Bessel

R Base 3.6.2

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 \(K_{\nu}\).

Syntax

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

Arguments

Parameter	Description
`x`	numeric, \(\ge 0\).
`nu`	numeric; The order (maybe fractional and negative) 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.

Return 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.

Details

If expon.scaled = TRUE, \(e^{-x} I_{\nu}(x)\), or \(e^{x} K_{\nu}(x)\) 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. The current algorithms will give warnings about accuracy loss for large arguments. In some cases, these warnings are exaggerated, and the precision is perfect. For large nu, say in the order of millions, the current algorithms are rarely useful.

Examples

Example

# NOT RUN {
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(-.5, 1)
plot(x, x, ylim = yl, ylab = "", type = "n",
     main = "Bessel Functions  J_nu(x)")
abline(h=0, v=0, lty=3)
for(nu in nus) lines(x, besselJ(x, nu = nu), col = nu + 2)
legend("topright", legend = paste("nu=", nus), col = nus + 2, lwd = 1, bty="n")

## Negative nu's --------------------------------------------------
xx <- 2:7
nu <- seq(-10, 9, length.out = 2001)
## --- I() --- --- --- ---
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=1:5)

## --- J() --- --- --- ---
bJ <- t(outer(xx, nu, besselJ))
matplot(nu, bJ, type = "l", ylim = c(-500, 200),
        xlab = quote(nu), ylab = quote(J[nu](x)),
        main = expression(paste("Bessel ", J[nu](x), " for fixed ", x)))
abline(v = 0, col = "light gray", lty = 3)
legend("topright", legend = paste("x=", xx), col=seq(xx), lty=1:5)

## ZOOM into right part:
matplot(nu[nu > -2], bJ[nu > -2,], type = "l",
        xlab = quote(nu), ylab = quote(J[nu](x)),
        main = expression(paste("Bessel ", J[nu](x), " for fixed ", x)))
abline(h=0, v = 0, col = "gray60", lty = 3)
legend("topright", legend = paste("x=", xx), col=seq(xx), lty=1:5)


##---------------  x --> 0  -----------------------------
x0 <- 2^seq(-16, 5, length.out=256)
plot(range(x0), c(1e-40, 1), log = "xy", xlab = "x", ylab = "", type = "n",
     main = "Bessel Functions  J_nu(x)  near 0\n log - log  scale") ; axis(2, at=1)
for(nu in sort(c(nus, nus+0.5)))
    lines(x0, besselJ(x0, nu = nu), col = nu + 2, lty= 1+ (nu%%1 > 0))
legend("right", legend = paste("nu=", paste(nus, nus+0.5, sep=", ")),
       col = nus + 2, lwd = 1, bty="n")

x0 <- 2^seq(-10, 8, length.out=256)
plot(range(x0), 10^c(-100, 80), log = "xy", xlab = "x", ylab = "", type = "n",
     main = "Bessel Functions  K_nu(x)  near 0\n log - log  scale") ; axis(2, at=1)
for(nu in sort(c(nus, nus+0.5)))
    lines(x0, besselK(x0, nu = nu), col = nu + 2, lty= 1+ (nu%%1 > 0))
legend("topright", legend = paste("nu=", paste(nus, nus + 0.5, sep = ", ")),
       col = nus + 2, lwd = 1, bty="n")

x <- x[x > 0]
plot(x, x, ylim = c(1e-18, 1e11), log = "y", ylab = "", type = "n",
     main = "Bessel Functions  K_nu(x)"); axis(2, at=1)
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)}))
# }