A complete (both amplitude and phase) measurement of the femtosecond linear free induction decay on a Raman active dye in solution (IR144 in methanol) is demonstrated. For weak femtosecond pulses passing through a homogeneous material, Beer's Law can be used to predict the spectral amplitude changes, and dispersion relations can be used to calculate the spectral phase change. A modified fast Fourier transform algorithm calculates the phase change by applying dispersion relations to a symmetrical absorption spectrum spanning positive and negative frequencies. A reformulation of the rotating wave approximation in the frequency domain shows that related limits on the accuracy of the rotating wave approximation in nonlinear optics arise from the width of the linear spectrum relative to the center frequency, and not from the excitation pulses. (C) 2001 American Institute of Physics.