A rigorous treatment of image formation, taking into account inelastic scattering and partially coherent illumination, is given for the CTEM. The results also hold for a STEM if the momentum vectors of the incident and outgoing waves are interchanged and the flight direction of the electrons is reversed. A generalized optical theorem is found which connects the anti-Friedel part of the elastic scattering amplitude with the squares of the elastic and inelastic scattering amplitudes. This theorem has important consequences for all approximation methods. For example, it shows that inelastic scattering influences the elastic scattering amplitude. It is further shown that the observed wings in optical diffractograms, obtained from micrographs of amorphous material taken using tilted beam illumination, can only be explained if double scattering (elastic and inelastic) is assumed. Hollow-cone illumination strongly suppresses the structural noise in the phase contrast image. The corresponding phase contrast transfer function (PCTF) is calculated for two different cone angles, assuming that no beam-limiting aperture is present. In addition, the PCTFs of a corrected microscope are calculated for two different illumination modes, one yielding negative and the other positive phase contrast. Finally, the contrast transfer has been investigated for a stem operating with a split detector.