Longitudinal space-charge geometric factor for an elliptical beam

The longitudinal space-charge impedance for a particle beam, for which the azimuthal extent is much longer than the transverse, is typically expressed in terms of a geometrical factor, g₀, which depends on the transverse charge distribution and the geometry of the vacuum chamber. Expressions for g₀ for a round beam in a concentric round pipe are well known; and values may be given for the on-axis impedance or for the impedance ensemble-averaged over the beam cross-section. We have obtained analogous expressions for the on-axis and ensemble-average factors g₀ for the case of an elliptical charge distribution ρ(m,x,y) =[1-(x/a)²-(y/b)²]^m inside a confocal elliptic metallic pipe for the indices m = 0, 1, 2. However, our results are not completely general; for example, the pipe cannot have the same eccentricity as the beam