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We present the rest-frame V-band luminosity function (LF) of galaxies at0.4<z<4.0, measured from a near-infrared selected sample constructed from theNMBS, the FIRES, the FIREWORKS, and the ultra-deep NICMOS and WFC3 observationsin the HDFN, HUDF, and GOODS-CDFS, all having high-quality optical tomid-infrared data. This unique sample combines data from surveys with a largerange of depths and areas in a self-consistent way, allowing us to (1) minimizethe uncertainties due to cosmic variance; and (2) simultaneously constrain thebright and faint ends with unprecedented accuracy over the targeted redshiftrange, probing the LF down to 0.1 L* at z~3.9. We find that (1) the faint endis fairly flat and with a constant slope from z=4, with alpha= -1.27 +/- 0.05;(2) the characteristic magnitude has dimmed by 1.3 mag from z~3.7 to z=0.1; (3)the characteristic density has increased by a factor of ~8 from z~3.7 to z=0.1,with 50% of this increase from z~4 to z~1.8; and (4) the luminosity densitypeaks at z~1-1.5, increasing by a factor of ~4 from z=4.0 to z~1-1.5, andsubsequently decreasing by a factor of ~1.5 by z=0.1. We find no evidence for asteepening of the faint-end slope with redshift out to z=4, in contrast withprevious observational claims and theoretical predictions. The constantfaint-end slope suggests that the efficiency of stellar feedback may evolvewith redshift. Alternative interpretations are discussed, such as differentmasses of the halos hosting faint galaxies at low and high redshifts and/orenvironmental effects.

The Evolution of the Rest-frame V-band Luminosity Function from z=4: A Constant Faint-end Slope over the Last 12 Gyr of Cosmic History