The longitudinal coupling impedance of a cylindrical beam pipe for arbitrary relativistic energy and mode frequency is obtained analytically for finite wall conductivity and finite wall thickness. Closed form expressions for the electromagnetic fields excited by a beam perturbation are derived analytically. General expressions for the resistive-wall impedance in the presence of a metallic shield and for the rf shielding effectiveness of the beam pipe have been obtained. The results are applied

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... o the GSI synchrotron SIS, where the thickness of the vacuum chamber in the dipole magnets is much smaller than the skin depth at injection energy. In addition, the transverse space-charge and resistive-wall impedances have been investigated analytically of a smooth cylindrical beam pipe of finite conductivity. Transverse beam charge distributions of a hollow beam and of a uniform beam are considered yielding different results. Closed form expressions for the excited electromagnetic fields in the beam-pipe-wall regions and for the corresponding total transverse impedance can be derived analytically for high energy beams.