We present estimates of the GALEX NUV and FUV luminosity functions (LFs) of the Coma cluster, over a total area of ~9 deg^2 (~25 Mpc^2), i.e. from the cluster center to the virial radius. Our analysis represents the widest and deepest UV investigation of a nearby cluster of galaxies made to date. The Coma UV LFs show a faint-end slope steeper than the one observed in the local field. This difference, more evident in NUV, is entirely due to the contribution of massive quiescent systems (e.g. ellipticals, lenticulars and passive spirals), more frequent in high density environments. On the contrary, the shape of the UV LFs for Coma star-forming galaxies does not appear to be significantly different from that of the field, consistently with previous studies of local and high redshift clusters. We demonstrate that such similarity is only a selection effect, not providing any information on the role of the environment on the star formation history of cluster galaxies. By integrating the UV LFs for star-forming galaxies (corrected for the first time for internal dust attenuation), we show that the specific star formation rate of Coma is significantly lower than the integrated SSFR of the field and that Coma-like clusters contribute only <7% of the total SFR density of the local universe. Approximately 2/3 of the whole star-formation in Coma is occurring in galaxies with M_star < 10^10 M_sol. The vast majority of star-forming galaxies has likely just started its first dive into the cluster core and has not yet been affected by the cluster environment. The total stellar mass accretion rate of Coma is ~(0.6-1.8) x 10^12 M_sol Gyr^-1, suggesting that a significant fraction of the population of lenticular and passive spirals observed today in Coma could originate from infalling galaxies accreted between z~1 and z~0.
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