[en] A new method for calculating the incoherent neutron scattering function for solids is introduced, based on the traditional phonon expansion, where the terms are calculated using the saddle point approximation for each term. The method is simple enough to be used in calculation systems of practical application. It provides very accurate results even for the tails of the scattering function, and is superior to the Gaussian approximation to the phonon terms. Numerical and experimental comparisons are presented in different types of magnitudes derived from the double differential cross section, using vanadium as a test case.