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Spherical Hankel Function of the Second Kind

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The spherical Hankel function of the second kind h_n^((1))(z) is defined by

h_n^((2))(z)=sqrt(pi/(2x))H_(n+1/2)^((2))(z)
(1)
=j_n(z)-in_n(z),
(2)

where H_n^((2))(z) is the Hankel function of the second kind and j_n(z) and n_n(z) are the spherical Bessel functions of the first and second kinds.

It is implemented in Wolfram Language Version 6 as SphericalHankelH2[n, z].

Explicitly, the first few are given by

h_0^((2))(z)=(ie^(-iz))/z
(3)
h_1^((2))(z)=-e^(-iz)(z-i)/(z^2)
(4)
h_2^((2))(z)=-ie^(-iz)(z^2-3iz-3)/(z^3)
(5)
h_3^((2))(z)=e^(-iz)(z^3-6iz^2-15z+15i)/(z^4).
(6)

The derivative is given by

d/(dz)h_n^((2))(z)=1/2[h_(n-1)^((2))(z)-(h_n^((2))(z)+zh_(n+1)^((2))(z))/z].
(7)
SphericalHankelH2

The plot above shows the real and imaginary parts of h_n^((2))(z) on the real axis for n=0, 1, ..., 5.

SphericalHankelH2ReIm
SphericalHankelH2Contours

The plots above shows the real and imaginary parts of h_0^((2))(z) in the complex plane.

See also

Hankel Function of the Second Kind, Spherical Hankel Function of the First Kind

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Spherical Bessel Functions." §10.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 437-442, 1972.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 623, 1985.

Referenced on Wolfram|Alpha

Spherical Hankel Function of the Second Kind

Cite this as:

Weisstein, Eric W. "Spherical Hankel Function of the Second Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalHankelFunctionoftheSecondKind.html

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