= Spherical far-field range condition
{title2=$\sum_n(2n/(ekR))^{2n}\sum_m|a_n^m|^2<\infty$}
For an outgoing field outside a sphere of radius $R$, write $u_s=k\sum_{n,m}i^{n+1}a_n^m h_n^{(1)}(kr)Y_n^m$. Its <far-field pattern> is $\sum a_n^mY_n^m$. If its trace on that sphere lies in <L2 space>, Parseval's identity gives $k^2\sum|h_n^{(1)}(kR)|^2|a_n^m|^2<\infty$. The <fixed-argument growth of spherical Hankel functions> yields the displayed equivalent range condition, with the low-order weights defined finitely. Square summability of the far-field coefficients alone does not ensure continuation to the chosen sphere.
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