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This paper builds further rigorous analysis on the root-exponentialconvergence for lightning schemes approximating corner singularity problems. Byutilizing Poisson summation formula, Runge's approximation theorem and Cauchy'sintegral theorem, the optimal rate is obtained for efficient lightning pluspolynomial schemes, newly developed by Herremans, Huybrechs and Trefethen\cite{Herremans2023}, for approximation of $g(z)z^\alpha$ or $g(z)z^\alpha\logz$ in a sector-shaped domain with tapered exponentially clustering poles, where$g(z)$ is analytic on the sector domain. From these results, Conjecture 5.3 in\cite{Herremans2023} on the root-exponential convergence rate is confirmed andthe choice of the parameter$\sigma_{opt}=\frac{\sqrt{2(2-\beta)}\pi}{\sqrt{\alpha}}$ may achieve thefastest convergence rate among all $\sigma>0$. Furthermore, based on Lehman andWasow's study of corner singularities \cite{Lehman1954DevelopmentsIT, Wasow},together with the decomposition of Gopal and Trefethen \cite{Gopal2019},root-exponential rates for lightning plus polynomial schemes in corner domains$\Omega$ are validated, and the best choice of lightning clustering parameter$\sigma$ for $\Omega$ is also obtained explicitly. The thorough analysisprovides a solid foundation for lightning schemes.

The root-exponential convergence of lightning plus polynomial approximation on corner domains