Over the past decade, the subject of asymptotic analysis has seen a sudden expansion of both applications and theory, as both superasymptotic and hyperasymptotic improvements have been developed. Several distinguished applied mathematicians and mathematical physicists have questioned the usefulness of these higher approximations. This lengthy and painstaking review (98 pages, 328 references) seeks to answer this question by producing an amazing variety of problems where such knowledge of higher order is necessary. The style of the review is lively and informal, as we have come to expect from this source. The author offers “not theorems, but expertise” and signals his intention by beginning with four heuristic physicists’ rules for predicating divergence.
His development starts with standard examples of the Stieltjes function (a transformation of the exponential integral) and Stieltjes series, before giving a very full review of Stokes’ lines, integrals (especially saddlepoint methods), differential equations (including complex matching), special functions and the relationship between numerical and asymptotic approximation. He shows that in the last case, numerical methods will usually give an approximation to the desired level of accuracy more quickly, but points out that the proper use of superasymptotics is to provide insight. “A sensible application is to compute a small term that is also the leading term to some crucial feature of a [physical] problem.”
The review identifies nine parallel lines of development in asymptotics over the past century. It attributes the recent “explosion” of ideas to the fact that these parallel threads have ceased to be parallel and begun to converge. As one of those who has contributed to this explosion, I found this a fascinating read, and can recommend it strongly, both to those already in this exciting field, and to those seeking a lively introduction which makes clear the full scope of these methods.