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 are implemented in the Wolfram
Language as HankelH1[n,
z].
For 
,
Hankel functions of the first kind have the contour
integral representation
 |
(2)
|
where the path runs from 0 along the positive imaginary axis, using the principal branch of 
when 
is not an integer. The function extends to other values
of 
by analytic continuation (NIST DLMF, eqn.
10.9.10).
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
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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.National Institute of Standards and Technology. "Integral
Representations." §10.9 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.9.E10.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 Resource. https://mathworld.wolfram.com/HankelFunctionoftheFirstKind.html
Subject classifications
