# Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral

## TL;DR

This paper develops a method for the analytic continuation and efficient numerical evaluation of elliptic Feynman integrals, specifically the kite and equal mass sunrise integrals, using elliptic polylogarithms and their periods.

## Contribution

It provides explicit formulas for the elliptic periods and demonstrates the convergence of the q-series, enabling fast numerical evaluation of these complex integrals.

## Key findings

- Explicit formulas for elliptic periods across all real t values
- Proved the q-series converges for all t except at singular points
- Achieved efficient numerical evaluation of the kite and sunrise integrals

## Abstract

We study the analytic continuation of Feynman integrals from the kite family, expressed in terms of elliptic generalisations of (multiple) polylogarithms. Expressed in this way, the Feynman integrals are functions of two periods of an elliptic curve. We show that all what is required is just the analytic continuation of these two periods. We present an explicit formula for the two periods for all values of $t \in {\mathbb R}$. Furthermore, the nome $q$ of the elliptic curve satisfies over the complete range in $t$ the inequality $|q|\le 1$, where $|q|=1$ is attained only at the singular points $t\in\{m^2,9m^2,\infty\}$. This ensures the convergence of the $q$-series expansion of the $\mathrm{ELi}$-functions and provides a fast and efficient evaluation of these Feynman integrals.