besselI.nuAsym: Asymptotic Expansion of Bessel I(x,nu) and K(x,nu) for Large... In Bessel: Bessel -- Bessel Functions Computations and Approximations

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

 ```1 2``` ```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, >= 0. `nu` numeric; The order (maybe fractional!) of the corresponding Bessel function. `k.max` integer number of terms in the expansion. Must be in `0:4`, currently. `expon.scaled` logical; if `TRUE`, the results are exponentially scaled in order to avoid overflow (I(nu)) or underflow (K(nu)), respectively. `log` logical; if TRUE, \log(f(.)) is returned instead of f.

Details

Abramowitz & Stegun , page 378, has formula 9.7.7 and 9.7.8 for the asymptotic expansions of I_{ν}(x) and K_{ν}(x), respectively.

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.)

Martin Maechler

References

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, `besselIasym()` for the case when x is large and ν is small or moderate; further base `besselI`, etc
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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))) ```