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We apply a recently suggested new strategy to solve differential equationsfor master integrals for families of Feynman integrals. After a set of masterintegrals has been found using the integration-by-parts method, the crucialpoint of this strategy is to introduce a new basis where all master integralsare pure functions of uniform transcendentality. In this paper, we apply thismethod to all planar three-loop four-point massless on-shell master integrals.We explicitly find such a basis, and show that the differential equations areof the Knizhnik-Zamolodchikov type. We explain how to solve the latter to allorders in the dimensional regularization parameter epsilon, including allboundary constants, in a purely algebraic way. The solution is expressed interms of harmonic polylogarithms. We explicitly write out the Laurent expansionin epsilon for all master integrals up to weight six.

Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation