The Approximation of Cauchy-Stieltjes and Laplace-Stieltjes Functions

The treatment of fractional diffusion operators leads to the rational approximation of Cauchy–Stieltjes functions. Similarly the computation of some high-dimensional integrals in quantum chemistry leads to the approximation of Lebesgue–Stieltjes functions by exponential sums. In both cases the exponential rate of approximation is simply computed via Gauss’ arithmetic-geometric iteration and an approximation procedure of the square root function. Surprisingly the convergence rates are different in the two cases although the derivations of the two results are very similar.