On the basis of standard nuclear evaporation theory, we calculate the prompt fission neutron spectrum N(E) as a function of both the fissioning nucleus and its excitation energy. To simulate the initial distribution of fission-fragment excitation energy and the subsequent cooling of the fragments as neutrons are emitted, we take the distribution of fission-fragment residual nuclear temperature to be triangular in shape, extending linearly from zero to a maximum value Tm. This maximum temperature is determined from the average energy release, the separation energy and kinetic energy of the neutron inducing fission, the total average fission-fragment kinetic energy, and the level density parameter of the Fermi gas model. The neutron energy spectrum for fixed residual nuclear temperature is integrated over this triangular distribution to obtain the neutron energy spectrum in the center-of-mass system of a given fission fragment, which is then transformed to the laboratory system. When the cross section σc for the inverse process of compound nucleus formation is assumed constant, N(E) is the sum of a four-term closed expression involving the exponential integral and the incomplete gamma function for the light fragment and an analogous result for the heavy fragment. We also calculate N(E) by numerical integration for an energy-dependent cross section σc that is obtained from an optical model; this shifts the peak in N(E) to somewhat lower neutron energy and changes the overall shape slightly. The spectra calculated for both a constant cross section and an energy-dependent cross section reproduce recent experimental data for several fissioning nuclei and excitation energies for a single choice of the nuclear level density parameter and without the use of any further adjustable parameters. However, the spectra calculated with an energy-dependent cross section agree somewhat better with the experimental data than do those calculated with a constant cross section. Our approach is also used to calculate , the average number of prompt neutrons per fission, as a function of excitation energy for several fissioning nuclei. At high excitation energy, where fission following the emission of one or more neutrons is possible, we take into account the effects of and competition between first-, second-, and third-chance fission when calculating both N(E) and . For ease of computation, we present finally an approximate way to simulate the energy dependence of the compound nucleus cross section through a slight readjustment of the value of the level density parameter.