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

In Bessel: Computations and Approximations for Bessel Functions

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

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), as for besselI() etc.
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.

Details

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

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 R interface.

References

Abramowitz, M., and Stegun, I. A. (1955, etc). Handbook of mathematical functions (NBS AMS series 55, U.S. Dept. of Commerce).

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.

For large x and/or nu arguments, algorithm AS~644 is not good enough, and the results may overflow to Infor underflow to zero, such that direct computation of \log(I_ν(x)) and \log(K_ν(x)) are desirable. For this, we provide besselI.nuAsym() 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

Bessel documentation built on May 2, 2019, 4:38 p.m.