The regular Coulomb wave function method for investigation of the circular waveg- uides, comprising a co-axially positioned solid ferrite cylinder of azimuthal magnetization that support normal TE 0n modes is regarded and put into practice to the simplest case in which the anisotropic medium fills them completely. Its main points are: i) the pertinent wave equation is a form of the Coulomb wave equation; ii ) the field components are expressed by the real Coulomb wave functions FL (η, ρ) where L = −0.5 or L = 0.5, η and ρ — real, ρ > 0, −∞ < η < +∞; iii ) the solution of propagation problem needs a detailed numerical study of the functions and of their zeros in ρ. The Georgiev and Georgieva’s pioneering idea is adopted and advanced to extend with new (in the case considered — with the electromagnetism and in particular with the theory of waveguides) the traditionally established field of application of the Coulomb wave functions FL (η, ρ) and GL (η, ρ) — the atomic physics and quantum mechanics. It stands on the generalized for all complex ρ, η and L Thompson and Barnett’s definition of the aforesaid func- tions, using their relations with the confluent hypergeometric ones. It is proved numerically that the Abramowitz and Stegun’s series, expressing the regular function, though initially constructed for real ρ and η (ρ > 0, −∞ < η < ∞) and a non-negative integer L (L = 0, 1, 2, . . .): i) is applicable also for certain fractional positive and negative real L (e.g., L = ±0.5), on condition that the set of allowable values of the other two parameters is unchanged; ii ) is preferable in the computations, compared to the one, harnessing the Thompson and Barnett’s representation of FL (η, ρ) in terms of the complex Kummer function. Its advantage consists in: i) it is more rapidly convergent; ii ) real parameters are used only; iii ) the independent variable in it is twice smaller. In view of this, employing the expansion of functions around zero, allows practically to get outcomes for much larger values of the latter than the Thompson and Barnett’s formula. The mathematical approach described is richly illustrated by graphs.