Hankel Function of the Second Kind
| H_n^((2))(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 second kind is implemented in the Wolfram Language as HankelH2 [n, z].
Hankel functions of the second kind can be represented as a contour integral using
![画像: H_n^((2))(z)=1/(ipi)int_(-infty [lower half plane])^0(e^((z/2)(t-1/t)))/(t^(n+1))dt.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HankelFunctionoftheSecondKind/NumberedEquation2.svg){kind=link}
The derivative of H_n^((2))(z) is given by
![画像: d/(dz)H_n^((2))(z)=1/2[H_(n-1)^((2))(z)-H_(n+1)^((2))(z)].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HankelFunctionoftheSecondKind/NumberedEquation3.svg){kind=link}
HankelH2ReIm
HankelH2Contours
The plots above show the structure of H_0^((2))(z) in the complex plane.
See also
Bessel Function of the First Kind, Bessel Function of the Second Kind, Hankel Function of the First Kind, Spherical Hankel Function of the First Kind, Watson-Nicholson 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 Second KindCite this as:
Weisstein, Eric W. "Hankel Function of the Second Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HankelFunctionoftheSecondKind.html