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Choosing the phase for the spin-weighted spheroidal functions

Gregory B. Cook* and Xiyue Wang†

  • *Contact author: cookgb@wfu.edu
  • †Contact author: xiyuew@caltech.edu
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Phys. Rev. D 114, 064052 – Published 10 September, 2026

DOI: https://doi.org/10.1103/ffpl-dyld

Abstract

The spin-weighted spheroidal functions are the eigenfunctions of the angular Teukolsky equation. They are a generalization of the widely used spin-weighted spherical functions, and are extremely important in the area of black-hole perturbation theory. Like other special functions, they have an inherent phase ambiguity and need to be phase fixed to be uniquely defined. Clearly, such a phase choice does not have a direct physical impact. But, without specifying phase choices, meaningful comparison between results reported in different works become more difficult. To date, possible phase choices for the spin-weighted spheroidal functions have received little attention. Here, we clearly define and extensively explore two useful phase fixing schemes, and we propose that the spherical-limit phase-fixing scheme be adopted as the default phase-fixing scheme for the spin-weighted spheroidal functions.

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Phys. Rev. D 114, 064052– Published 10 September, 2026
  • Received 26 March 2026
  • Accepted 6 August 2026
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