  The need for analytic continuation arises frequently in the context of
inverse problems. Notwithstanding the uniqueness theorems, such problems are
notoriously ill-posed without additional regularizing constraints. We consider
several analytic continuation problems with typical global boundedness
constraints that restore well-posedness. We show that all such problems exhibit
a power law precision deterioration as one moves away from the source of data.
In this paper we demonstrate the effectiveness of our general Hilbert
space-based approach for determining these exponents. The method identifies the
"worst case" function as a solution of a linear integral equation of Fredholm
type. In special geometries, such as the circular annulus or upper half-plane
this equation can be solved explicitly. The obtained solution in the annulus is
then used to determine the exact power law exponent for the analytic
continuation from an interval between the foci of an ellipse to an arbitrary
point inside the ellipse. Our formulas are consistent with results obtained in
prior work in those special cases when such exponents have been determined.
