  We study the symmetric square of Picard-Fuchs operators of genus one curves
and the thereby induced generalized Clausen identities. This allows the
computation of analytic expressions for the periods of all one-parameter K3
manifolds in terms of elliptic integrals. The resulting expressions are
globally valid throughout the moduli space and allow the explicit inversion of
the mirror map and the exact computation of distances, useful for checks of the
Swampland Distance Conjecture. We comment on the generalization to
multi-parameter models and provide a two-parameter example.
