# besselI.nuAsym: Asymptotic Expansion of Bessel I(x,nu) and K(x,nu) for Large... In 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))) ```