  [Abridged] We determined the luminosity function (LF) of the Coma cluster
galaxies selected by luminosity, and the LF bi-variate in central brightness.
The Coma cluster and control fields were imaged using the CFH12K (42x28 arcmin)
and UH8K (28x28 arcmin) wide-field cameras at the CFHT. Selected HST images
were used for testing. Quantities were derived from measurements in at least
two colors, which have the following features: (1) Galaxies as faint as three
times the luminosity of the brightest globular clusters are in the completeness
region of our data. (2) We have a complete census (in the explored region) of
low surface brightness galaxies with central surface brightness galaxies almost
as low as the faintest so far cataloged ones. (3) The explored area is among
the largest ever sampled with CCDs at comparable depth for any cluster of
galaxies. (4) The error budget includes all sources of errors known to date.
Using HST images we also discovered that blends of globular clusters, not
resolved in individual components due to seeing, look like dwarf galaxies when
observed from the ground and are numerous and bright. The derived Coma LF is
relatively steep (alpha=-1.4) over the 11 magnitudes sampled, but the slope and
shape depend on color. A large population of faint low surface brightness
galaxies was discovered, representing the largest contributor (in number) to
the LF at faint magnitudes. We found a clear progression for a faintening of
the LF from high surface brightness galaxies (mu~20 mag/arcsec2) to galaxies of
very faint central brightnesses (mu~24.5 mag/arcsec2), and some evidence for a
steepening. Compact galaxies, usually classified as stars and therefore not
included in the LF, are found to be a minor population in Coma.
