Lee Segel - The Importance of Asymptotic Analysis
The following is a modified version of a few of the paragraphs of Lee Segel's paper 'The Importance of Asymptotic Analysis in Applied Mathematics', The American Mathematical Monthly 73 (1) (1966), 7-14:
Kurt Friedrichs began his 1954 Gibbs lecture on asymptotic phenomena in mathematical physics [Asymptotic phenomena in mathematical physics, Bull. Amer. Math. Soc. 61 (1955), 217-260] by saying "the problems I intend to speak about belong to the somewhat undefined and disputed region at the border between mathematics and physics." In the ten years since his lecture there has been a growing tendency to define this region and some others as the domain of applied mathematics. A number of explicit university applied mathematics programs have been started, the Society for Industrial and Applied Mathematics has been founded in the United States and the Institute of Mathematics and its Applications in England.
It ought to be helpful to have on record various opinions on what this new, or renewed, activity is all about. The general nature of applied mathematics has been ably discussed [H Greenberg, Applied mathematics: what is needed in research and education, SIAM Review 4 (1962), 297-319], [H Greenspan, 'Applied mathematics as a science', American Mathematical Monthly 68 (1961), 872-880] and the time now appears ripe for more detailed remarks on the elements of an applied mathematical outlook. The purpose of the following supplement to Friedrichs' remarks and references is to stress the importance of asymptotic analysis in applied mathematics. Although they will not be mentioned here, there are of course several other important aspects of the subject. One of these is the use of computing machines. Murray [F J Murray, 'Education for applied mathematics', American Mathematical Monthly 69 (1962), 347-357] concentrates so exclusively on this one aspect that he gives the impression that applied mathematics is a problem solving service rather than the independent science envisioned here and elsewhere. An indication of the central position of asymptotics is that in concentrating on it we do not present an unbalanced picture of applied mathematics.
Another important aspect of the subject is regular perturbation theory, which gives rise to convergent series solutions to problems. A convergent power series about a point is asymptotic as the independent variable approaches the point, but we wish to show the importance of "genuine" or non-convergent asymptotic series. To do this we shall briefly examine (i) certain basic theorems connected with asymptotic expansions, (ii) a differential equation in the neighbourhood of an essential singularity, (iii) a very simple singular perturbation problem, and (iv) the lack of genuine distinction between a large variable and a small one - all in order to emphasize the connection between asymptotic approximations and essential singularities. By means of some examples, we shall try to show that the principal goal of the applied mathematician, basic understanding of physical phenomena, is most likely to be attained when asymptotic analysis is used to obtain the qualitative behavior of a solution near its worst (essential) singularity.
In his lecture, Friedrichs discussed two permissible views of Stokes-phenomenon-like discontinuities - the genuine boundary of a natural object or an approximation to a region of rapid change. He mentioned the paradox that classical mechanics is the first asymptotic approximation to quantum mechanics yet the latter cannot be defined without reference to the former. He thereby gave physical support for the view that asymptotic methods offer more than another approximation technique but rather have a fundamental role in the mathematical description of physical phenomena. This same view is supported here by stressing the connection between asymptotics and essential singularities.
To repeat the qualification made at the beginning, in stressing the importance of asymptotics we do not mean to make an invidious comparison with other weapons of the applied mathematician not treated in this note. For example, only a few point masses need be involved in a gravitational interaction for useful results to be derivable by the asymptotic methods of statistical mechanics. Of great interest, however, are three-body problems where such many-particle methods are inapplicable. In any given situation, then, search for a "large" variable and an accompanying singular perturbation problem may be in vain. This is illustrated by the tale of the applied mathematician who resigned from a meteorology research group as soon as he discovered that all terms in the relevant equations were of the same order of magnitude. On the other hand, a skilled applied mathematician, aware of the power of asymptotic methods, can frequently formulate a problem in such a way that they can be used to reveal the essential features of the phenomenon (see [G Carrier, 'Boundary layer problems in applied mechanics', in R von Mises and T von Karman, (eds.), Advances in Applied Mechanics (Academic Press, New York, 1953), 1-19]). One is reminded here of a remark concerning a certain distinguished scientist, "All he can do is solve boundary layer problems. ... But of course he can turn all problems into boundary layer problems!" To sum up, since he is most often interested in the qualitative behaviour of the solution to an idealized problem, the applied mathematician frequently uses asymptotic expansions which give this behaviour because they describe functions near their worst (essential) singularities. The applied mathematician has the technical ability to use asymptotic methods when the occasion demands. More important, he possesses the power of formulating problems so that asymptotic methods can be used when the occasion permits.