  We present the first detailed photometric and spectroscopic study of the
white dwarfs (WDs) in the field of the ~225 Myr old (log tau_cl = 8.35) open
cluster NGC 1039 (M34) as part of the ongoing Lick-Arizona White Dwarf Survey.
Using wide-field UBV imaging, we photometrically select 44 WD candidates in
this field. We spectroscopically identify 19 of these objects as WDs; 17 are
hydrogen-atmosphere DA WDs, one is a helium-atmosphere DB WD, and one is a cool
DC WD that exhibits no detectable absorption lines. We find an effective
temperature (T_eff) and surface gravity (log g) for each DA WD by fitting
Balmer-line profiles from model atmospheres to the observed spectra. WD
evolutionary models are then invoked to derive masses and cooling times for
each DA WD. Of the 17 DAs, five are at the approximate distance modulus of the
cluster. Another WD with a distance modulus 0.45 mag brighter than that of the
cluster could be a double-degenerate binary cluster member, but is more likely
to be a field WD. We place the five single cluster member WDs in the empirical
initial-final mass relation and find that three of them lie very close to the
previously derived linear relation; two have WD masses significantly below the
relation. These outliers may have experienced some sort of enhanced mass loss
or binary evolution; however, it is quite possible that these WDs are simply
interlopers from the field WD population. Eight of the 17 DA WDs show
significant CaII K absorption; comparison of the absorption strength with the
WD distances suggests that the absorption is interstellar, though this cannot
be confirmed with the current data.
