The quasinormal-mode frequencies of a Schwarzschild black hole are calculated within an accurate phase-integral analysis. Two different phase-integral formulas are derived by means of uniform approximations using parabolic Weber functions and Coulomb wave functions, respectively. These formulas are valid when clusters of possibly close-lying transition points in the complex coordinate plane must be considered. By comparison with results of exact phase-amplitude calculations the phase-integral results are proved to be of high accuracy. Conclusively, the improved phase-integral method so far provides the most efficient way to determine approximate values for the characteristic frequencies of the lowest-lying, as well as highly damped, quasinormal modes of a Schwarzschild black hole.
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Andersson et al. (1992) studied this question.
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