In this paper we present the detailed results of an evaluation of the multipole amplitudes for photoproduction of pions on nucleons. We have solved the dispersion relations for the multipoles E 0 +, M 1_, M 1+, E 1 +, E 2_ and M 2_ by the Muskhelishvili-Omn6s method using an appropriate ansatz for the inelastic multipole amplitudes. Four phenomenological parameters occuring in our theory have been determined by a least squares fit of the s ~ experimental different cross section data. Except at pion CM-angles between 50 ~ and 80 ~ and at 180 ~ in n o production there is satisfactory agreement between theory and experiment for all measured differential cross sections, polarizations and asymmetries of s ~ and n+-photoproduction.
With this paper we start a systematic study of the multipole amplitudes for electroand photoproduction of pions in the low energy region. It is based on the integral equations following from fixed t dispersion relations and the Watson theorem. We start with a discussion of general mathematical problems one encounters when solving the singular integral equations. Especially the non-uniqueness of their solutions poses difficult problems which cannot be solved within the framework of the dispersion theory. As boundary conditions "from outside" we use at threshold the predictions of the static theory. But thereby only the magnetic dipole excitation of the first isobar is determined. For the electric quadrupole amplitude one needs additional conditions.
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