Series analysis of multivalued functions

A recurrent problem in mathematical physics, for example, in the theory of critical phenomena, is the need to study the structure of physically interesting functions at a branch point of a complex structure usually called a ‘‘confluent singularity.’’ In such a neighborhood the function is necessarily multivalued. In addition, the value of such a function is sometimes required on a branch cut or even off the first Riemann sheet. Our approach to this problem is inspired by the Riemann (global) monodromy theorem and consists of using series expansions to form integral approximants (special case of Hermite-Padé approximants) to represent multivalued functions on multiple Riemann sheets. We prove an analogous local-monodromy-theorem, functional-representation results. We further identify the important ‘‘separation property’’ and use it to prove a convergence theorem for horizontal sequences of integral approximants. We make an extensive numerical investigation, using horizontal, diagonal, and constrained diagonal sequences, and find that these methods give excellent results on a wide variety of test functions of rather complex structure.