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This paper aims at giving an as complete and detailed as possible derivation of the six electromagnetic field components created by an offset point charge travelling at any speed in an infinitely long circular multilayer beam pipe. Outcomes from this study are a novel and efficient matrix method for the field matching determination of all the constants involved in the field components, and the generalization to any azimuthal mode together with the final summation on all such modes in the impedance formulas. In particular the multimode direct space-charge impedances (both longitudinal and transverse) are given, as well as the wall impedance to any order of precision. New quadrupolar terms for the transverse wall impedance are found, which look negligible in the ultrarelativistic case but might be of significance for low-energy beams. In principle from this analysis the electromagnetic fields created by any particular source, with a finite transverse shape, can then be computed using convolutions.