R: Bessel Functions

Bessel Functions

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; >= 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.

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.

Details

The underlying code for these functions stem 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.

Value

numeric of the same length of x with the (scaled, if expon.scale=T) values of the corresponding Bessel function.

References

Abramowitz, M. and Stegun, I. A. (1972).
Handbook of Mathematical Functions, Dover, New York;
Chapter 9: Bessel Functions of Integer Order.

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)

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)


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)

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