On the Approximation of Algebraic Irrational Functions by Means of Rational Fractions, with Special Reference to Network Problems

It is shown that the branch cuts of any one-valued branch of certain algebraic irrational functions can be interpreted in the potential analog model as dipole chains. From this interpretation rational approximants are deduced which are characterized by poles and zeros alternating along the branch cut(s), the detailed character of their distribution being related to the order of the branch cut in question. It is shown that the theory can account for the common features of approximants reported in the literature. Although it does not provide a straightforward recipe for tackling new problems, the theory is believed to be of potential usefulness in that it indicates the general nature of the required approximants.