In a free electron laser (FEL), the electron bunch energy profile at the undulator entrance can have temporal structures. In this paper, we derive analytical expressions for the FEL in the undulator, in the case of the electron bunch having both energy chirp and energy curvature. The FEL properties are studied analytically by convoluting a Gaussian seed laser with the FEL Green's function obtained by solving the coupled Vlasov–Maxwell equations. In particular, for different ratios of the temporal duration of the seed laser and that of the Green's function, interesting behavior is revealed.