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3tt contributions analytic continuation analytic properties barycentric system branch cut calculation Chew and Mandelstam complex plane constant cross section crossing relation 26a crossing symmetry cut by poles decay rate derive determined from experiment dispersion relations ds'j effective-range formula energy formula 12 four-vector fundamental Higher partial waves homogeneous integral equation interchange intermediate isospin-independent function isotopic spin Lawrence Radiation Laboratory left cut left-hand cut Legendre polynomials locate the singularities LOW-ENERGY PHOTOPION PRODUCTION Mandelstam representation matrix element meson method Mi(si momenta momentum-transfer variable nearby singularities neutral pions NEUTRAL-PION DECAY nucleon obtain once the parameters p-wave amplitude p-wave integral equation p-wave solution partial-wave amplitudes partial-wave decomposition photon physical region polynomial expansion production from pions pseudoscalar q plane real axis region of convergence Relations for Partial-Wave scalar transition amplitude sign function spectral function symmetry point tt-tt two-pole formula unitary condition unknown parameter unphysical vanish zero