Numerical evaluation of iterated integrals related to elliptic Feynman integrals

TL;DR

This paper presents a numerical implementation within GiNaC for evaluating elliptic Feynman integrals and related iterated integrals involving modular forms and Kronecker functions with arbitrary precision.

Contribution

It introduces a new computational approach for numerically evaluating complex elliptic integrals, including elliptic multiple polylogarithms, within the convergence region.

Findings

Implementation allows arbitrary precision evaluation

Includes integrals of modular forms and Kronecker functions

Enables numerical analysis of elliptic Feynman integrals

Abstract

We report on an implementation within GiNaC to evaluate iterated integrals related to elliptic Feynman integrals numerically to arbitrary precision within the region of convergence of the series expansion of the integrand. The implementation includes iterated integrals of modular forms as well as iterated integrals involving the Kronecker coefficient functions $g^{(k)}(z,\tau)$. For the Kronecker coefficient functions iterated integrals in $d\tau$ and $dz$ are implemented. This includes elliptic multiple polylogarithms.