  We consider the local analytic behavior for a family of holomorphic
differentials on a family of degenerating annuli. Three results and discussion
are presented. The first is the normal families Lemma 1. The second is an
isomorphism of sheaves, formula (3), giving a direct description of families of
regular $k$-differentials (sections of powers of the relative dualizing sheaf)
in terms of $k$-canonical forms on the total space of the family. The third is
a general holomorphic extension property, Lemma 3, for families given on smooth
Riemann surfaces/curves to extend to the limiting nodal Riemann
surfaces/curves. Divisors of families of holomorphic differentials are also
discussed.
