SIAM REVIEWVol. 17, No. 3, July 1975

BOOK REVIEWS

EDITED BY ARTHUR WOUK

Publishers are invited to send books for review to Professor Arthur Wouk, Department of Com-puting Science, University of Alberta, Edmonton, 7, Alberta Canada.

Asymptotics and Special Functions. By F. W. J. OLVER. Academic Press, New York,1974. xvi + 572 pp., $39.50 or 18.95.Is part of the appeal of asymptotic analysis the aura of paradox, of exotic

indiscretion, surrounding it? The British analyst G. H. Hardy thought so.Mathematicians, perhaps, feel towards asymptotic analysis the same way physicistsfeel towards the strong nuclear interaction.

250 years ago, mathematicians were more stolid, or less romantic. 250 yearsago, Euler could integrate the following integral repeatedly by parts to get aseries

re -xt 2 3dt

x2 + "-(*) G(x) + x x x4

and despite the fact the series converges for no finite value of x (as the ratio testshows), Euler would not have hesitated to write, for instance,

G(1)-- 1- +2!-3! +-..,

because the result would have seemed "natural", having been derived throughthe use of familiar and straightforward processes. In fact, if set x 10 and addup just the first four terms of the series on the right, get the value .09140. Thecorrect value, obtainable by numerical integration, is .09156. Larger values of xproduce even more accurate answers. The series, though it diverges, has somehowmanaged to mimic the behavior of the function which generated it.

It is the very usefulness of asymptotic series (for that is what the above seriesis) that has made them so continually intriguing. Asymptotic series--or expansionsor expressions (whatever term one prefers)--tend to describe "limiting" or"ultimate" values of a function, precisely the values that are both important andintractable to other methods of computation, for instance, the behavior of a wavefar from the propagating source. Physical scientists, being reasonable men, con-tinued to use asymptotic expansions even while mathematicians viewed them withincreasing skepticism. Abel, Cauchy and Weierstrass had completed their pioneer-ing work--the era of mathematical rigor was to come--and in the early 1800’sthere was no way manipulation with asymptotic series could be strictly justified.

In the latter part of the 19th century and the early part of the 20th century,the concept of asymptotic series was placed on a firm logical basis. Poincar6 andStieltjes carefully defined such series and described many of their most usefulproperties. Although there remains an enormous reservoir of difficult, unsolvedproblems in the area, it is not from a lack of rigor. And the expansions are still

eminently useful. It is just that for a long time, mathematicians made unfair569

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570 BOOK REVIEWS

demands on asymptotic expansions--demanding, in fact, that they behave likethe comfortably familiar convergent expansions. If we use the perfect hindsightvouchsafed each new generation of analysts, few of us will have any difficulty inanalyzing the character of the above expansion. Let me write

G(x) g,(x) + e,(x),

where

1! 2! (- 1) l(n 1)!gn(X)X X2-

+ + +X

Now, if take the truncated binomial series

=l-t+t +--, t-C-l,l+t l+t

and substitute it in (*), find

e- xt

en(X) (-- )" ----+--{

in other words, have expressed G as the truncated first n terms of the series onthe right of (*) plus an "error" term e,(x). It is a simple matter to show that

le,(x)l < f t" e-x, dtn

xn+This tells me, for example, that if x is 10, the error committed by using just the firstfour terms of the series in (*) is less than 4 !/105 .00024. It also tells me the errorgoes to zero not as n goes to infinity, but as the variable x goes to infinity, whichillustrates what meant when said that asymptotic expressions describe limitingbehavior. This property holds for even more general asymptotic expansions, forinstance, those of the kind

The remainder

G(x) c/),(x).n=0

e,,(x) G(x)- dp,(x)k=O

will go to zero as the variable x goes to infinity (often in some sector S in the complexplane). In an abundance of physical problems, the functions qS,(x)= b,(x, 2)will also depend on some parameter 2, 2 belonging, say, to some set A in the complexplane. A question one is often required to ask is, "Does e,(x) e,(x, 2) go uniformlyto zero for 2 e A as x- o in S?" It is on the question of uniform convergence ofasymptotic expansions, indeed on the problem of describing the error of broadclasses of asymptotic expansions, that the author of the present book has dis-tinquished himself.

