Error Functions¶
The error function is described in Abramowitz & Stegun, Chapter 7.
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gsl_sf_erf(x)¶
This routine computes the error function \(\operatorname{erf}(x)\), where
\[\operatorname{erf}(x) = (2/\sqrt{\pi}) \int_0^x \exp(-t^2) dt.\]
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gsl_sf_erfc(x)¶
This routine computes the complementary error function
\[\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = (2/\sqrt{\pi}) \int_x^\infty \exp(-t^2) dt.\]
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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.
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gsl_sf_erf_Z(x)¶
This routine computes the Gaussian probability density function \(Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)\).
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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 \(+\infty\).
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gsl_sf_hazard(x)¶
This routine computes the hazard function for the normal distribution.