Transformation methods in the analysis of series for critical properties

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Abstract
During the past twenty years, considerable use has been made of conformal transformations as an aid to series analysis in the study of critical phenomena; however, there has been no evident systematic approach to the task of choosing the most suitable transformation for a given series. In this review we discuss the theory of transformations and provide such a rational and systematic method of approach. The purpose of transformation is to map the singularity of interest (usually the critical point) significantly closer to the origin than any other singularity, so that it dominates the later coefficients of the series; extrapolation methods based on Darboux's theorems may then be employed. We show that, for series likely to arise in thermodynamics, it is always possible to find transformations which achieve this purpose. We provide a set of conditions, which should be satisfied by any transformation function, to ensure straightforward and valid analysis, and we discuss the basic types of transformation and their selection in practice. In the final sections, we illustrate the approach with some important examples, and show that it leads to transformed series which are much smoother than those previously obtained.