An all-analytic optical transfer function (OTF) tool for characterizing the performance of a large segmented telescope with/without adaptive optics (AO) correction of the telescope dynamical aberrations is presented. This tool is to be applied to the determination of the Thirty Meter Telescope (TMT) optical budget error, for both telescope aberrations and AO systems specifications. It takes into account the effect of the dynamical aberrations of all optical surfaces from all the hexagonal segments to the tertiary mirror, and includes as an option AO correction of these errors. Here we present the mathematical development of the method, and give an example of application to a 73 segments 10-m telescope, without AO correction.