  We present luminosity and surface brightness distributions of 40,111 galaxies
with K-band photometry from the United Kingdom Infrared Telescope (UKIRT)
Infrared Deep Sky Survey (UKIDSS) Large Area Survey (LAS), Data Release 3 and
optical photometry from Data Release 5 of the Sloan Digital Sky Survey (SDSS).
Various features and limitations of the new UKIDSS data are examined, such as a
problem affecting Petrosian magnitudes of extended sources. Selection limits in
K- and r-band magnitude, K-band surface brightness and K-band radius are
included explicitly in the 1/Vmax estimate of the space density and luminosity
function. The bivariate brightness distribution in K-band absolute magnitude
and surface brightness is presented and found to display a clear
luminosity--surface brightness correlation that flattens at high luminosity and
broadens at low luminosity, consistent with similar analyses at optical
wavelengths. Best fitting Schechter function parameters for the K-band
luminosity function are found to be M*-5 log h=-23.19 +/- 0.04, alpha=-0.81 +/-
0.04 and phi*=(0.0166 +/- 0.0008)h^3 Mpc^{-3}, although the Schechter function
provides a poor fit to the data at high and low luminosity, while the
luminosity density in the K band is found to be j = (6.305 +/- 0.067) x 10^8
L_sun h Mpc^{-3}. However, we caution that there are various known sources of
incompleteness and uncertainty in our results. Using mass-to-light ratios
determined from the optical colours we estimate the stellar mass function,
finding good agreement with previous results. Possible improvements are
discussed that could be implemented when extending this analysis to the full
LAS.
