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amplitude analytic continuation analytic properties barycentric system branch cut calculation Chew and Mandelstam constant cross section crossing relation 26a crossing symmetry decay rate derive determined from experiment dispersion relations effective-range formula entum formula 12 four-vector Frazer and Fulco fundamental gauge invariant Higher partial waves homogeneous integral equation interaction interchange invariance isotopic spin Lawrence Radiation Laboratory left cut left-hand cut Legendre polynomials locate the singularities low-energy Mandel Mandelstam representation matrix element meson method momentum Mſs nearby singularities neutral pions nucleon obtain once the parameters p-wave annplitude p-wave solution partial-wave annplitudes particles photon physical region plane polynomial expansion problem production from pions pseudoscalar real axis region of convergence scalar transition annplitude sign function sl s1 spectral function stan representation symmetry point transfer variables two-dimensional representation proposed two-pole formula unitary condition unknown parameter unphysical vanish variables s1 zero