Abstract
A systematic classification of phases contours and zeros is introduced and their effect on asymptotic behavior investigated. A constraint is found on the spacing of closed contours if they are not to contribute to asymptotic behavior. Use is made of a theorem due to P\'olya and Szeg\"o about entire functions with real negative zeros. The concept of topological equivalence of phase contour maps is discussed. The theorem due to P\'olya and Szeg\"o is applied to the fixed-angle scattering amplitude with the assumption that certain kinds of contours or zeros dominate this leads to a constraint on the order of the entire function by which the amplitude may be approximated.
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