besselI.nuAsym {Bessel}R Documentation

Asymptotic Expansion of Bessel I(x,nu) and K(x,nu) for Large nu (and x)


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)



numeric or complex, with real part \ge 0.


numeric; The order (maybe fractional!) of the corresponding Bessel function.


integer number of terms in the expansion. Must be in 0:5, currently.


logical; if TRUE, the results are exponentially scaled, the same as in the corresponding BesselI() and BesselK() functions in order to avoid overflow (I_{\nu}) or underflow (K_{\nu}), respectively.


logical; if TRUE, \log(f(.)) is returned instead of f.


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. (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.


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)))

[Package Bessel version 0.6-0 Index]