Numerical Evaluation Of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity

TL;DR

This paper accurately computes the real and imaginary parts of a challenging oscillatory integral involving a complex exponential and a logarithmic function, achieving high precision through specialized numerical methods.

Contribution

It introduces a combination of partial integration, Simpson's rule, and series acceleration techniques to evaluate a complex oscillatory integral with high precision.

Findings

Achieved 20-digit accuracy in integral evaluation

Demonstrated effectiveness of combining partial integration with series acceleration

Addressed challenges posed by the integrand's branch cut and logarithmic nature

Abstract

Real and imaginary part of the limit 2N->infinity of the integral int_{x=1..2N} exp(i*pi*x)*x^(1/x) dx are evaluated to 20 digits with brute force methods after multiple partial integration, or combining a standard Simpson integration over the first halve wave with series acceleration techniques for the alternating series co-phased to each of its points. The integrand is of the logarithmic kind; its branch cut limits the performance of integration techniques that rely on smooth higher order derivatives.