# Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations

## TL;DR

This paper reviews the higher order WKB method for calculating black hole quasinormal modes and grey-body factors, providing practical recipes, error estimates, and an automatic code for quick and accurate computations.

## Contribution

It introduces improved recipes and an automatic code for higher order WKB calculations, enhancing accuracy and reliability in quasinormal mode and grey-body factor estimations.

## Key findings

- Averaging Pade approximations improves WKB accuracy.
- Practical recipes for error estimation are proposed.
- Higher order WKB can be reliably used with these methods.

## Abstract

The WKB approach for finding quasinormal modes of black holes, suggested in [1] by Schutz and Will at the first order and later developed to higher orders [2-4], became popular during the past decades, because, unlike more sophisticated numerical approaches, it is automatic for different effective potentials and mostly provides sufficient accuracy. At the same time, the seeming simplicity of the WKB approach resulted in appearance of a big number of partially misleading papers, where the WKB formula was used beyond its scope of applicability. Here we review various situations in which the WKB formula can or cannot bring us to reliable conclusions. As the WKB series converges only asymptotically, there is no mathematically strict criterium for evaluation of an error. Therefore, here we are trying to introduce a number of practical recipes instead and summarize cases in which higher WKB orders improve accuracy. We show that averaging of the Pade approximations, suggested first by J. Matyjasek and M. Opala [4], leads to much higher accuracy of the WKB approach, estimate the error and present the automatic code [5] which computes quasinormal modes and grey-body factors.