Resolution of singularities by rational functions

Results on the rational approximation of functions containing singularities are presented. We build further on the "lightning method," recently proposed by Trefethen and Gopal [SIAM J. Numer. Anal., 57 (2019), pp. 2074-2094], based on exponentially clustering poles close to the singularities. Our results are obtained by augmenting the lightning approximation set with either a low-degree polynomial basis or partial fractions with poles clustering toward infinity in order to obtain a robust approximation of the smooth behavior of the function. This leads to a significant increase in the achievable accuracy as well as the convergence rate of the numerical scheme. For the approximation of x\alpha on [0, 1], the optimal convergence rate as shown by Stahl [Bull. Amer. Math. Soc., 28 (1993), pp. 116--122] is now achieved simply by least-squares fitting.