| Bessel {base} | R Documentation |
Bessel Functions of integer and fractional order, of first and second kind, J(nu) and Y(nu), and Modified Bessel functions (of first and third kind), I(nu) and K(nu).
besselI(x, nu, expon.scaled = FALSE) besselK(x, nu, expon.scaled = FALSE) besselJ(x, nu) besselY(x, nu)
x |
numeric, >= 0. |
nu |
numeric; The order (maybe fractional!) of the corresponding Bessel function. |
expon.scaled |
logical; if TRUE, the results are
exponentially scaled in order to avoid overflow
(I(nu)) or underflow (K(nu)),
respectively. |
If expon.scaled = TRUE, exp(-x) I(x;nu),
or exp(x) K(x;nu) are returned.
For nu < 0, formulae 9.1.2 and 9.6.2 from Abramowitz &
Stegun are applied (which is probably suboptimal), except for
besselK which is symmetric in nu.
Numeric vector of the same length of x with the (scaled, if
expon.scale=TRUE) values of the corresponding Bessel function.
Original Fortran code:
W. J. Cody, Argonne National Laboratory
Translation to C and adaption to R:
Martin Maechler maechler@stat.math.ethz.ch.
The C code is a translation of Fortran routines from http://www.netlib.org/specfun/r[ijky]besl.
Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. Dover, New York; Chapter 9: Bessel Functions of Integer Order.
Other special mathematical functions, such as
gamma, Γ(x), and beta,
B(x).
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 bogus 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)}))