Review of longitudinal perturbation formalism

The foundation of the beam perturbation formalism was established on the Sacherer integral equation. The original solution was obtained by using simplified radial functions, and the results can be used to explain several important beam instability problems. To solve the instabilities caused by high beam intensities calls for more complete solution and the extension to the formalism as well. For the non-coupled perturbation problem without frequency spread, two different eigenvalue type solutions were proposed. The formalism is also modified to solve the azimuthal mode coupling instability and the beam instability with synchrotron frequency spread, where both dispersion type and eigenvalue type solutions were proposed. In this article, a brief review of longitudinal perturbation formalism based on Sacherer integral equation is given. Some perspectives will also be discussed