We present panoramic Spitzer/MIPS mid- and far-infrared and GALEX ultraviolet imaging of the the most massive and dynamically active system in the local Universe, the Shapley supercluster at z=0.048, covering the 5 clusters which make up the supercluster core. We combine these data with existing spectroscopic data from 814 confirmed supercluster members to produce the first study of a local rich cluster including both ultraviolet and infrared luminosity functions (LFs). This joint analysis allows us to produce a complete census of star-formation (both obscured and unobscured), extending down to SFRs~0.02-0.05Msun/yr, and quantify the level of obscuration of star formation among cluster galaxies, providing a local benchmark for comparison to ongoing and future studies of cluster galaxies at higher redshifts with Spitzer and Herschel. The GALEX NUV and FUV LFs obtained have steeper faint-end slopes than the local field population, due largely to the contribution of massive, quiescent galaxies at M_FUV>-16. The 24um and 70um galaxy LFs for the Shapley supercluster instead have shapes fully consistent with those obtained for the Coma cluster and for the local field galaxy population. This apparent lack of environmental dependence for the shape of the FIR luminosity function suggests that the bulk of the star-forming galaxies that make up the observed cluster infrared LF have been recently accreted from the field and have yet to have their star formation activity significantly affected by the cluster environment. We estimate a global SFR of 327 Msun/yr over the whole supercluster core, of which just ~20% is visible directly in the UV continuum and ~80% is reprocessed by dust and emitted in the infrared. The level of obscuration (L_IR/L_FUV) in star-forming galaxies is seen to increase linearly with L_K over two orders of magnitude in stellar mass.
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