Gamma-ray bursts ( GRBs ) are brief but intense emission of soft \gamma - rays , mostly lasting from a few seconds to a few thousand seconds .
For such kind of high energy transients , their isotropic-equivalent-energy ( E _ { iso } ) function may be more scientifically meaningful when compared with GRB isotropic-equivalent-luminosity function ( L _ { iso } ) , as the traditional luminosity function refers to steady emission much longer than a few thousand seconds .
In this work we for the first time construct the isotropic-equivalent-energy function for a sample of 95 bursts with measured redshifts ( z ) and find an excess of high- z GRBs .
Assuming that the excess is caused by a GRB luminosity function evolution in a power-law form , we find a cosmic evolution of E _ { iso } \propto ( 1 + z ) ^ { 1.80 ^ { +0.36 } _ { -0.63 } } , which is comparable to that between L _ { iso } and z , i.e. , L _ { iso } \propto ( 1 + z ) ^ { 2.30 ^ { +0.56 } _ { -0.51 } } ( both 1 \sigma ) .
The evolution-removed isotropic-equivalent-energy function can be reasonably fitted by a broken power-law , in which the dim and bright segments are \psi ( E _ { iso } ) \propto E _ { iso } ^ { -0.27 \pm 0.01 } and \psi ( E _ { iso } ) \propto E _ { iso } ^ { -0.87 \pm 0.07 } , respectively ( 1 \sigma ) .
For the cosmic GRB formation rate , it increases quickly in the region of 0 \leq z \lesssim 1 , and roughly keeps constant for 1 \lesssim z \lesssim 4 , and finally falls with a power index of -3.80 \pm 2.16 for z \gtrsim 4 , in good agreement with the observed cosmic star formation rate so far .