The subject of asymptotic analysis has produced a number of superb books.Perhaps as in the literature of no other field of mathematics, these books reveal

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BOOK REVIEWS 571

the individuality, taste, and mathematical acculturation of each author. Althoughcertain topics are common to them, each book seems to display its own privateexpertise. own almost a dozen of these books and find many of them essential tomy research.

Few books in any one area of mathematics are more dissimilar in theirapproaches than Arthur Erd61yi’s book, Asymptotic Expansions [1] and N. C. deBruijn’s Asymptotic Methods in Analysis [2]. The former is thin, coolly andelegantly reasoned, tightly organized. Treating continuous functions of a simplevariable, it squeezes out of an integral the maximum amount of information bythe use of a minimum number of very simple tools--integration by parts, forinstance, de Bruijn’s book is quirky, eclectic, fitfully brilliant--full of exotic results,most of them obtained on an ad hoc basis. For instance, take Xo real, so that0 < Xo < rt, and define

Xn+ sin x,, n 0, 1,2,

(the iterated sine function). Who would suspect that

x x10 n- +0 H -- ’

Or, let f’ be continuous and nondecreasing for x large enough, and f(x)= x2

+ O(x), x . Can you show f’(x) 2x + O(x/2)? Such are the types of resultsde Bruijn gives.

There is Wolfgang Wasow’s impressive Asymptotic Expansions for OrdinaryD(fferential Equations 3. Full of important material, it gains much of its effective-ness from the author’s use of a matrix theoretic formulation. Conceptually difficultand demanding as the book must be for the student, it is so beautifully organizedthat have been able to use it successfully as a basis of an advanced undergraduatecourse.

Then there are books which are long on physical intuition and short on rigor,such as J. Heading’s n Introduction to Phase-Integral Methods [4], which is toWasow as Walter Lantz is to Ingmar Bergman; books which concentrate on oneor another problem area (M. A. Evgrafov, symptotic Estimates and EntireFunctions 5]; B. L. J. Braaksma, symptotic Expansions and Analytic Continua-tions for a Class of Barnes Integrals [6); papers long enough to be books (e.g.,Erd41yi and Wyman 7 or Meier’s series of papers on the G-function [8]); andbooks on other subjects with enough subsidiary material on asymptotic methodsto serve as texts (Jeffreys and Jeffreys [9] and Watson [10]).

In one sense, Frank Olver’s remarkable book is more dissimilar to thesebooks than they are dissimilar to one another. It is oriented towards the study ofthe error of asymptotic approximations. No other book treats this problem, andthe reason is simple: it has always been considered too difficult. It is a simplematter to describe Laplace’s method for integrals. It is quite another matter togive a realistic estimate of how close the first term of the expansion given byLaplace’s method is to the given function. As far as am aware, nothing is knownabout such things, at least not until the publication of this book. Mathematicians

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572 BOOK REVIEWS

have a way of shrugging such questions off--one in my hearing referred to theentire subject of error bounds as a "bottomless pit".

But the problem is one of extraordinary practical interest, and the presentbook tackles it head on. You will not have to read very far before you becomeconvinced that this is the work of a professional in the very best sense of that word.The author has acquired an enormous amount of practical experience--he knowswhere the problems are and what needs to be done. He is ferociously adept atshowing how a facile application of a classical method can give a ludicrous result.

He has realized that every mathematical problem reduces, ultimately, to aproblem of form. Finding the error of an asymptotic expansion is never, really,the problem. The problem is to choose the form of the error, the form of its struc-tural components, in such a way that things came to a natural, clear, easily com-putable end. The author has found that two unifying concepts--the total variationofa function and (for problems in differential equations) the error control function--are often the very tools needed to produce a tractable error formula.

In planning his strategy, the author decided to make the book self-contained.In other words, all necessary material on the special functions of mathematicalphysics is developed as it is needed. The second, fifth and seventh chapters aredevoted almost entirely to the properties of special functions:the gamma and psifunctions, orthogonal polynomials, Gauss’s hypergeometric function, Besselfunctions, and the confluent hypergeometric function. think he has made a wisedecision, and not only because it makes the book suitable for use as an advancedtext, although it does that. The alternative to making the material self-contained isto write a disjointed and ad hoc book which sends the reader scurrying to half adozen different books for the necessary background material.

