  We present a current catalog of 21 cm HI line sources extracted from the
Arecibo Legacy Fast Arecibo L-band Feed Array (ALFALFA) survey over ~2800
square degrees of sky: the alpha.40 catalog. Covering 40% of the final survey
area, the alpha.40 catalog contains 15855 sources in the regions 07h30m < R.A.
< 16h30m, +04 deg < Dec. < +16 deg and +24 deg < Dec. < +28 deg and 22h < R.A.
< 03h, +14 deg < Dec. < +16 deg and +24 deg < Dec. < +32 deg. Of those, 15041
are certainly extragalactic, yielding a source density of 5.3 galaxies per
square degree, a factor of 29 improvement over the catalog extracted from the
HI Parkes All Sky Survey. In addition to the source centroid positions, HI line
flux densities, recessional velocities and line widths, the catalog includes
the coordinates of the most probable optical counterpart of each HI line
detection, and a separate compilation provides a crossmatch to identifications
given in the photometric and spectroscopic catalogs associated with the Sloan
Digital Sky Survey Data Release 7. Fewer than 2% of the extragalactic HI line
sources cannot be identified with a feasible optical counterpart; some of those
may be rare OH megamasers at 0.16 < z < 0.25. A detailed analysis is presented
of the completeness, width dependent sensitivity function and bias inherent in
the current alpha.40 catalog. The impact of survey selection, distance errors,
current volume coverage and local large scale structure on the derivation of
the HI mass function is assessed. While alpha.40 does not yet provide a
completely representative sampling of cosmological volume, derivations of the
HI mass function using future data releases from ALFALFA will further improve
both statistical and systematic uncertainties.
