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.. index::
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single: error function
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single: erf(x)
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single: erfc(x)
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The error function is described in Abramowitz & Stegun, Chapter 7. The
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functions in this section are declared in the header file
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:file:`gsl_sf_erf.h`.
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Error Function
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--------------
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.. function:: double gsl_sf_erf (double x)
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int gsl_sf_erf_e (double x, gsl_sf_result * result)
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These routines compute the error function :math:`\erf(x)`,
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where
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:math:`\erf(x) = (2/\sqrt{\pi}) \int_0^x dt \exp(-t^2)`.
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.. Exceptional Return Values: none
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Complementary Error Function
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----------------------------
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.. function:: double gsl_sf_erfc (double x)
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int gsl_sf_erfc_e (double x, gsl_sf_result * result)
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These routines compute the complementary error function
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:math:`\erfc(x) = 1 - \erf(x) = (2/\sqrt{\pi}) \int_x^\infty \exp(-t^2)`
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.. Exceptional Return Values: none
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Log Complementary Error Function
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--------------------------------
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.. function:: double gsl_sf_log_erfc (double x)
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int gsl_sf_log_erfc_e (double x, gsl_sf_result * result)
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These routines compute the logarithm of the complementary error function
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:math:`\log(\erfc(x))`.
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.. Exceptional Return Values: none
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Probability functions
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---------------------
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The probability functions for the Normal or Gaussian distribution are
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described in Abramowitz & Stegun, Section 26.2.
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.. function:: double gsl_sf_erf_Z (double x)
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int gsl_sf_erf_Z_e (double x, gsl_sf_result * result)
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These routines compute the Gaussian probability density function
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:math:`Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)`
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.. function:: double gsl_sf_erf_Q (double x)
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int gsl_sf_erf_Q_e (double x, gsl_sf_result * result)
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These routines compute the upper tail of the Gaussian probability function
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:math:`Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2)`
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.. Exceptional Return Values: none
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.. index::
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single: hazard function, normal distribution
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single: Mills' ratio, inverse
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The *hazard function* for the normal distribution,
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also known as the inverse Mills' ratio, is defined as,
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.. only:: not texinfo
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.. math:: h(x) = {Z(x) \over Q(x)} = \sqrt{2 \over \pi} {\exp(-x^2 / 2) \over \erfc(x/\sqrt 2)}
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.. only:: texinfo
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::
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h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)
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It decreases rapidly as :math:`x` approaches :math:`-\infty` and asymptotes
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to :math:`h(x) \sim x` as :math:`x` approaches :math:`+\infty`.
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.. function:: double gsl_sf_hazard (double x)
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int gsl_sf_hazard_e (double x, gsl_sf_result * result)
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These routines compute the hazard function for the normal distribution.
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.. Exceptional Return Values: GSL_EUNDRFLW
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