The Extremely Luminous Quasar Survey in the Sloan Digital Sky Survey
  footprint. III. The South Galactic Cap Sample and the Quasar Luminosity
  Function at Cosmic Noon
  We have designed the Extremely Luminous Quasar Survey (ELQS) to provide a
highly complete census of unobscured UV-bright quasars during the cosmic noon,
$z=2.8-5.0$. Here we report the discovery of 70 new quasars in the ELQS South
Galactic Cap (ELQS-S) quasar sample, doubling the number of known extremely
luminous quasars in $4,237.3\,\rm{deg}^2$ of the SDSS footprint. These
observations conclude the ELQS and we present the properties of the full ELQS
quasar catalog, containing 407 quasars over $11,838.5\,\rm{deg}^2$. Our novel
ELQS quasar selection strategy resulted in unprecedented completeness at the
bright end and allowed us to discover 109 new quasars in total. This marks an
increase of $\sim36\%$ (109/298) to the known population at these redshifts and
magnitudes, while we further are able to retain a selection efficiency of
$\sim80\%$. On the basis of 166 quasars from the full ELQS quasar catalog, who
adhere to the uniform criteria of the 2MASS point source catalog, we measure
the bright-end quasar luminosity function (QLF) and extend it one magnitude
brighter than previous studies. Assuming a single power law with exponential
density evolution for the functional form of the QLF, we retrieve the best fit
parameters from a maximum likelihood analysis. We find a steep bright-end slope
of $\beta\approx-4.1$ and we can constrain the bright-end slope to
$\beta\leq-3.4$ with $99\%$ confidence. The density is well modeled by the
exponential redshift evolution, resulting in a moderate decrease with redshift
($\gamma\approx-0.4$).
