Bessel {base}R Documentation

Bessel Functions

Description

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).

Usage

besselI(x, nu, expon.scaled = FALSE)
besselK(x, nu, expon.scaled = FALSE)
besselJ(x, nu)
besselY(x, nu)

Arguments

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.

Details

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.

The current algorithms will give warnings about accuracy loss for large arguments. In some cases, these warnings are exagerated, and the precision is perfect. For large nu, say in the order of millions, the current algorithms are rarely useful.

Value

Numeric vector with the (scaled, if expon.scaled = TRUE) values of the corresponding Bessel function.

The length of the result is the maximum of the lengths of the parameters. All parameters are recycled to that length.

Author(s)

Original Fortran code: W. J. Cody, Argonne National Laboratory
Translation to C and adaption to R: Martin Maechler maechler@stat.math.ethz.ch.

Source

The C code is a translation of Fortran routines from http://www.netlib.org/specfun/ribesl, ../rjbesl, etc.

References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. Dover, New York; Chapter 9: Bessel Functions of Integer Order.

See Also

Other special mathematical functions, such as gamma, Γ(x), and beta, B(x).

Examples

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+0.5)))
    lines(x0, besselJ(x0, nu = nu), col = nu + 2)
legend(3, 1e50, legend = paste("nu=", paste(nus, nus+0.5, 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+0.5)))
    lines(x0, besselK(x0, nu = nu), col = nu + 2)
legend(3, 1e50, legend = paste("nu=", paste(nus, nus + 0.5, 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)}))

[Package base version 3.1.3 Index]