  Building on introducing exponentially clustered poles, Trefethen and his
collaborators introduced lightning algorithms for approximating functions of
singularities. These schemes may achieve root-exponential convergence rates. In
particular, based on a specific choice of the parameter of the tapered
exponentially clustered poles, the lightning approximation with either a
low-degree polynomial basis may achieve the optimal convergence rate simply as
the best rational approximation for prototype $x^\alpha$ on $[0,1]$, which was
illustrated through delicate numerical experiments and conjectured in [SIAM J.
Numer. Anal., 61:2580-2600, 2023]. By utilizing Poisson's summation formula and
results akin to Paley-Wiener Theorem, we rigorously show that all these schemes
with a low-degree polynomial basis achieve root-exponential convergence rates
with exact orders in approximating $x^\alpha$ for arbitrary clustered
parameters theoretically, and provide the best choices of the parameter to
achieve the fastest convergence rate for each type of clustered poles, from
which the conjecture is confirmed as a special case. Ample numerical evidences
demonstrate the optimality and sharpness of the estimates.
