base (version 3.4.1)

Bessel: 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, \(\ge 0\).

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.

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.

Details

If expon.scaled = TRUE, \(e^{-x} I_{\nu}(x)\), or \(e^{x} K_{\nu}(x)\) 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 exaggerated, and the precision is perfect. For large nu, say in the order of millions, the current algorithms are rarely useful.

References

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

In order of “Source” citation above:

Sockne, David J. (1973) Bessel Functions of Real Argument and Integer Order. NBS Jour. of Res. B. 77B, 125--132.

Cody, William J. (1983) Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Transactions on Mathematical Software 9(2), 242--245.

Campbell, J.B. (1980) On Temme's algorithm for the modified Bessel function of the third kind. ACM Transactions on Mathematical Software 6(4), 581--586.

Campbell, J.B. (1979) Bessel functions J_nu(x) and Y_nu(x) of float order and float argument. Comp. Phy. Comm. 18, 133--142.

Temme, Nico M. (1976) On the numerical evaluation of the ordinary Bessel function of the second kind. J. Comput. Phys. 21, 343--350.

See Also

Other special mathematical functions, such as gamma, \(\Gamma(x)\), 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)}))