Stability diagrams for Landau damping

Stability diagrams allow one to determine whether a system is stable due to the presence of incoherent tune spread in a beam, a phenomenon known as Landau damping. This paper presents an overview of the mathematical foundations that underpin stability diagrams. I first describe stability diagrams as a mapping between two complex planes: the space of eigenvalues of the underlying Vlasov equation, and a space that can most easily be described as the product of beam current and an effective impedance. I go on to describe the circumstances when the Vlasov description of impedance-driven instabilities can or can not be formulated to construct a stability diagram. Finally I outline how this is applied to impedance-driven collective effects in particle accelerators.