besselI.nuAsym {Bessel} | R Documentation |
Asymptotic Expansion of Bessel I(x,nu) and K(x,nu) for Large nu (and x)
Description
Compute Bessel functions and
for large
and possibly large
,
using asymptotic expansions in Debye polynomials.
Usage
besselI.nuAsym(x, nu, k.max, expon.scaled = FALSE, log = FALSE)
besselK.nuAsym(x, nu, k.max, expon.scaled = FALSE, log = FALSE)
Arguments
x |
numeric or |
nu |
numeric; The order (maybe fractional!) of the corresponding Bessel function. |
k.max |
integer number of terms in the expansion. Must be in
|
expon.scaled |
logical; if |
log |
logical; if TRUE, |
Details
Abramowitz & Stegun , page 378, has formula 9.7.7 and 9.7.8 for the asymptotic
expansions of and
, respectively,
also saying
When
, these expansions
(of
and
)
hold uniformly with respect to
in the sector
,
where
iw qn arbitrary positive number.
and for this reason, we require
.
The Debye polynomials are defined in 9.3.9 and 9.3.10 (page 366).
Value
a numeric vector of the same length as the long of x
and
nu
. (usual argument recycling is applied implicitly.)
Author(s)
Martin Maechler
References
Abramowitz, M., and Stegun, I. A. (1964, etc). Handbook of mathematical functions, pp. 366, 378.
See Also
From this package Bessel: BesselI()
; further,
besselIasym()
for the case when is large and
is small or moderate.
Further, from base: besselI
, etc.
Examples
x <- c(1:10, 20, 50, 100, 100000)
nu <- c(1, 10, 20, 50, 10^(2:10))
sapply(0:4, function(k.)
sapply(nu, function(n.)
besselI.nuAsym(x, nu=n., k.max = k., log = TRUE)))
sapply(0:4, function(k.)
sapply(nu, function(n.)
besselK.nuAsym(x, nu=n., k.max = k., log = TRUE)))