Dingle R. Asymptotic Expansions - Their Derivation and Interpretation

London and New York: Academic Press, 1973. — XV, 521 p.Classical Textbook on Asymptotic Expansions.In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by Dingle (1973) revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.Only now we come to understand how profound were the concepts that Dingle had developed in the 1950s. Nowadays, systematic approximations beyond the least terms of asymptotic series are being developed by several groups of scientists worldwide. Nevertheless, Dingle’s inimitable original exposition deserves to be better known.R.B. Dingle describes how divergent series originate, how their terms can be calculated, and above all how they can be regarded as exact coded representations of functions, yielding extraordinarily accurate approximations after clever decoding. When the book was first published in 1973, Dingle’s startlingly original perspective was largely unappreciated. In later developments, some of the ideas have been rediscovered, but not the complete vision; in particular, the fact that the divergences typically take a universal form, and the implications of this fact, are wholly Dingle’s.Throughout this book, the designation "asymptotic series" will be reserved for those series in which for large values of the variable at all phases the terms first progressively decrease in magnitude, then reach a minimum and thereafter increase. The theory of such ultimately divergent series will be called the discipline of "asymptotics". (Frequently, in both mathematics and physics, these phrases have been allowed wider meanings; see footnote to Section 5 of Chapter I). Some examples of such series were discovered in the early eighteenth century by James Stirling, Leonard Euler and Colin Maclaurin, but for over a hundred years were rarely regarded seriously by pure mathematicians. The pure mathematicians of the early nineteenth century, and above all Augustin-Louis Cauchy and Neils Abel, placed too great a stress on convergence to trust such runaway expansions. However, notwithstanding the continued absence of a thorough general investigation, a number of important asymptotic expansions were soon being found. Outstanding discoveries of this period were Pierre Laplace's (1812) evaluation of integrals by expanding the integrand about its limits or stationary points, Joseph Liouville and George Green's solutions to second-order linear differential equations (both 1837), and the observation made by Sir George Stokes (1864) that "constant" multiplying factors in asymptotic expansions can jump discontinuously as the phase of the variable is changed.The present book is in large measure restricted to the classic problems of asymptotics, namely the investigation of asymptotic expansions of various types - power, large-order, transitional and uniform - which can be derived from convergent series, integral representations and second-order linear differential equations.