| Bessel {base} | R Documentation |
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.
besselI(x, nu, expon.scaled = FALSE) besselK(x, nu, expon.scaled = FALSE) besselJ(x, nu) besselY(x, nu) gammaCody(x)
x |
numeric, >= 0. |
nu |
numeric; >= 0 unless in besselK which
is symmetric in nu. The order 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. |
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.
Numeric vector of the same length of x with the (scaled, if
expon.scale=TRUE) values of the corresponding Bessel function.
Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions, Dover, New York; Chapter 9: Bessel Functions of Integer Order.
Other special mathematical functions, as the
gamma, &Gamma(x), and beta,
B(x).
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)
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)