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 I_{\nu}(x)
and
K_{\nu}(x)
for large \nu
and possibly large
x
,
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 I_{\nu}(x)
and K_{\nu}(x)
, respectively,
also saying
When \nu \to +\infty
, these expansions
(of I_{\nu}(\nu z)
and
K_{\nu}(\nu z)
)
hold uniformly with respect to z
in the sector
arg z \le \frac{1}{2} \pi  \epsilon
,
where \epsilon
iw qn arbitrary positive number.
and for this reason, we require \Re(x) \ge 0
.
The Debye polynomials u_k(x)
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. (1955, etc). Handbook of mathematical functions (NBS AMS series 55, U.S. Dept. of Commerce), pp. 366, 378.
See Also
From this package Bessel: BesselI()
; further,
besselIasym()
for the case when x
is large and
\nu
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)))