besselI.nuAsym | R Documentation |
Compute Bessel functions I_{\nu}(x)
and
K_{\nu}(x)
for large \nu
and possibly large
x
,
using asymptotic expansions in Debye polynomials.
besselI.nuAsym(x, nu, k.max, expon.scaled = FALSE, log = FALSE)
besselK.nuAsym(x, nu, k.max, expon.scaled = FALSE, log = FALSE)
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, |
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).
a numeric vector of the same length as the long of x
and
nu
. (usual argument recycling is applied implicitly.)
Martin Maechler
Abramowitz, M., and Stegun, I. A. (1964, etc). Handbook of mathematical functions, pp. 366, 378.
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.
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)))
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