We present the X-ray luminosity function (XLF) for clusters of galaxies der
ived from the RASS1 Bright Sample. The sample, selected from the ROSAT All-
Sky Survey in a region of 2.5 sr within the southern Galactic cap, contains
130 clusters with flux limits in the range similar to 3-4 x 10(12) ergs cm
(-2) s(-1) in the 0.5-2.0 keV band. A maximum likelihood fit with a Schecht
er function of the XLF over the entire range of luminosities (0.045-28.0 x
10(44) ergs s(-1)) gives alpha = 1.52(-0.11)(+0.11), L* = 3.80(-0.55)(+0.70
) x 10(44) ergs s(-1), and A = 5.07 +/- 0.45 x 10(-7) Mpc(-3) (10(44) ergs
s(-1))(alpha-1). We investigate possible evolutionary effects within the sa
mple, out to our redshift limit (z similar to 0.3), finding no evidence for
evolution. Our results are in good agreement with other local estimates of
the XLF, implying that this statistic for the local universe is now well d
etermined. Comparison with XLFs for distant clusters (0.3 < z < 0.6) shows
that no evolution is present for L-X less than or similar to 10(44) ergs s(
-1). However, we detect differences at the 3 sigma level between our local
XLF and the distant one estimated by Henry et al. for the Einstein Extended
Medium-Sensitivity Survey (EMSS) sample. This difference is still present
when considering the EMSS sample revised by Nichol et al.