We recently put out a paper to compute the high-order, large-frequency expansions for the eigenvalue and the eigenfunction of the spin-weighted spheroidal equation. The spin-weighted spheroidal equation turns up in a number of places in physics, and I most often encounter it using the Teukolsky formalism to model gravitational wave emission. A great strength of our paper is the combination of analytic and numerical results. The topic of high-frequency expansion of spin-weighted harmonics has been addressed before. Evaluating the results of that work relied on the results of an earlier work which had an error in it. Careful comparison with numerical calculations brought this to light and our recent paper corrects and extends the literature.
Our work provides formula to compute a high-frequency expansion of the eigenvalue of the spin-weighted spheroidal equation. Code to compute this series expansion has also been made publicly available in the SpinWeightedSpheroidalHarmonics Mathematica package of the Black Hole Perturbation Toolkit. Details are provided in the paper. We also provide an example notebook which shows how to use the new feature and gives code to compute the coefficients of the expansion of the harmonic.