\HeaderA{Cauchy}{The Cauchy Distribution}{Cauchy}
\aliasA{dcauchy}{Cauchy}{dcauchy}
\aliasA{pcauchy}{Cauchy}{pcauchy}
\aliasA{qcauchy}{Cauchy}{qcauchy}
\aliasA{rcauchy}{Cauchy}{rcauchy}
\keyword{distribution}{Cauchy}
\begin{Description}\relax
Density, distribution function, quantile function and random
generation for the Cauchy distribution with location parameter
\code{location} and scale parameter \code{scale}.
\end{Description}
\begin{Usage}
\begin{verbatim}
dcauchy(x, location = 0, scale = 1, log = FALSE)
pcauchy(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qcauchy(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rcauchy(n, location = 0, scale = 1)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x, q}] vector of quantiles.
\item[\code{p}] vector of probabilities.
\item[\code{n}] number of observations. If \code{length(n) > 1}, the length
is taken to be the number required.
\item[\code{location, scale}] location and scale parameters.
\item[\code{log, log.p}] logical; if TRUE, probabilities p are given as log(p).
\item[\code{lower.tail}] logical; if TRUE (default), probabilities are
\eqn{P[X \le x]}{P[X <= x]}, otherwise, \eqn{P[X > x]}{P[X > x]}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
If \code{location} or \code{scale} are not specified, they assume
the default values of \code{0} and \code{1} respectively.

The Cauchy distribution with location \eqn{l}{} and scale \eqn{s}{} has
density
\deqn{f(x) = \frac{1}{\pi s}
\left( 1 + \left(\frac{x - l}{s}\right)^2 \right)^{-1}
}{f(x) = 1 / (pi s (1 + ((x-l)/s)^2))}
for all \eqn{x}{}.
\end{Details}
\begin{Value}
\code{dcauchy}, \code{pcauchy}, and \code{qcauchy} are respectively
the density, distribution function and quantile function of the Cauchy
distribution.  \code{rcauchy} generates random deviates from the
Cauchy.
\end{Value}
\begin{Source}\relax
\code{dcauchy}, \code{pcauchy} and \code{qcauchy} are all calculated
from numerically stable versions of the definitions.

\code{rcauchy} uses inversion.
\end{Source}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
\emph{Continuous Univariate Distributions}, volume 1, chapter 16.
Wiley, New York.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{dt}{dt}} for the t distribution which generalizes
\code{dcauchy(*, l = 0, s = 1)}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
dcauchy(-1:4)
\end{ExampleCode}
\end{Examples}

