Property of rational functions related to band-pass transformation

It is shown that if a rational function F (s) does not change when 1/ s is substituted instead of s then F (s) is the rational function of s + (1/ s). If the transfer function is a ratio of recursive (symmetric) polynomials, the application of this property can simplify the network synthesis, and, in case of filter synthesis, may result in new filter configurations. In addition, the method chosen for the proof can be used to create new filter transfer functions with useful characteristics. Examples are provided.