Analytic continuation of the kite family

We consider results for the master integrals of the kite family, given in terms of ELi-functions which are power series in the nome \(q\) of an elliptic curve. The analytic continuation of these results beyond the Euclidean region is reduced to the analytic continuation of the two period integrals which define \(q.\) We discuss the solution to the latter problem from the perspective of the Picard-Lefschetz formula.