  We solve crossing equations analytically in the deep Euclidean regime. Large
scaling dimension $\Delta$ tails of the weighted spectral density of primary
operators of given spin in one channel are matched to the Euclidean OPE data in
the other channel. Subleading $1\over \Delta$ tails are systematically captured
by including more operators in the Euclidean OPE in the dual channel. We use
dispersion relations for conformal partial waves in the complex $\Delta$ plane,
the Lorentzian inversion formula and complex tauberian theorems to derive this
result. We check our formulas in a few examples (for CFTs and scattering
amplitudes) and find perfect agreement. Moreover, in these examples we observe
that the large $\Delta$ expansion works very well already for small $\Delta
\sim 1$. We make predictions for the 3d Ising model. Our analysis of dispersion
relations via complex tauberian theorems is very general and could be useful in
many other contexts.
