Some mathematical properties of root loci for control system design

IN the study of a linear lumped system one is confronted with a linear differential equation $a_{0} i + a_1{di over dt} + cdots a_n {d^{n} i over dt^{n}} = b_{0}e + b_1 {de over dt} + cdots b_{m} {d^{m} e over dt^m} eqno{hbox{(1)}}$ where the applied stimulus and system response are e(t) and i(t), or vice versa, and t is the independent variable.