erfc(x) = 1 - \erf(x) = (2/\sqrt{\pi}) \int_x^\infty \exp(-t^2)`
.. Exceptional Return Values: none
Log Complementary Error Function
--------------------------------
.. function:: double gsl_sf_log_erfc (double x)
int gsl_sf_log_erfc_e (double x, gsl_sf_result * result)
These routines compute the logarithm of the complementary error function
:math:`\log(\erfc(x))`.
.. Exceptional Return Values: none
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)
int gsl_sf_erf_Z_e (double x, gsl_sf_result * result)
These routines compute the Gaussian probability density function
:math:`Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)`
.. function:: double gsl_sf_erf_Q (double x)
int gsl_sf_erf_Q_e (double x, gsl_sf_result * result)
These routines compute the upper tail of the Gaussian probability function
:math:`Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2)`
.. Exceptional Return Values: none
.. index::
single: hazard function, normal distribution
single: Mills' ratio, inverse
The *hazard function* for the normal distribution,
also known as the inverse Mills' ratio, is defined as,
.. only:: not texinfo
.. math:: h(x) = {Z(x) \over Q(x)} = \sqrt{2 \over \pi} {\exp(-x^2 / 2) \over \erfc(x/\sqrt 2)}
.. only:: texinfo
::
h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)
It decreases rapidly as :math:`x` approaches :math:`-\infty` and asymptotes
to :math:`h(x) \sim x` as :math:`x` approaches :math:`+\infty`.
.. function:: double gsl_sf_hazard (double x)
int gsl_sf_hazard_e (double x, gsl_sf_result * result)
These routines compute the hazard function for the normal distribution.
.. Exceptional Return Values: GSL_EUNDRFLW