Spherical Hankel Function of the First Kind
The spherical Hankel function of the first kind h_n^((1))(z) is defined by
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
The derivative is given by
![画像: d/(dz)h_n^((1))(z)=1/2[h_(n-1)^((1))(z)-(h_n^((1))(z)+zh_(n+1)^((1))(z))/z].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/SphericalHankelFunctionoftheFirstKind/NumberedEquation1.svg){kind=link}
The plot above shows the real and imaginary parts of h_n^((1))(z) on the real axis for n=0, 1, ..., 5.
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 KindExplore with Wolfram|Alpha
More things to try:
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 KindCite this as:
Weisstein, Eric W. "Spherical Hankel Function of the First Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SphericalHankelFunctionoftheFirstKind.html