We have compiled a complete extragalactic sample based on \sim 25 , 000 deg ^ { 2 } to a limiting flux of 3 \times 10 ^ { -11 } ergs cm ^ { -2 } s ^ { -1 } ( \sim 7 , 000 deg ^ { 2 } 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 non-uniform exposure .
The flux-number relation is best described by a power-law with a slope of \alpha = 1.66 \pm 0.11 .
The integration of the cumulative flux per unit area leads to f _ { 20 - 40 keV } = 2.6 \times 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 AGN 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 two power-law form , with an index of \gamma _ { 1 } = 0.8 below , and \gamma _ { 2 } = 2.1 above the turn-over luminosity of L _ { * } = 2.4 \times 10 ^ { 43 } ergs s ^ { -1 } .
The emissivity of all INTEGRAL AGNs per unit volume is W _ { 20 - 40 keV } ( > 10 ^ { 41 } ergs s ^ { -1 } ) = 2.8 \times 10 ^ { 38 } ergs s ^ { -1 % } h _ { 70 } ^ { 3 } 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 ( { \bar { z } } = 0.022 ) , only limited conclusions can be drawn for the evolution of AGNs in this energy band .