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 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.

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.
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.
Author(s)
Original Fortran code:
W. J. Cody, Argonne National Laboratory
Translation to C and adaptation to R:
Martin Maechler [email protected].
Source
The C code is a translation of Fortran routines from
https://netlib.org/specfun/ribesl, ‘../rjbesl’, etc.
The four source code files for bessel[IJKY] each contain a paragraph
“Acknowledgement” and “References”, a short summary of
which is
 besselI

based on (code) by David J. Sookne, see Sookne (1973)...
Modifications... An earlier version was published in Cody (1983).
 besselJ

as besselI
 besselK

based on (code) by J. B. Campbell (1980)... Modifications...
 besselY

draws heavily on Temme's Algol program for
$Y$
... and on Campbell's programs for $Y_\nu(x)$
.... ... heavily modified.
References
Abramowitz, M. and Stegun, I. A. (1972).
Handbook of Mathematical Functions.
Dover, New York;
Chapter 9: Bessel Functions of Integer Order.
In order of “Source” citation above:
Sookne, David J. (1973).
Bessel Functions of Real Argument and Integer Order.
Journal of Research of the National Bureau of Standards,
77B, 125–132.
doi:10.6028/jres.077B.012.
Cody, William J. (1983).
Algorithm 597: Sequence of modified Bessel functions of the first kind.
ACM Transactions on Mathematical Software, 9(2), 242–245.
doi:10.1145/357456.357462.
Campbell, J.B. (1980).
On Temme's algorithm for the modified Bessel function of the third kind.
ACM Transactions on Mathematical Software, 6(4), 581–586.
doi:10.1145/355921.355928.
Campbell, J.B. (1979).
Bessel functions J_nu(x) and Y_nu(x) of float order and float argument.
Computer Physics Communications, 18, 133–142.
doi:10.1016/00104655(79)900304.
Temme, Nico M. (1976).
On the numerical evaluation of the ordinary Bessel function of the
second kind.
Journal of Computational Physics, 21, 343–350.
doi:10.1016/00219991(76)900322.
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(.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")
xx < 2:7
nu < seq(10, 9, length.out = 2001)
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)
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)
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)
x0 < 2^seq(16, 5, length.out=256)
plot(range(x0), c(1e40, 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(1e18, 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, 1e5, 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)
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)}))