BesselH {Bessel}R Documentation

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


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.


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



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


complex or numeric vector of values different from 0.


numeric, must currently be non-negative.


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


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


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


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.


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


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


see BesselI.

See Also

BesselI etc; the Airy function Airy.


##------------------ 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 ..

[Package Bessel version 0.6-0 Index]