The Franke filtration of the space of automorphic forms on the adelic points of a reductive group over a number field, with a fixed cuspidal support, is a finite descending filtration whose successive quotients are defined in terms of main values of derivatives of Eisenstein series. The key observation is that, whenever an Eisenstein series has a pole at certain point of evaluation, the residues should contribute to a deeper quotient of the filtration than the Eisenstein series in question. The new method presented in this talk exploits this observation and the explicit descriptions of the Franke filtration to determine the regions of holomorphy of degenerate Eisenstein series. In particular, the application of the new method in the case of Eisenstein series on the general linear group is presented. The first part of the talk, in which the fundamental notions of automorphic forms, Eisenstein series and the Franke filtration will be explained in some detail, also serves as the introduction to the entire workshop. This work is supported by the Croatian Science Foundation under the project HRZZ-IP-2022-4615.