The electronic structure and x-ray magnetic circular dichroism (XMCD) spectra of the ferromagnetic superconductor UCoGe at the U N4,5, Ge and Co K and Co L2,3 edges were investigated theoretically from first principles, using the fully relativistic Dirac linear muffin-tin orbital band structure method. The electronic structure is obtained with the local spin-density approximation (LSDA), as well as with a generalization of the LSDA+U method which takes into account the non-diagonal occupation matrix (in spin indexes) of localized electrons. A stable ferromagnetic ground state was found. The uranium total magnetic moment is quite small (about -0,171 <$E mu sub B>) in the LSDA approximation as a result of almost complete cancellation between the spin magnetic moment of 0,657 <$E mu sub B> and the opposite orbital magnetic moment of -0,828 <$E mu sub B>, resulting from strong spin-orbit coupling at the uranium site. Valency of U ion in UCoGe is close to 3+. The ratio orbital and spin magnetic moments Ml/Ms ranged from 1,163 in the GGA approach up to 2,456 for the LSDA+U calculations is smaller than the corresponding ratio for the free ion U<^>3+ value (2,60), it can indicate a significant delocalization of the 5f-electron states due to the hybridization of the U 5f electrons with the conduction band and Co 3d electrons. The line shape of the dichroic spectra at the U M5 and M4 edges predicted by considering the magneto-optical selection rules as well as the occupation and the energy sequence of the mj-projected partial densities of states. The theoretically calculated XMCD spectra at the U M4,5, Ge and Co K and Co L2,3 edges are in good agreement with the experimentally measured spectra.

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