Conformal Pseudo-Asymptotics and Special Functions

Abstract

We develop a general simple methodology for very fast and accurate evaluation of special functions of several classes. We use a family of fractional-parabolic deformations of the contours of integration to appropriate Riemann surfaces, make the corresponding conformal changes of variables and use the simplified trapezoid rule. Simple approximate recommendations for the choice of parameters of the scheme lead to programs that are competitive with Mathematica and Python. In a number of situations, for the error tolerance E-15, the programs are dozens, hundreds and even thousands times faster than functions implemented in MATLAB.