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.](/images/equations/HankelFunctionoftheFirstKind/NumberedEquation2.svg) |
(2)
|
The derivative of 
is given by
 |
(3)
|
The plots above show the structure of 
in the complex plane.
See also
Bessel Function of the First Kind,
Bessel Function
of the Second Kind,
Debye's Asymptotic
Representation,
Hankel Function
of the Second Kind,
Spherical
Hankel Function of the First Kind,
Watson-Nicholson
Formula,
Weyrich's Formula
Explore with Wolfram|Alpha
References
Arfken, G. "Hankel Functions." §11.4 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 604-610,
1985.Morse, P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 623-624,
1953.Referenced on Wolfram|Alpha
Hankel Function of the
First Kind
Cite this as:
Weisstein, Eric W. "Hankel Function of the First Kind." From MathWorld--A Wolfram Web Resource.
https://mathworld.wolfram.com/HankelFunctionoftheFirstKind.html
Subject classifications
