Analytic properties of the longitudinal beam transfer function

The frequency dependence of the longitudinal beam transfer function (BTF) in a storage ring, when expressed in a basis of azimuthal harmonics of the line density, is the Fourier transform of a causal function that depends on the radio-frequency potential well in which the bunch moves. The effect of all synchrotron harmonics are included in this function, which is derived from Krinsky and Wang´s expression for the BTF expressed in the same basis (S. Krinsky and J.-M. Wang, Part. Accel. 17, 109-139 (1985)). Analytic properties of the terms of the BTF expressed in a series of synchrotron harmonics, which are approximately Shaposhnikova´s BTF matrix elements (E. Shaposhnikova, CERN Report No. SL-94-19-RF (1994)), are studied through the large-argument asymptotics of corresponding causal functions