Hankel Function of the Second Kind
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 [n,
z].
Hankel functions of the second kind can be represented as a contour integral using
![👁 H_n^((2))(z)=1/(ipi)int_(-infty [lower half plane])^0(e^((z/2)(t-1/t)))/(t^(n+1))dt.](https://mathworld.wolfram.com/images/equations/HankelFunctionoftheSecondKind/NumberedEquation2.svg){kind=link}
The derivative of 👁 H_n^((2))(z)
is given by
![👁 d/(dz)H_n^((2))(z)=1/2[H_(n-1)^((2))(z)-H_(n+1)^((2))(z)].](https://mathworld.wolfram.com/images/equations/HankelFunctionoftheSecondKind/NumberedEquation3.svg){kind=link}
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
More things to try:
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