\Header{Bessel}{Bessel Functions}
\alias{bessel}{Bessel}
\alias{besselI}{Bessel}
\alias{besselJ}{Bessel}
\alias{besselK}{Bessel}
\alias{besselY}{Bessel}
\alias{gammaCody}{Bessel}
\keyword{math}{Bessel}
\begin{Description}\relax
Bessel Functions of integer and fractional order, of first
and second kind, \eqn{J_{\nu}}{J(nu)} and \eqn{Y_{\nu}}{Y(nu)}, and
Modified Bessel functions (of first and third kind),
\eqn{I_{\nu}}{I(nu)} and \eqn{K_{\nu}}{K(nu)}. 

\code{gammaCody} is the \eqn{(\Gamma)}{} function as from the Specfun
package and originally used in the Bessel code.\end{Description}
\begin{Usage}
\begin{verbatim}
besselI(x, nu, expon.scaled = FALSE)
besselK(x, nu, expon.scaled = FALSE)
besselJ(x, nu)
besselY(x, nu)
gammaCody(x)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] numeric, \eqn{\ge 0}{>= 0}.
\item[\code{nu}] numeric; The \emph{order} (maybe fractional!) of the
corresponding Bessel function.
\item[\code{expon.scaled}] logical; if \code{TRUE}, the results are
exponentially scaled in order to avoid overflow
(\eqn{I_{\nu}}{I(nu)}) or underflow (\eqn{K_{\nu}}{K(nu)}),
respectively.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The underlying C code stems from \emph{Netlib}
(\url{http://www.netlib.org/specfun/r[ijky]besl}).

If \code{expon.scaled = TRUE}, \eqn{e^{-x} I_{\nu}(x)}{exp(-x) I(x;nu)},
or \eqn{e^{x} K_{\nu}(x)}{exp(x) K(x;nu)} are returned.

\code{gammaCody} may be somewhat faster but less precise and/or robust
than \R's standard \code{\Link{gamma}}.  It is here for experimental
purpose mainly, and \emph{may be defunct very soon}.

For \eqn{\nu < 0}{nu < 0}, formulae 9.1.2 and 9.6.2 from the
reference below are applied (which is probably suboptimal), unless for
\code{besselK} which is symmetric in \code{nu}.\end{Details}
\begin{Value}
Numeric vector of the same length of \code{x} with the (scaled, if
\code{expon.scale=TRUE}) values of the corresponding Bessel function.\end{Value}
\begin{Author}\relax
Original Fortran code:
W. J. Cody, Argonne National Laboratory \\
Translation to C and adaption to \R:
Martin Maechler \email{maechler@stat.math.ethz.ch.}\end{Author}
\begin{References}\relax
Abramowitz, M. and Stegun, I. A. (1972)
\emph{Handbook of Mathematical Functions.} Dover, New York;
Chapter 9: Bessel Functions of Integer Order.\end{References}
\begin{SeeAlso}\relax
Other special mathematical functions, as the
\code{\Link{gamma}}, \eqn{\Gamma(x)}{}, and \code{\Link{beta}},
\eqn{B(x)}{}.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
nus <- c(0:5,10,20)

x <- seq(0,4, len= 501)
plot(x,x, ylim = c(0,6), ylab="",type='n', main = "Bessel Functions  I_nu(x)")
for(nu in nus) lines(x,besselI(x,nu=nu), col = nu+2)
legend(0,6, leg=paste("nu=",nus), col = nus+2, lwd=1)

x <- seq(0,40,len=801); yl <- c(-.8,.8)
plot(x,x, ylim = yl, ylab="",type='n', main = "Bessel Functions  J_nu(x)")
for(nu in nus) lines(x,besselJ(x,nu=nu), col = nu+2)
legend(32,-.18, leg=paste("nu=",nus), col = nus+2, lwd=1)

## Negative nu's :
xx <- 2:7
nu <- seq(-10,9, len = 2001)
op <- par(lab = c(16,5,7))
matplot(nu, t(outer(xx,nu, besselI)), type = 'l', ylim = c(-50,200),
        main = expression(paste("Bessel ",I[nu](x)," for fixed ", x,
                                ",  as ",f(nu))),
        xlab = expression(nu))
abline(v=0, col = "light gray", lty = 3)
legend(5,200, leg = paste("x=",xx), col=seq(xx), lty=seq(xx))
par(op)

x0 <- 2^(-20:10)
plot(x0,x0^-8, log='xy', ylab="",type='n',
     main = "Bessel Functions  J_nu(x)  near 0\n log - log  scale")
for(nu in sort(c(nus,nus+.5))) lines(x0,besselJ(x0,nu=nu), col = nu+2)
legend(3,1e50, leg=paste("nu=", paste(nus,nus+.5, sep=",")), col=nus+2, lwd=1)

plot(x0,x0^-8, log='xy', ylab="",type='n',
     main = "Bessel Functions  K_nu(x)  near 0\n log - log  scale")
for(nu in sort(c(nus,nus+.5))) lines(x0,besselK(x0,nu=nu), col = nu+2)
legend(3,1e50, leg=paste("nu=", paste(nus,nus+.5, sep=",")), col=nus+2, lwd=1)

x <- x[x > 0]
plot(x,x, ylim=c(1e-18,1e11),log="y", ylab="",type='n',
     main = "Bessel Functions  K_nu(x)")
for(nu in nus) lines(x,besselK(x,nu=nu), col = nu+2)
legend(0,1e-5, leg=paste("nu=",nus), col = nus+2, lwd=1)

## Check the Scaling :
for(nu in nus)
   print(all(abs(1- besselK(x,nu)*exp( x) / besselK(x,nu,expo=TRUE)) < 2e-15))
for(nu in nus)
   print(all(abs(1- besselI(x,nu)*exp(-x) / besselI(x,nu,expo=TRUE)) < 1e-15))

yl <- c(-1.6, .6)
plot(x,x, ylim = yl, ylab="",type='n', main = "Bessel Functions  Y_nu(x)")
for(nu in nus){xx <- x[x > .6*nu]; lines(xx,besselY(xx,nu=nu), col = nu+2)}
legend(25,-.5, leg=paste("nu=",nus), col = nus+2, lwd=1)



\end{ExampleCode}
\end{Examples}

