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 / V _ { \mathrm { max } } 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 \pm 0.04 , \alpha = -0.81 \pm 0.04 and \phi ^ { * } = ( 0.0166 \pm 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 \pm 0.067 ) \times 10 ^ { 8 } L _ { \odot } 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 .