1. G. Veneziano:Nuovo Cimento,57 A, 190 (1968);M. Suzuki: unpublished. For other models see,e.g.,N. N. Khuri:Phys. Rev.,176. 2026 (1968);R. Jengo: CERN preprint Th. 948 (1968);E. Predazzi: Indiana University preprint (1969);J. C. Botke andR. Blankenbecler: U.C. Santa Barbara preprint (1969). See alsoF. Arbab:Phys. Rev.,183, 1207 (1969);D. D. Coon:Phys. Lett.,29 B, 669 (1969).
2. A. Erdelyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Higher Transcendental Functions, vol.1, Sect. 6.7 (7) (1953).
3. A. Eedelyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Higher Transcendental Functions, vol.1, Sect. 6.13 (1) (1953). We choose the minus sign in eq. (1.3) to have the phase −π. Then, with Im α≥0 and |x|≫|a|, eq. (6.13) (22) can be shown to reduce to eq. (1.9) to first order ina/x. The asymptotic behavior of ϕ for |argx|<π is obtained by plugging eq. (1.9) into (1.8). Since ϕ is single valued inx, the same behavior is obtained when |argx=2nπ|<π.
4. H. Harari:Phys. Rev. Lett.,20, 1395 (1968);S. Y. Chu, C. I. Tan andP. Ting:Phys. Rev.,181, 2079 (1969).
5. It is probable that the extra $$(ln s)^{\alpha _0 } $$ factor in thes-channel Regge behavior could be eliminated via an extension of the theorem proved in Sect.2 through the choice of a more general meromorphic function having its average asymptotic zeropole spacing given by α0+ɛ. This function would replace the ratio of the two gamma-function in eq. (1.5). Since any change in the average asymptotic zero-pole spacing leads to a change in the power ofs lns, it is probable that changes on the order of lns could be achieved by changing the average asymptotic spacing of an infinite subset of the zeros and poles. Of course we could remove the $$(ln s)^{\alpha _0 } $$ factor by settingf e proportional to an extra factor, $$[ln(s_c - s)]^{ - \alpha _0 } $$ , but a logarithmic cut ats e is then introduced.