Analytic Evaluation of the Series over Azimuthal Harmonics at the Analysis of the Stability of Bunched Beams Coherent Oscillations

At the analysis of the stability of coherent motion of multibunch beams including counterrotating beams) one should deal with expressions analogous to the effective impedance - the serieses over harmonics of revolution frequency of the RF structure impedance at the side frequencies to these harmonics, with certain factors depending on the harmonic number, such as the bunch line density spectrum, the phase factor and the factor describing the order of multipole synchrotron oscillations. In this paper, we present the method for analytic summation of these serieses for resonant impedance, which seems not to be made before in the general case including all mentioned factors. Comparison of obtained expressions with formulae used in previous papers shows the limits of validity of simpler approaches. The obtained expressions are used in the computer codes MBIM1 and MBIM2 presented at this conference, which calculate coherent oscillations stability for arbitrary multibunch beams.