Weinberg's book is nice, but it's nothing like a research monograph. He gives his preferred version of how to treat infrared divergences, but that was already textbook material by the time his book came out, and he leaves plenty of sub-leading stuff out. The main results aren't even Weinberg's, only the simplified formalism. The real legwork was started by Bloch and Nordsieck, [Phys. Rev. 52, 54 (all the way back in1937!)] and completed by Yennie, Frautschi, and Suura [Annals of Physics 13, 379 (1961)] before Weinberg even touched the problem.

There might be a simple formula if one of the Bessel functions is switched to a Neumann function instead.

The boundary would have run through Purgatory, which I recall had something of its own peculiar topology.