Resumming Kerr Quasinormal-Mode Frequencies: Accuracy and Breakdown Near Extremality

Kerr black-hole quasinormal modes are usually computed with numerical methods, but analytic approximations remain useful for identifying the physics that controls different parts of the spectrum. In this paper, we ask whether the divergent, high-order Wentzel-Kramers-Brillouin (WKB) expansion about the peak of the Chandrasekhar-Detweiler potential can be made predictive through Pad\'e and Borel-Pad\'e resummation. We develop two complementary implementations: a semi-analytic slow-rotation expansion in the dimensionless spin $a$ (carried out through 21th WKB order), and a fixed-spin Pad\'e-WKB implementation for the resummed frequency equation (carried out through 41st WKB order). In the slow-rotation regime, the 21th-order resummed expansion is significantly more accurate than the fourth-order approximation found previously. For damped modes at larger spins, the fixed-point iteration agrees well with Leaver's method, reaching fractional errors below $10^{-7}$ in the real part of the fundamental $m=0$ mode at $a=0.99$. The same strategy fails for modes that approach the zero-damped branch near extremality. We trace this breakdown to the near-horizon structure of the Chandrasekhar-Detweiler potential. As the extremal limit is approached, nearby poles produce rapid variation on the throat scale, so a local Taylor expansion about the potential peak no longer uniformly captures the relevant region.