We have compiled a complete extragalactic sample based on ~25,000 deg2 to a limiting flux of 3 × 10-11 ergs cm-2 s-1 (~7000 deg2 to a flux limit of 10-11 ergs cm-2 s-1) in the 20-40 keV band with INTEGRAL. We have constructed a detailed exposure map to compensate for effects of nonuniform exposure. The flux-number relation is best described by a power law with a slope of α = 1.66 ± 0.11. The integration of the cumulative flux per unit area leads to f20-40 keV = 2.6 × 10-10 ergs cm-2 s-1 sr-1, which is about 1% of the known 20-40 keV X-ray background. We present the first luminosity function of AGNs in the 20-40 keV energy range, based on 38 extragalactic objects detected by the imager IBIS-ISGRI on board INTEGRAL. The luminosity function shows a smoothly connected double-power-law form with an index of γ1 = 0.8 below and γ2 = 2.1 above the turnover luminosity of L* = 2.4 × 1043 ergs s-1. The emissivity of all INTEGRAL AGNs per unit volume is W20-40 keV( 1041 ergs s-1) = 2.8 × 1038 ergs s-1 h Mpc-3. These results are consistent with those derived in the 2-20 keV energy band and do not show a significant contribution by Compton-thick objects. Because the sample used in this study is truly local ( = 0.022), only limited conclusions can be drawn for the evolution of AGNs in this energy band.
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