Phase-contrast imaging is one of the most important emerging x-ray imaging techniques. In this work we analyse, from a theoretical point of view, the in-line phase-contrast image formation under general assumptions. The approach is based on wave-optical theory (Fresnel/Kirchoff diffraction integrals) and on the formalism of the mutual coherence function for the evolution of the coherence wavefield properties. Our theoretical model can be applied to phase-contrast imaging realized both by using highly coherent synchrotron radiation and micro-focus x-ray laboratory sources. Thus, the model is suitable for widespread applications, ranging from material science to medical imaging of human body parts. However, it cannot be applied to polychromatic sources, although the validity of the model does not require particularly demanding characteristics of monochromaticity. In addition, for moderate phase gradients, a useful analytical formula of the phase-contrast visibility is derived, based on the a priori knowledge of source size and distance, pixel detector size, defocus distance, material/tissue dielectric susceptibility and characteristic scales of transversal and longitudinal non-uniformities of the material/tissue dielectric susceptibility. Comparisons both with experimental results published by other authors and with simulations based on a Fourier optics approach have been reported, to confirm the validity of the proposed analytical formula