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

In Bessel: Computations and Approximations for Bessel Functions

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

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

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 or complex, with real part \ge 0.
nu	numeric; The order (maybe fractional!) of the corresponding Bessel function.
k.max	integer number of terms in the expansion. Must be in 0:5, currently.
expon.scaled	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.
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_{\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 is an 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).

Value

a numeric vector of the same length as the long of x and nu. (usual argument recycling is applied implicitly.)

Author(s)

Martin Maechler

References

Abramowitz, M., and Stegun, I. A. (1964, etc). Handbook of mathematical functions, pp. 366, 378, e.g.,

See Also

From this package Bessel: BesselI(); further, besselIasym() for the case when x is large and \nu is small or moderate (using A.&S. (9.7.1) and (9.7.2)).

Further, from base: besselI, etc.

Examples

x <- c(1,2, 10, 20, 50, 100, 1000, 1e4)
nu <- c(outer(c(1,2,5), 10^(0:9)))
ks <- 0:4; ks <- setNames(ks, paste0("k=",ks))

if(requireNamespace("sfsmisc")) { # Mostly for plotting
    formatN <- if(packageVersion("sfsmisc") >= "1.1.26")
                   sfsmisc::frmtNum else sfsmisc::formatN
    eaxis    <- sfsmisc::eaxis
    mult.fig <- sfsmisc::mult.fig
} else { ## no {sfsmisc}
    formatN <- function(x) formatC(x, width=1)
    eaxis <- axis
    mult.fig <- function(nr.plots, main=NULL, ...) { # cheap but working substitute
        mfrow <- n2mfrow(nr.plots)
        tit.wid <- if(is.null(main)) 0 else 1 + 1.5 * par("cex.main")
        old.par <- par(mfrow = mfrow, oma = c(0, 0, tit.wid, 0),
                       mar = 0.1 + c(4, 4, 2, 1), mgp = c(1.5, 0.6, 0))
        if (!is.null(main)) {
            plot.new(); mtext(main, side = 3, outer = TRUE, cex=par("cex.main")); par(new = TRUE)
        }
        invisible(list(old.par = old.par))
      }
}

nus <- setNames(nu, paste0("nu=",formatN(nu)))

mI <- sapply(simplify = "array", ks, function(k.)
            sapply(nus, function(n.)
                   besselI.nuAsym(x, nu=n., k.max = k., log = TRUE)))

mK <- sapply(simplify = "array", ks, function(k.)
            sapply(nus, function(n.)
                   besselK.nuAsym(x, nu=n., k.max = k., log = TRUE)))
str(mK) # 7 x 30 x 5 array

##------------ Plotting  besselI :  I_[nu](x = x0) vs nu -----------------------------------

str(op <- mult.fig(length(x))$old.par); par("mfrow")
for(ix in seq_along(x)) {
    matplot(nu, mI[ix,,], type = "l", log = "x",
            xaxt = "n", xlab = quote(nu), ylab = quote(I[nu](x)),
            main = substitute(I[nu](x==X, nu, k.max == 0:4), list(X = (xi <- x[ix]))))
    eaxis(1)
}
par(op)

##------------ Plotting  besselK :  K_[nu](x = x0) vs nu -----------------------------------

str(mult.fig(length(x), main = expression(log(K[nu](x)) ~ vs ~ nu ~~ "x in log-scale"))$old.par)
par("mfrow")
for(ix in seq_along(x)) {
    y <- mK[ix,,]
    matplot(nu, y, type = "l", log = "x", ylim = c(min(y), min(max(y), 3*abs(min(y)))),
            xaxt = "n", xlab = quote(nu), ylab = quote(K[nu](x)),
            main = substitute(K[nu](x==X, nu, k.max == 0:4), list(X = (xi <- x[ix]))))
    abline(h = 0, col = "gray20", lty = 3)
    eaxis(1)
}
par(op)

str(mult.fig(length(x), main = expression(log(K[nu](x)) ~ vs ~ nu ~~ "in log-log"))$old.par)
for(ix in seq_along(x)) {
    matplot(nu, mK[ix,,], type = "l", log = "xy", # --> 14 warnings: each plot h
            xaxt = "n", yaxt = "n", xlab = quote(nu), ylab = quote(K[nu](x)),
            main = substitute(K[nu](x==X, nu, k.max == 0:4), list(X = (xi <- x[ix]))))
    eaxis(1); eaxis(2)
}
par(op)

## Compare with  besselK() -- which I can use expon.scaled !
dimnames(mI)[[1]] <- dimnames(mK)[[1]] <- paste0("x=", formatN(x))
str(mK)

rbind(t(mK[,"nu=200",]), log.besselK = log(besselK(x, nu=200, expon.scaled=TRUE)) - x)
##--> apart from k=0  the asymptotic approximation seems good already for k=1 here

rbind(t(mK[,"nu=500",]), log.besselK = log(besselK(x, nu=500, expon.scaled=TRUE)) - x)
##--> apart from k=0  the asymptotic approximation seems good already for k=1 here
##---> need other, smaller x[] range -  to get non-Inf besselK()

Bessel documentation built on Sept. 28, 2026, 5:10 p.m.