  Starting from the Lorentzian inversion formula, we derive a dispersion
relation which computes a four-point function in 1d CFTs as an integral over
its double discontinuity. The crossing symmetric kernel of the integral is
given explicitly for the case of identical operators with integer or
half-integer scaling dimension. This derivation complements the one that uses
analytic functionals. We use the dispersion relation to evaluate holographic
correlators defined on the half-BPS Wilson line of planar $\mathcal{N}=4$ super
Yang-Mills, reproducing results up to fourth order in an expansion at large
t'Hooft coupling.
