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).

`gammaCody` is the (Γ) function as from the Specfun package and originally used in the Bessel code.

### Usage

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

### 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

The underlying C code stems from Netlib (http://www.netlib.org/specfun/r[ijky]besl).

If `expon.scaled = TRUE`, exp(-x) I(x;nu), or exp(x) K(x;nu) are returned.

`gammaCody` may be somewhat faster but less precise and/or robust than R's standard `gamma`. It is here for experimental purpose mainly, and may be defunct very soon.

For nu < 0, formulae 9.1.2 and 9.6.2 from the reference below are applied (which is probably suboptimal), unless for `besselK` which is symmetric in `nu`.

### Value

Numeric vector of the same length of `x` with the (scaled, if `expon.scale=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.

### 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, as the `gamma`, Γ(x), and `beta`, B(x).

### Examples

    nus <- c(0:5,10,20)

    x <- seq(0,4, len= 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, leg=paste("nu=",nus), col = nus+2, lwd=1)

    x <- seq(0,40,len=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, leg=paste("nu=",nus), col = nus+2, lwd=1)

    ## Negative nu's :
    xx <- 2:7
    nu <- seq(-10,9, len = 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, leg = 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, leg=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, leg=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, leg=paste("nu=",nus), col = nus+2, lwd=1)

    ## Check the Scaling :
    for(nu in nus)
       print(all(abs(1- besselK(x,nu)*exp( x) / besselK(x,nu,expo=TRUE)) < 2e-15))
    for(nu in nus)
       print(all(abs(1- besselI(x,nu)*exp(-x) / besselI(x,nu,expo=TRUE)) < 1e-15))

    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, leg=paste("nu=",nus), col = nus+2, lwd=1)

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