  The initial-final mass relationship connects the mass of a white dwarf with
the mass of its progenitor in the main-sequence. Although this function is of
fundamental importance to several fields in modern astrophysics, it is not well
constrained either from the theoretical or the observational points of view. In
this work we revise the present semi-empirical initial-final mass relationship
by re-evaluating the available data. The distribution obtained from grouping
all our results presents a considerable dispersion, which is larger than the
uncertainties. We have carried out a weighted least-squares linear fit of these
data and a careful analysis to give some clues on the dependence of this
relationship on some parameters such as metallicity or rotation. The
semi-empirical initial-final mass relationship arising from our study covers
the range of initial masses from 1.0 to 6.5 M_sun, including in this way the
low-mass domain, poorly studied until recently. Finally, we have also performed
a test of the initial-final mass relationship by studying its effect on the
luminosity function and on the mass distribution of white dwarfs. This was done
by using different initial-final mass relationships from the literature,
including the expression derived in this work, and comparing the results
obtained with the observational data from the Palomar Green Survey and the
Sloan Digital Sky Survey (SDSS). We find that the semi-empirical initial-final
mass relationship derived here gives results in good agreement with the
observational data, especially in the case of the white dwarf mass
distribution.
