# Asymptotic Expansions: Their Derivation and Interpretation

**by R B Dingle**

I have decided to host Dingle’s book on my home page because it is long out of print and much in demand. He 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.

R B (‘Bob’) Dingle was my Ph.D supervisor in the early 1960s. From him I was benignly infected by asymptotics, which I came to regard as central in physical mathematics and which is echoed in most of the papers I have written. But although I was aware of his seminal discoveries, I was a ‘first terms man’ for many years. Only in the late 1980s, when I started working on the high orders of divergent series, did I come to understand how profound were the concepts that Dingle had developed in the 1950s. Many of my papers thereafter can be regarded as developments of those in his book – see in particular papers 181, 190, 197, 201, 208, 222, 223, 225, 234, 241, 244, 245, 249, 260, 261 and 265. Of these, the most far-reaching were written in collaboration with Christopher Howls: http://www.soton.ac.uk/maths/people/profiles/applied/cjh.html

Nowadays, systematic approximation 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. Hosting it on my home page is an act of homage to my asymptotics teacher.

The book can be downloaded from here – Dingle.pdf (5Mb pdf file)