Rademacher expansion of modular integrals

We develop a method to evaluate integrals of non-holomorphic modular functions over the fundamental domain of the torus with modular parameter τ analytically. It proceeds in two steps: first the integral is transformed to a Lorentzian contour by the same strategy that leads to the Lorentzian inversion formula in CFT, and then we apply a two-dimensional version of the Rademacher expansion. This computes the integral in terms of an expansion sensitive to the singular behaviour of the integrand near all the Lorentzian cuspsτ→i∞, ¯τ→x∈Q. We apply this technique to a variety of examples such as the evaluation of string one-loop partition functions, where it leads to the first analytic formula for the cosmological constants of the bosonic string and the SO(16) × SO(16) string.