Bessel {Bessel}

Bessel Functions of Complex Arguments I(), J(), K(), and Y()

Description

Compute the Bessel functions I(), J(), K(), and Y(), of complex arguments z and real nu,

Usage

BesselI(z, nu, expon.scaled = FALSE, nSeq = 1, verbose = 0)
BesselJ(z, nu, expon.scaled = FALSE, nSeq = 1, verbose = 0)
BesselK(z, nu, expon.scaled = FALSE, nSeq = 1, verbose = 0)
BesselY(z, nu, expon.scaled = FALSE, nSeq = 1, verbose = 0)

Arguments

z

complex or numeric vector.

nu

numeric (scalar).

expon.scaled

logical indicating if the result should be scaled by an exponential factor, typically to avoid under- or over-flow. See the ‘Details’ about the specific scaling.

nSeq

positive integer; if > 1, computes the result for a whole sequence of nu values;
if nu >= 0,nu, nu+1, ..., nu+nSeq-1,
if nu < 0, nu, nu-1, ..., nu-nSeq+1.

verbose

integer defaulting to 0, indicating the level of verbosity notably from C code.

Details

The case nu < 0 is handled by using simple formula from Abramowitz and Stegun, see details in besselI().

The scaling activated by expon.scaled = TRUE depends on the function and the scaled versions are

J():

BesselJ(z, nu, expo=TRUE):= \exp(-\left|\Im(z)\right|) J_{\nu}(z)

Y():

BesselY(z, nu, expo=TRUE) := \exp(-\left|\Im(z)\right|) Y_{\nu}(z)

I():

BesselI(z, nu, expo=TRUE) := \exp(-\left|\Re(z)\right|) I_{\nu}(z)

K():

BesselK(z, nu, expo=TRUE) := \exp(z) K_{\nu}(z)

Value

a complex or numeric vector (or matrix with nSeq columns if nSeq > 1) of the same length (or nrow when nSeq > 1) and mode as z.

Author(s)

Donald E. Amos, Sandia National Laboratories, wrote the original fortran code. Martin Maechler did the translation to C, and partial cleanup (replacing goto's), in addition to the R interface.

References

Abramowitz, M., and Stegun, I. A. (1964, etc). Handbook of mathematical functions (NBS AMS series 55, U.S. Dept. of Commerce), https://personal.math.ubc.ca/~cbm/aands/

Wikipedia (20nn). Bessel Function, https://en.wikipedia.org/wiki/Bessel_function

D. E. Amos (1986) Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order; ACM Trans. Math. Software 12, 3, 265–273.

D. E. Amos (1983) Computation of Bessel Functions of Complex Argument; Sand83-0083.

D. E. Amos (1983) Computation of Bessel Functions of Complex Argument and Large Order; Sand83-0643.

D. E. Amos (1985) A subroutine package for Bessel functions of a complex argument and nonnegative order; Sand85-1018.

Olver, F.W.J. (1974). Asymptotics and Special Functions; Academic Press, N.Y., p.420

See Also

The base R functions besselI(), besselK(), etc.

The Hankel functions (of first and second kind), H_{\nu}^{(1)}(z) and H_{\nu}^{(2)}(z): Hankel.

The Airy functions Ai() and Bi() and their first derivatives, Airy.

For large x and/or nu arguments, algorithm AS~644 is not good enough, and the results may overflow to Inf or underflow to zero, such that direct computation of \log(I_\nu(x)) and \log(K_\nu(x)) are desirable. For this, we provide besselI.nuAsym(), besselIasym() and besselK.nuAsym(*, log= *), based on asymptotic expansions.

Examples

## For real small arguments, BesselI() gives the same as base::besselI() :
set.seed(47); x <- sort(round(rlnorm(20), 2))
M <- cbind(x, b = besselI(x, 3), B = BesselI(x, 3))
stopifnot(all.equal(M[,"b"], M[,"B"], tol = 2e-15)) # ~4e-16 even
M

## and this is true also for the 'exponentially scaled' version:
Mx <- cbind(x, b = besselI(x, 3, expon.scaled=TRUE),
               B = BesselI(x, 3, expon.scaled=TRUE))
stopifnot(all.equal(Mx[,"b"], Mx[,"B"], tol = 2e-15)) # ~4e-16 even