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Review of "Symbolic Asymptotics by John R. Shackell", Springer Verlag, 243 pages, $79.95

Everyone who has taken a first year University course of analysis is familiar with the notion of how fast functions of a real variable x ‘grow’ as we let x tend to infinity. For example, one recalls that, ‘the exponential function grows faster than any fixed, power of x’. Loosely speaking, asymptotics can be viewed as the study of questions such as, given two functions of a real variable f and g, how does one measure the speed of growth of each of these functions, and then, how does one measure these two quantities relative to one another? Beyond undergraduate mathematics, such questions have long since been important in the physical sciences, where the variable in question is thought of as time, and one is interested in how a given system will change as a function of it; often for arbitrarily far periods in the future— as in the study of radioactive decay for example. It is therefore unsurprising that today, with the ubiquitous role that computers play in scientific development, that there is now great interest in the problem of how to to automate the processes which can answer questions such as those mentioned above. This is exactly what this book is about: how does one go about doing this within a symbolic environment, that is, when using a computer algebra system such as Maple, Mathematica, or one of the many others that are now readily available to people working within Universities or industry alike. It discusses ways of representing asymptotic information, and algorithms to calculate it, for a wide class of functions. Practically all of this book comprises the research of Professor Shackell in this field over the last 20 years; previously this was only available in various conference papers and journals.