The need for analytic continuation arises frequently in many applications, such as the extrapolation of complex electromagnetic permittivity from a given band of frequencies or the determination of geometric features of microstructure of a composite based on measurements of its effective properties. In a joint work with Yury Grabovsky we consider a large class of such problems where analytic continuation exhibits a power law precision deterioration as one moves away from the source of data. We introduce a 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, an ellipse or an upper half-plane the solution of the integral equation and the corresponding exponent can be found explicitly. In more general geometries numerical solution of the integral equation supports the power law precision decay.