Partial fraction expansion of electron optical cardinal elements

Studying the behavior of virtual or asymptotic cardinal elements (position of focal points, focal widths) of electron optical systems during variation of the acceleration voltage U characteristic discontinuities (poles) are encountered. Partial fraction expansion of the quantities in question with regard to U-1/2 for magnetic systems (round lenses and quadrupoles, straight axis) are investigated. Convergence of the series is established by methods of complex analysis, especially by use of the concept of R-functions introduced by E. Wigner. The results can be used also for systems consisting only of electric quadrupoles. The relations with the terminology introduced by Otto Scherzer are worked out. Applications in analytical field models are suggested. Connections with the problem of determination of the lens field from optical data and with inverse eigenvalue problems are discussed.