Both the light curve and spectral evolution of the radiation from a relativistic fireball with extremely short duration are studied , in order to examine the curvature effect for different forms of the radiation spectrum .
Assuming a \delta function emission we get formulas that get rid of the impacts from the intrinsic emission duration , applicable to any forms of spectrum .
It shows that the same form of spectrum could be observed at different times , with the peak energy of the spectrum shifting from higher energy bands to lower bands following E _ { peak } \propto t ^ { -1 } .
When the emission is early enough the t ^ { 2 } f _ { \nu } ( t ) form as a function of time will possess exactly the same form that the intrinsic spectrum as a function of frequency has .
Assuming f _ { \nu } \propto \nu ^ { - \beta } t ^ { - \alpha } one finds \alpha = 2 + \beta which holds for any intrinsic spectral forms .
This relation will be broken down and \alpha > 2 + \beta or \alpha \gg 2 + \beta will hold at much later time when the angle between the moving direction of the emission area and the line of sight is large .
An intrinsic spectrum in the form of the Band function is employed to display the light curve and spectral evolution .
Caused by the shifting of the Band function spectrum , a temporal steep decay phase and a spectral softening appear simultaneously .
The softening phenomenon will appear at different frequencies .
It occurs earlier for higher frequencies and later for lower frequencies .
The terminating softening time t _ { s,max } depends on the observation frequency , following t _ { s,max } \propto \nu ^ { -1 } .
This model predicts that the softening duration would be linearly correlated with t _ { s,max } ; the observed \beta _ { min } and \beta _ { max } are determined by the low and high energy indexes of the Band function ; both \beta _ { min } and \beta _ { max } are independent of the observation frequency .