Hankel Function of the First Kind
المؤلف:
Arfken, G.
المصدر:
"Hankel Functions." §11.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press
الجزء والصفحة:
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24-3-2019
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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.](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.
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.
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