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Three-mass triangle integrals and single-valued polylogarithms

We study one and two-loop triangle integrals with massless propagators and all external legs off shell. We show that there is a kinematic region where the results can be expressed in terms of a basis of single-valued polylogarithms in one complex variable. The relevant space of single-valued functions can be determined a priori and the results take strikingly a simple and compact form when written in terms of this basis. We study the properties of the basis functions and illustrate how one can easily analytically continue our results to all kinematic regions where the external masses have the same sign.