Hankel Function of the First Kind
The Hankel functions of the first kind are defined as
| H_n^((1))(z)=J_n(z)+iY_n(z), |
(1)
|
where J_n(z) is a Bessel function of the first kind and Y_n(z) 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
![画像: H_n^((1))(z)=1/(ipi)int_(0 [upper half plane])^infty(e^((z/2)(t-1/t)))/(t^(n+1))dt.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HankelFunctionoftheFirstKind/NumberedEquation2.svg){kind=link}
The derivative of H_n^((1))(z) is given by
HankelH1ReIm
HankelH1Contours
The plots above show the structure of H_0^((1))(z) 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 FormulaExplore with Wolfram|Alpha
WolframAlpha
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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.Referenced on Wolfram|Alpha
Hankel Function of the First KindCite this as:
Weisstein, Eric W. "Hankel Function of the First Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HankelFunctionoftheFirstKind.html