3.2.1.9. SphHankelH_Asym.m¶
Calculate the spherical Hankel function \(\mathrm{h}_n(z)\) using the asymptotic expansion.
3.2.1.9.1. Syntax¶
h = SphHankelH_Asym(n, z)
3.2.1.9.2. Output arguments¶
h — \(\mathrm{h}_n(z)\).
3.2.1.9.3. Input arguments¶
3.2.1.9.3.1. Mandatory arguments¶
n — order
z — the argument \(z\)
3.2.1.9.3.2. Optional arguments¶
approx_order
\(K\in \mathbb{N}\) — Calculate the spherical Hankel function using the asymptotic expansion.
0— The returned result reduces to Eq. (3.2.4), see Eq. (10.52.4) in Olver et al. [5].
(3.2.4)¶\[\mathrm{h}_n^{(1)}(z)
\sim
\frac{\mathrm{e}^{\mathrm{i}z}}{\mathrm{i}^{n+1}z}\]
3.2.1.9.4. Dependencies¶
The result is obtained by calculating the Hankel function using HankelH_Asym.m with the relation by, see Eq. (10.47.5) in Olver et al. [5].
(3.2.5)¶\[\mathrm{h}_n^{(1)}(z)
=
\sqrt{\frac{\pi}{2z}}
H_{n+1/2}^{(1)}(z)\]