Description Usage Arguments Details Value Author(s) References See Also Examples
Compute Bessel function I[nu](x)
and K[nu](x)
for large x and small or moderate
nu, using the asymptotic expansions (9.7.1) and (9.7.2), p.377-8 of
Abramowitz & Stegun, for x -> Inf, even valid for
complex
x,
I_a(x) = exp(x) / sqrt(2*pi*x)* f(x, a),
where
f(x,a) = 1 - (mu-1) / (8x) + (mu-1)(mu-9) / (2! (8x)^2) - ...,
and mu = 4*a^2 and |arg(x)| < π/2.
Whereas besselIasym(x,a)
computes a possibly exponentially scaled
and/or log
ged version of I_a(x),
besselI.ftrms
returns the corresponding terms in the
series expansion of f(x,a) above.
1 2 3 |
x |
numeric or complex (with real part) >= 0. |
nu |
numeric; the order (maybe fractional!) of the corresponding Bessel function. |
k.max, K |
integer number of terms in the expansion. |
expon.scaled |
logical; if |
log |
logical; if TRUE, \log(f(.)) is returned instead of f. |
Even though the reference (A. & S.) requires
|\arg(z)| < pi/2 for I() and
|\arg(z)| < 3 pi/2 for K(),
where arg(z) := Arg(z)
,
the zero-th order term seems correct also for negative (real) numbers.
a numeric (or complex) vector of the same length as x
.
Martin Maechler
Abramowitz, M., and Stegun, I. A. (1955, etc). Handbook of mathematical functions (NBS AMS series 55, U.S. Dept. of Commerce).
From this package Bessel() BesselI()
; further,
besselI.nuAsym()
which is useful when ν is large
(as well); further base besselI
, etc
1 2 3 4 5 6 7 8 9 10 11 | x <- c(1:10, 20, 50, 100^(2:10))
nu <- c(1, 10, 20, 50, 100)
r <- lapply(c(0:4,10,20), function(k.)
sapply(nu, function(n.)
besselIasym(x, nu=n., k.max = k., log = TRUE)))
warnings()
try( # needs improvement in R [or a local workaround]
besselIasym(10000*(1+1i), nu=200, k.max=20, log=TRUE)
) # Error in log1p(-d) : unimplemented complex function
|
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