BesselH {Bessel}

Hankel (H-Bessel) Function (of Complex Argument)

Description

Compute the Hankel functions H(1,*) and H(2,*), also called ‘H-Bessel’ function (of the third kind), of complex arguments. They are defined as

H(1,\nu, z) := H_{\nu}^{(1)}(z) = J_{\nu}(z) + i Y_{\nu}(z),

H(2,\nu, z) := H_{\nu}^{(2)}(z) = J_{\nu}(z) - i Y_{\nu}(z),

where J_{\nu}(z) and Y_{\nu}(z) are the Bessel functions of the first and second kind, see BesselJ, etc.

Usage

BesselH(m, z, nu, expon.scaled = FALSE, nSeq = 1, verbose = 0)

Arguments

m

integer, either 1 or 2, indicating the kind of Hankel function.

z

complex or numeric vector of values different from 0.

nu

numeric, must currently be non-negative.

expon.scaled

logical indicating if the result should be scaled by an exponential factor (typically to avoid under- or over-flow).

nSeq

positive integer; if > 1, computes the result for a whole sequence of nu values of length nSeq, see ‘Details’ below.

verbose

integer defaulting to 0, indicating the level of verbosity notably from C code.

Details

By default (when expon.scaled is false), the resulting sequence (of length nSeq) is for m = 1,2,

y_j = H(m, \nu+j-1, z),

computed for j=1,\dots,nSeq.

If expon.scaled is true, the sequence is for m = 1,2

y_j = \exp(-\tilde{m} z i) \cdot H(m, \nu+j-1, z),

where \tilde{m} = 3-2m (and i^2 = -1), for j=1,\dots,nSeq.

Value

a complex or numeric vector (or matrix if nSeq > 1) of the same length and mode as z.

Author(s)

Donald E. Amos, Sandia National Laboratories, wrote the original fortran code. Martin Maechler did the R interface.

References

see BesselI.

See Also

BesselI etc; the Airy function Airy.

Examples

##------------------ H(1, *) ----------------
nus <- c(1,2,5,10)
for(i in seq_along(nus))
   curve(BesselH(1, x, nu=nus[i]), -10, 10, add= i > 1, col=i, n=1000)
legend("topleft", paste("nu = ", format(nus)), col = seq_along(nus), lty=1)

## nu = 10 looks a bit  "special" ...   hmm...
curve(BesselH(1, x, nu=10), -.3, .3, col=4,
      ylim = c(-10,10), n=1000)

##------------------ H(2, *) ----------------
for(i in seq_along(nus))
   curve(BesselH(2, x, nu=nus[i]), -10, 10, add= i > 1, col=i, n=1000)
legend("bottomright", paste("nu = ", format(nus)), col = seq_along(nus), lty=1)
## the same nu = 10 behavior ..