In scattering theory, dispersion relations of the Kramers-Kronig form are often used to reconstruct the real part of an amplitude from its imaginary part, which is typically easier to measure and/or to compute.  Since this technique is based on very general principles such as causality, it is broadly applicable to other contexts. In conformal theories it turns out to imply that local operators are not independent of each other, but rather organize into analytic functions of spin. In this talk I will review a recent formula which quantifies this statement, and present some of its applications to the gauge-gravity (AdS/CFT) correspondence, where it provides a uniquely efficient tool for the computation of AdS Witten diagrams. Based on 1703.00278 and 1711.02031.