  Recently, a new method for calculating the Cherenkov field acting on a
point-like electron bunch passing through longitudinally homogeneous structures
lined with arbitrary slowdown layers was proposed, where the formalism was
obtained though consideration of a general integral relation that allows
calculation of the fields at the vicinity of a point-like bunch. It
demonstrates that the Cherenkov field at the point of the short relativistic
bunch does not depend on the waveguide system material and is a constant for
any given transverse dimensions and cross-section shapes of waveguides. With
this paper we present a strict derivation of the fields formulas valid at the
cross-section of a bunch on the basis of a conformal mapping method. We
generalize the results of the previous paper to the case of transversely
distributed bunch by deriving a two-dimensional Green function at the
cross-section of a bunch. Comparison of the results proving validity of the
method is given in the appendix.
