High-precision computation of uniform asymptotic expansions for special functions

Resumo

In this dissertation, we investigate new methods to obtain uniform asymptotic expansions for the numerical evaluation of special functions to high-precision. We shall first present the theoretical and computational fundamental aspects required for the development and ultimately implementation of such methods. Applying some of these methods, we obtain efficient new convergent and uniform expansions for numerically evaluating the confluent hypergeometric functions and the Lerch transcendent at high-precision. In addition, we also investigate a new scheme of computation for the generalized exponential integral, obtaining on the fastest and most robust implementations in double-precision floating-point arithmetic. In this work, we aim to combine new developments in asymptotic analysis with fast and effective open-source implementations. These implementations are comparable and often faster than current open-source and commercial stateof-the-art software for the evaluation of special functions.