Signed Shintani cones for number fields with one complex place

We give a signed fundamental domain for the action on C⁎×R+n−2 of the totally positive units E(k)+E(k)+ of a number field k of degree n and having exactly one pair of complex embeddings. This signed fundamental domain, built of k-rational simplicial cones, is as convenient as a true fundamental domain for the purpose of studying Dedekind zeta functions. However, while there is no general construction of a true fundamental domain, we construct a signed fundamental domain from any set of fundamental units of k.