Evaluates the Bessel function of the third kind, the Hankel function for any real order.
BESSELH(v, K, z, opt)
v 
 
A real or real series, the order. The order must be real but need not be an integer.  
K 
 
Optional. An integer, either 1 or 2. Specifies the type of the Hankel function. Defaults to 1.  
z 
 
Any scalar or series. The input value.  
opt 
 
Optional. An integer, the scaling method:

A scalar or series, The value of H_{ν}^{(K)}(z) where ν is the order, z is the input and K is either 1 or 2, the type of Hankel transform.
besselh(0, 3)
returns 0.260052 + 0.376850i, the value of H_{0}^{(1)}(3).
W1: 0..0.2..1;
W2: besselh(1, 1, w1)
Returns H_{1}^{(1)}(z) for z between 0 and 1. W2 contains the series:
{NA,
0.099501  3.323825i,
0.196027  1.780872i,
0.286701  1.260391i,
0.368842  0.978144i,
0.440051  0.781213i}
besselh(3..9, 0..0.2..10)
Evaluates the Bessel function of the third kind of type 1 for orders 3 through 9 with inputs from 0 to 10. Each column of the result contains the output for the specified order.
Bessel functions are solutions to the differential equation:
where ν is the order, J_{ν}(z), is a solution of the first kind and Y_{ν}(z) is a linearly independent solution of the second kind.
The Hankel functions are related to the Bessel functions by:
Hankel functions are also known as Bessel functions of the third kind.
BESSELH is based on a FORTRAN library written by D. E. Amos.
[1] Abramowitz and Stegun
Handbook of Mathematical Functions (9th printing 1970)
US Gov. Printing Office
Section 9.1.1, 9.1.89, 9.12
[2] Amos, D.E.
A Subroutine Package for Bessel Functions of a Complex
Argument and Nonnegative Order
Sandia National Laboratory Report
SAND851018, May, 1985.
[3] Amos, D.E.
A Portable Package for Bessel Functions of a Complex
Argument and Nonnegative Order
Trans. Math. Software, 1986