Hankel Function of the First Kind
The Hankel functions of the first kind are defined as
 |
(1)
|
where 
is a Bessel
function of the first kind and 
is a Bessel
function of the second kind. Hankel functions of the first kind is implemented
in the Wolfram Language as HankelH1[n,
z].
Hankel functions of the first kind can be represented as a contour integral over the upper half-plane using
![H_n^((1))(z)=1/(ipi)int_(0 [upper half plane])^infty(e^((z/2)(t-1/t)))/(t^(n+1))dt.](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/HankelFunctionoftheFirstKind/NumberedEquation2.gif) |
(2)
|
The derivative of 
is
given by
 |
(3)
|
The plots above show the structure of 
in
the complex plane.
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Bessel functions
