\HeaderA{Bessel}{Bessel Functions}{Bessel}
\aliasA{bessel}{Bessel}{bessel}
\aliasA{besselI}{Bessel}{besselI}
\aliasA{besselJ}{Bessel}{besselJ}
\aliasA{besselK}{Bessel}{besselK}
\aliasA{besselY}{Bessel}{besselY}
\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)}.
\end{Description}
\begin{Usage}
\begin{verbatim}
besselI(x, nu, expon.scaled = FALSE)
besselK(x, nu, expon.scaled = FALSE)
besselJ(x, nu)
besselY(x, nu)
\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
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.

For \eqn{\nu < 0}{nu < 0}, formulae 9.1.2 and 9.6.2 from Abramowitz \&
Stegun  are applied (which is probably suboptimal), except 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{Source}\relax
The C code is a translation of Fortran routines from
\url{http://www.netlib.org/specfun/r[ijky]besl}.
\end{Source}
\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, such as
\code{\LinkA{gamma}{gamma}}, \eqn{\Gamma(x)}{}, and \code{\LinkA{beta}{beta}},
\eqn{B(x)}{}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
require(graphics)

nus <- c(0:5, 10, 20)

x <- seq(0, 4, length.out = 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, legend = paste("nu=", nus), col = nus+2, lwd = 1)

x <- seq(0, 40, length.out = 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, legend = paste("nu=", nus), col = nus+2, lwd = 1)

## Negative nu's :
xx <- 2:7
nu <- seq(-10, 9, length.out = 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, legend = 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, legend = 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, legend = 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, legend=paste("nu=", nus), col = nus+2, lwd = 1)

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, legend = paste("nu=", nus), col = nus+2, lwd = 1)

## negative nu in bessel_Y -- was bogous for a long time
curve(besselY(x, -0.1), 0, 10, ylim = c(-3,1), ylab = '')
for(nu in c(seq(-0.2, -2, by = -0.1)))
  curve(besselY(x, nu), add = TRUE)
title(expression(besselY(x, nu) * "   " *
                 {nu == list(-0.1, -0.2, ..., -2)}))
\end{ExampleCode}
\end{Examples}

