The short-distance expansion of the tau function of the radial
sine-Gordon/Painlev\'e III equation is given by a convergent series which
involves irregular c=1 conformal blocks and possesses certain periodicity
properties with respect to monodromy data. The long-distance irregular
expansion exhibits a similar periodicity with respect to a different pair of
coordinates on the monodromy manifold. This observation is used to conjecture
an exact expression for the connection constant providing relative
normalization of the two series. Up to an elementary prefactor, it is given by
the generating function of the canonical transformation between the two sets of
coordinates.Comment: 18 pages, 1 figur
