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

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.

Value

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

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+.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)}))