base (version 3.1.1)

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.

Source

The C code is a translation of Fortran routines from http://www.netlib.org/specfun/ribesl, ../rjbesl, etc.

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.

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, $\Gamma(x)$, 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)}))