Bessel
Bessel Functions
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 $K(nu)$.
 Keywords
 math
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.
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
.
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.
Source
The C code is a translation of Fortran routines from http://www.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:
Sockne, David J. (1973) Bessel Functions of Real Argument and Integer Order. NBS Jour. of Res. B. 77B, 125132.
Cody, William J. (1983) Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Transactions on Mathematical Software 9(2), 242245.
Campbell, J.B. (1980) On Temme's algorithm for the modified Bessel function of the third kind. ACM Transactions on Mathematical Software 6(4), 581586.
Campbell, J.B. (1979) Bessel functions J_nu(x) and Y_nu(x) of float order and float argument. Comp. Phy. Comm. 18, 133142.
Temme, Nico M. (1976) On the numerical evaluation of the ordinary Bessel function of the second kind. J. Comput. Phys. 21, 343350.
See Also
Other special mathematical functions, such as
gamma
, $\Gamma(x)$, and beta
,
$B(x)$.
Examples
library(base)
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(1e18, 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, 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)
## 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)}))