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`.

### 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)}))

[Package base version 2.15.1 Index]