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 (\Gamma) 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, \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.

Details

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

If expon.scaled = TRUE, e^{-x} I_{\nu}(x), or e^{x} K_{\nu}(x) 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, \Gamma(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)

---