  The limiting procedure of special Kahler manifolds to their rigid limit is
studied for moduli spaces of Calabi-Yau manifolds in the neighbourhood of
certain singularities. In two examples we consider all the periods in and
around the rigid limit, identifying the nontrivial ones in the limit as periods
of a meromorphic form on the relevant Riemann surfaces. We show how the Kahler
potential of the special Kahler manifold reduces to that of a rigid special
Kahler manifold. We extensively make use of the structure of these Calabi-Yau
manifolds as K3 fibrations, which is useful to obtain the periods even before
the K3 degenerates to an ALE manifold in the limit. We study various methods to
calculate the periods and their properties. The development of these methods is
an important step to obtain exact results from supergravity on Calabi-Yau
manifolds.
