Prologue: Regular Variation

Summary

Part of the motivation for this book was its role in solving open problems in regular variation – in brief, the study of limiting relations of the form f(λx)/f(x) → g(x) as x → ∞ for all λ > 0 and its relatives. This was the subject of the earlier book Regular Variation by N. H. Bingham, C. M. Goldie and J. L. Teugels (BGT). So to serve as prologue to the present book, a brief summary of the many uses of regular variation is included, to remind readers of BGT and spare others needing to consult it. Topics covered include: probability (weak law of large numbers, central limit theorem, stability, domains of attraction, etc.), complex analysis (Abelian, Tauberian and Mercerian theorems, Levin–Pfluger theory), analytic number theory (prime divisor functions; results of Hardy and Ramanujan, Erdős and Kac, Rényi and Turán); the Cauchy functional equation g(λμ) = g(λ)g(μ) for all λ; μ > 0; dichotomy – the solutions are either very nice (powers) or very nasty (pathological – the ‘Hamel pathology’).