  Although the WKB series converges only asymptotically and guarantees the
exact result solely in the eikonal regime, we have managed to derive concise
analytical expressions for the quasinormal modes and grey-body factors of black
holes, expanding beyond the eikonal approximation. Remarkably, these
expressions demonstrate unexpectedly strong accuracy. We suggest a
comprehensive approach for deriving analytical expressions for grey-body
factors and quasinormal modes at various orders beyond the eikonal
approximation. Two cases are examined as examples: the Schwarzschild-de Sitter
black hole and hairy black holes within the framework of Effective Field
Theory. We have publicly shared a generic code that calculates analytical
expressions for grey-body factors and quasinormal modes of spherical black
holes.
