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Revision 161411bb86f97e5a8bd89091cd61d03a33c2761a authored by Martin Maechler on 06 February 2012, 00:00:00 UTC, committed by Gabor Csardi on 06 February 2012, 00:00:00 UTC
1 parent 5bc804b
Tip revision: 161411b
math-fun.Rd
\name{math-fun}
\alias{A..Z}
\alias{sinc}
\title{Sinc, Zolotarev's, and Other Mathematical Utility Functions}
\description{
\code{sinc(x)} computes the \emph{sinc function}
\eqn{s(x)=\sin(x)/x}{s(x) = sin(x)/x} for \eqn{x\ne 0}{x != 0} and \eqn{s(0) = 1},
such that \eqn{s()} is continuous, also at \eqn{x = 0}.

\code{A..Z(x, a)} computes Zolotarev's function to
the power \code{1-a}.

}
\usage{
sinc(x)
A..Z(x, alpha, I.alpha = 1 - alpha)
}
\arguments{
\item{x}{\code{\link{numeric}} argument in \eqn{[0,\pi]}{[0,pi]}, typically a vector.}
\item{alpha}{parameter in (0,1].}
\item{I.alpha}{must be \code{ = 1 - alpha}, maybe more accurately
when \code{alpha} is very close to 1.}
}
\details{
For more details about Zolotarev's function, see, for example, Devroye (2009).
}
\value{
\code{A..Z(x,alpha)} is \eqn{\tilde A_{Z}(x,\alpha)}{A~Z(x,alpha)},
defined as
\deqn{\frac{\sin(\alpha x)^\alpha\sin((1-\alpha)x)^{1-\alpha}}{\sin(x)},\
x\in[0,\pi],}{%
sin(alpha*x)^alpha * sin((1-alpha)*x)^(1-alpha) / sin(x),  x in [0,pi],}

where \eqn{\alpha\in(0,1]}{alpha in (0,1]} is \code{alpha}.
}
\author{Martin Maechler}
\seealso{
\code{\link{retstable}} internally makes use of these functions.
}
\references{
Devroye, L. (2009)
Random variate generation for exponentially and polynomially tilted
stable distributions,
\emph{ACM Transactions on Modeling and Computer Simulation} \bold{19}, 18, 1--20.
}
\examples{
curve(sinc, -15,25); abline(h=0,v=0, lty=2)
curve(A..Z(x, 0.25), xlim = c(-4,4),
main = "Zolotarev's function A(x) ^ 1-alpha")
}
\keyword{math} Computing file changes ...