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.. _continuous-halfcauchy:
HalfCauchy Distribution
=======================
If :math:`Z` is Hyperbolic Secant distributed then :math:`e^{Z}` is
Half-Cauchy distributed. Also, if :math:`W` is (standard) Cauchy
distributed, then :math:`\left|W\right|` is Half-Cauchy distributed.
Special case of the Folded Cauchy distribution with :math:`c=0.`
The support is :math:`x\geq0`. The standard form is
.. math::
:nowrap:
\begin{eqnarray*} f\left(x\right) & = & \frac{2}{\pi\left(1+x^{2}\right)} \\
F\left(x\right) & = & \frac{2}{\pi}\arctan\left(x\right)\\
G\left(q\right) & = & \tan\left(\frac{\pi}{2}q\right)\end{eqnarray*}
.. math::
M\left(t\right)=\cos t+\frac{2}{\pi}\left[\mathrm{Si}\left(t\right)\cos t-\mathrm{Ci}\left(\mathrm{-}t\right)\sin t\right]
where :math:`\mathrm{Si}(t)=\int_0^t \frac{\sin x}{x} dx`, :math:`\mathrm{Ci}(t)=-\int_t^\infty \frac{\cos x}{x} dx`.
.. math::
:nowrap:
\begin{eqnarray*} m_{d} & = & 0\\
m_{n} & = & \tan\left(\frac{\pi}{4}\right)\end{eqnarray*}
No moments, as the integrals diverge.
.. math::
:nowrap:
\begin{eqnarray*} h\left[X\right] & = & \log\left(2\pi\right)\\ & \approx & 1.8378770664093454836.\end{eqnarray*}
Implementation: `scipy.stats.halfcauchy`
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