Space charge impedance and electromagnetic fields in elliptical vacuum chambers

Starting from the electric fields produced by a point charge and a dipole travelling inside a circular vacuum chamber, in this paper we derive a formalism for a complete set of equations that describe the electromagnetic fields and the longitudinal and transverse coupling impedance arising by the interactions of a beam with a perfectly conducting pipe in the case of elliptic geometry. The expressions, which are valid for any frequency and beam energy, are written in terms of expansion of Mathieu functions, allow to range from a circular geometry to the parallel plates, and show an interesting parallelism with the well known expressions for a circular pipe. We also obtain that, under the approximation of low frequency, the formalism allows us to derive the Laselett coefficients for parallel plates, circular and elliptic beam pipe.