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GeezerVison is showing mathworld.wolfram.com — Spherical Hankel Function of the First Kind -- from Wolfram MathWorld

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Spherical Hankel Function of the First Kind

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

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

where H_n^((1))(z) is the Hankel function of the first kind and j_n(z) and n_n(z) are the spherical Bessel functions of the first and second kinds.

It is implemented in the Wolfram Language as SphericalHankelH1[n, z].

Explicitly, the first few are

h_0^((1))(z)=-ie^(iz)1/z
(3)
h_1^((1))(z)=-e^(iz)(z+i)/(z^2)
(4)
h_2^((1))(z)=ie^(iz)(z^2+3iz-3)/(z^3)
(5)
h_3^((1))(z)=e^(iz)(z^3+6iz^2-15z-15i)/(z^4).
(6)

The derivative is given by

d/(dz)h_n^((1))(z)=1/2[h_(n-1)^((1))(z)-(h_n^((1))(z)+zh_(n+1)^((1))(z))/z].
(7)
SphericalHankelH1

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

SphericalHankelH1ReIm
SphericalHankelH1Contours

The plots above shows the real and imaginary parts of h_0^((1))(z) in the complex plane.

See also

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

Cite this as:

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

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