Establish the asymptotic structure of complex-frequency spheroidal-harmonic sectors

Determine the complete large-complex-frequency asymptotic sector structure of the Kerr spin-weighted spheroidal harmonics and derive the corresponding asymptotic behavior needed for contour deformation, including whether asymptotic sectors beyond the oblate- and prolate-type sectors exist and the next-to-leading oblate-sector expansion for general complex \(c=a\omega\).

Background

The appendix separates the fixed-mode angular spectrum into oblate-type branches with ${}_sA_{\ell m}\sim-c^2$ and prolate-type branches with ${}_sA_{\ell m}\sim\pm ic(2\bar L+1)$. These sectors lead to different angular localization and exponential behavior on large complex-frequency contours.

A complete description is required because different portions of a large semicircle can sample different sectors, preventing a single branch-independent exponential phase from being used to establish uniform contour decay. The paper states that the present analysis is restricted to the two sectors identified numerically and that even the next-to-leading oblate coefficient is unavailable in the required general-complex form.