  Angular power spectra are central to the study of our Universe. In this
paper, I develop a new method for the numeric evaluation and analytic
estimation of the angular cross-power spectrum of two random fields using
complex analysis and Picard- Lefschetz theory. The proposed continuous
deformation of the integration domain resums the highly oscillatory integral
into a convex integral whose integrand decays exponentially. This deformed
integral can be quickly evaluated with conventional integration techniques.
These methods can be used to quickly evaluate and estimate the angular power
spectrum from the three-dimensional power spectrum for all angles (or multipole
moments). This method is especially useful for narrow redshift bins, or samples
with small redshift overlap, for which the Limber approximation has a large
error.
