Spheroidal wave functions: which angular normalization?
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The issue proposes prolate and oblate spheroidal wave functions and asks maintainers to choose an angular normalization. Read the linked discussion in #303 and compare the stated Meixner–Schäfke and Flammer conventions, including the SciPy conversion and reference constraints. Done means a convention is agreed on; implementation and high-precision reference tests are described as follow-up work.
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描述
Following @dschmitz89's request in #303: we're going to add prolate and oblate spheroidal wave functions (characteristic value, angular function of the first kind, radial functions of the first and second kind, and derivatives), ported from Van Buren's MIT-licensed codes and tested against high-precision Arb/MPFR references.
The radial functions are normalized the same way in every convention. The angular function $S_{mn}(c,\eta)$ is not, and we need to pick one:
Meixner–Schäfke (DLMF 30.4, Mathematica's SpheroidalPS, Van Buren's default)
- Includes the Condon–Shortley phase, consistent with
boost::math::legendre_p. - $|S_{mn}| = |P_n^m|$ for every $c$, so the function stays bounded as $c$ grows.
Flammer (scipy.special.pro_ang1/obl_ang1)
- No Condon–Shortley phase; $S_{mn}(c,0) = P_n^m(0)$ for even $n-m$, $S_{mn}'(c,0) = P_n^{m\prime}(0)$ for odd.
- Drop-in for SciPy.
- For oblate functions the norm grows quickly with $c$: $|S_{00}|/|P_0^0|$ is about 1.2e3 at $c=10$, 3.4e11 at $c=30$ and 2.6e24 at $c=60$.
I lean towards Meixner–Schäfke. SciPy can convert with $S^{\text{Flammer}} = \frac{P_n^m(0)}{\mathrm{Ps}(0)},\mathrm{Ps}$ (and use derivatives at 0 for odd $n-m$), with the phase factor $(-1)^m$. The other option is to ship both. Opinions welcome, especially from the SciPy side.
Note that SciPy's current values can't serve as test references: the reference pro_rad2 satisfies the Wronskian $R^{(1)}R^{(2)\prime} - R^{(1)\prime}R^{(2)} = 1/(c(\xi^2-1))$ to 1e-47, while SciPy's differs from it by 1.1% at $(m,n,c,\xi) = (1,2,2,1.5)$ and by a factor of about 7 at $(3,7,10,1.2)$.
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