Phase Representation of Analytic Functions

The phase representation which expresses an analytic function essentially in terms of its phase (not imaginary part) along the cuts is discussed. In particular, the precise conditions under which this phase representation is valid and also the asymptotic behavior of the phase representation are studied in detail. It is proved that the asymptotic behavior is essentially the same at infinity in all directions. The derivation of the high-energy behavior of scattering amplitudes and the $\frac{N}{D}$ representation of the partial-wave amplitude are discussed as applications of the phase representation. Finally, the phase representation is used in determining the total numbers of zeros of the forward pion-nucleon scattering amplitudes. It is found that the charge nonexchange amplitude has either 2 or 4 zeros, depending upon the signs of the $S$-wave scattering lengths, while the charge exchange amplitude has 11 zeros.