Relationship between Paley-Wiener theorem and the stationary phase method?

For a given complex function , the Paley-Wiener theorem provides necessary and sufficient conditions, on the magnitude , that there exists an "appropriate phase" such that , the Fourier transform of , is a causal function. The theorem does not give a general way to derive this phase; it only proves its existence. The Fourier transform can also be calculated using asymptotic expansions and approximations such as the stationary phase method. That method can lead to a better knowledge of the conditions on the phase to be an "appropriate" one.