Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds, Double Exponential Transform, and Open-Source Implementation libkww

TL;DR

This paper introduces libkww, a C library for accurately computing the Fourier transform of the stretched exponential function, featuring analytic error bounds and accelerated numerical integration using double exponential transformation.

Contribution

The paper presents a new C library, libkww, with high-precision computation of the Laplace-Fourier transform of the stretched exponential, including analytic error bounds and an efficient double exponential integration method.

Findings

Achieves sixteen-digit accuracy in transform computations.

Provides analytic error bounds for series expansions.

Uses double exponential transformation for fast intermediate frequency integration.

Abstract

The C library \texttt{libkww} provides functions to compute the Kohlrausch-Williams-Watts function, i.e.\ the Laplace-Fourier transform of the stretched (or compressed) exponential function $\exp(-t^\beta)$ for exponents $\beta$ between 0.1 and 1.9 with sixteen-digits accuracy. Analytic error bounds are derived for the low and high frequency series expansions. For intermediate frequencies the numeric integration is enormously accelerated by using the Ooura-Mori double exponential transformation. The source code is available from the project home page \url{http://apps.jcns.fz-juelich.de/doku/sc/kww}.