Mathematical basis and usage

Use integer order and degree 0≤m≤n. The parameter c and evaluation coordinates may be real or complex. Write σ=1 for prolate and σ=-1 for oblate geometry.

Precision and outputs

s = smn(1, 2, 1.25, [-0.3, 0.3]; precision=:quad)
r = rmn(1, 2, 1.25, 2.0; precision=:quad)
s.value
s.derivative

value and derivative are vectors; scalar coordinates give length-one vectors. The derivative is with respect to the coordinate.

Parallel evaluation

Calls are thread-safe. Start Julia with julia --threads=auto --project:

parameters = [0.5, 1.0, 1.5, 2.0]
tasks = [Threads.@spawn smn(1, 2, c, [-0.3, 0.3]; precision=:quad)
         for c in parameters]
results = fetch.(tasks)

Keep shared inputs unchanged and use separate output buffers for each task.

Angular functions

smn(m,n,c,eta) solves

\[\frac{d}{d\eta}\left[(1-\eta^2)S'\right] +\left(\lambda_{mn}(c)-\sigma c^2\eta^2-\frac{m^2}{1-\eta^2}\right)S=0, \qquad -1<\eta<1.\]

The separation constant satisfies λₘₙ(0)=n(n+1). The DLMF parameter is γ²=σc²; see DLMF 30.2.

First kind

kind=1 is the default. It uses Meixner–Schäfke normalization and includes the Condon–Shortley phase (-1)^m:

\[S_{mn}(0,\eta)=\mathsf P_n^m(\eta),\qquad S_{mn}(c,-\eta)=(-1)^{n-m}S_{mn}(c,\eta).\]

For real c and fixed m:

\[\int_{-1}^{1}S_{mn}(c,\eta)S_{mk}(c,\eta)\,d\eta =\frac{2(n+m)!}{(2n+1)(n-m)!}\,\delta_{nk}.\]

normalize=true multiplies by the reciprocal square root of this norm, giving unit norm. For complex c, normalization is continued analytically; it is not a unit integral of the squared magnitude. See DLMF 30.4.

smn(1, 1, 0.0, 0.3).value  # approximately [-0.953939201416946]
smn(1, 2, 1.25, [-0.3, 0.3]; normalize=true, precision=:quad)

Second kind

kind=2 selects the Ferrers-branch Qs:

q = smn(1, 2, 1.25, [-0.3, 0.3]; kind=2, precision=:quad)

Its parity, zero-parameter limit, and normalization are

\[Q(c,-\eta)=(-1)^{n-m+1}Q(c,\eta),\qquad Q(0,\eta)=\mathsf Q_n^m(\eta),\]

\[(1-\eta^2)(P Q'-P'Q) =\frac{(n+m)!}{(n-m)!}A_n^m A_n^{-m}.\]

Here P is the default first-kind function, and A denotes the joining sums in DLMF 30.11.4. The second-kind convention is specified by DLMF 30.5 and 30.8(ii).

Radial functions

rmn(m,n,c,x) solves

\[\frac{d}{dx}\left[(x^2-\sigma)R'\right] +\left(c^2x^2-\lambda_{mn}(c)-\frac{\sigma m^2}{x^2-\sigma}\right)R=0.\]

rmn(1, 2, 1.25, [1.5, 2.0]; kind=2, precision=:quad)
rmn(1, 2, 1.25, [0.0, 1.0]; spheroid=:oblate, kind=3, precision=:quad)

normalization=:static evaluates $c^{-n}R_1$ or $c^{n+1}R_2$, including their finite limits at c=0. It supports kinds 1 and 2 in rmn, jacobian_rmn, radial_wronskian, and radial accuracy.

rmn(0, 1, 0, 2; normalization=:static) # value ≈ [2/3]

Complex branches

Complex coordinates continue the real-coordinate normalization on these cuts:

Angular z=±1 and prolate z=1 retain their one-sided endpoint conventions. Oblate continuation passes through z=0. Coordinate order does not change the branch. Complex coordinates support the same derivative, scaling, and degree-range options. Use accuracy(...; diagnostics=true) for convergence and conditioning estimates.

smn(1, 2, 1.25, 0.3 + 0.2im; derivatives=2)
rmn(1, 2, 1.25, [1.5 + 0.2im, 2.0 - 0.1im]; kind=3)

For complex c, scalar eigenvalue, smn, and rmn follow degree n from real(c) along the vertical segment to c. Angular normalization and phase follow that branch.

Further calculations

Mathematical tools covers eigenvalue operators, parameter derivatives, inverse bandwidth, sweeps, degree ranges, scaled values, logarithmic derivatives, higher coordinate derivatives, and accuracy diagnostics.