This is a sequel to the paper published under the same title [Acta Phys. Pol. B 35, 2249 (2004)] in which the integral representation of the matrix element  \langle u | exp (-\sigma C₁)| u \rangle , where |u \rangle = exp (-iS(u)) | 0 \rangle is the quantum Coulomb field and C₁ = -(1/2) M_{\mu \nu} M^{\mu \nu} is the first Casimir operator of the proper, orthochronous Lorentz group, was given. In this paper another integral representation of the same matrix element is given. In this new representation contributions from the bound state, which belongs to the supplementary series, and from the continuous spectrum, which belongs to the main series, are separated. This allows to calculate the asymptotic behaviour of the matrix element for \sigma \to \infty . The matrix element \langle u| exp (-\sigma C₁)| u \rangle is a non-analytic function of \sigma at \sigma = 0. The nature of this non-analyticity is clarified by means of a representation of the relevant integrals with the help of the function g(x) = \sum _n=-\infty ^+\infty  exp (-\pi n²x) which satisfies the well known functional equation g(x) = x^-1/2g(1/x),  x>0.

PACS numbers: 12.20.Ds, 11.10.Jj