The first chapter presents the usual background concepts: asymptoticsymbols, integration, differentiation of order relations, etc., with a fascinatingside excursion into the asymptotic solution of transcendental equations (much ofthis material comes from de Bruijn). The author establishes the basis of his ap-proach to error analysis by a discussion of the variational operator. There are, asin all the following chapters, copiously gratifying historical notes and additionalreferences. Failure to provide a historical context has been a grave defect of otherbooks on asymptotic methods, for instance, de Bruijn’s book, which does not evencontain a bibliography. consider such mathematical exposition the product ofa guild mentality. In contrast, Olver’s book has 12 closely printed pages of biblio-graphy. It also has a fine general index, by the way, providing a healthy alternativeto the petulant elitism of those British analysts who enjoy denying readers accessto their books.

In Chapter 3, we learn about Watson’s lemma, the Riemann-Lebesguelemma, Laplace’s method, the method of stationary phase. There is a beautifulsection on error bounds for Watso.n’s lemma and Laplace’s method--all new andexciting material. Olver, by the way, writes very well. He is careful to motivatewhat he does--often he shows by an example what difficulties arise, and thenstates and proves a theorem whose hypotheses enable one to circumvent thedifficulty. Nothing is ad hoc. His treatment of the asymptotic estimation ofFourier integrals is as neat as any other have seen. The same standard of clarityprevails throughout most of the book. It is only when the material becomes so

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BOOK REVIEWS 573

difficult and higb_ly specialized (Chapters 13 and 14)--differential equations withtwo turning points, the Gans-Jeffreys formulas, etc.--that his expository styleflags. This is, we must remember, hard analysis: the purview of the dedicated andobsessed few. And if it’s not neat, so what? At this stage of the game, it’s the resultsthat count.

Chapter 4 extends the work of Chapter 3 into the complex plane--saddlepoints and the method of steepest descent. have never seen this important methoddiscussed satisfactorily. What is one to do with the problem of path deformation’?One can either wave hands (Jeffreys and Jeffreys [91) or specify conditions ofvalidity which can, in practice, be verified for a pitifully small class of integrands,all of which have already been treated.

Olver starts off sensibly, with a treatment of Laplace integrals with a complexparameter

l(z) e-tq(t) dr,

then moves to the case where q is holomorphic in an appropriate sector in thecomplex plane and shows how the range of the validity of the asymptotic expan-sion for I can be extended. Next, he gives Watson’s lemma for a complex parameterand a series of intriguing exercises followed by a discussion of the Airy integral.Then he treats Laplace’s method for contour integrals.

I(z) e-P(’)q(t) dr,

applicable when Re[zp(t)] attains its minimum value at a or b. He gives a list offour assumptions the integral must satisfy for his analysis to apply, which seemnot unrealistically restrictive nor too difficult to verify.

Finally, he treats the method of steepest descents. His strategy is to devoteseveral paragraphs to the philosophy behind the method, state a theorem, andfollow with a number of nontrivial, well-worked-out examples. He sees, quiterightly, that using the method of steepest descents is a matter of craft. Each newproblem presents its own challenges. Whether our response produces success orfailure will depend entirely on our training and technique. In this discipline, thereare no clear-cut answers.

Chapter 6 treats the Liouville-Green approximation by first presenting thegeneralized Liouville transformation of the second order linear differential equa-tion in self-adjoint form. The author shows how a logical choice of the transforma-tion variable leads to the constant coefficient in the resulting equation;in otherwords, why the method might be expected to work. There are many treasures inthis chapter, including an error analysis that is new and an extension to complexvariables.

In Chapter 7, the author gives the usual asymptotic theory for differentialequations near irregular singular points--the so-called "sub-normal" solutions.His examples include detailed discussions of the Bessel functions and the confluenthypergeometric function. The error analysis in 12 of the chapter is simply

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574 BOOK REVIEWS

extraordinary. When f and g in the differential equation

w" + f(z)w’ + e,(z)w 0

themselves have Poincar6-type representations valid in some sector S, the analysisyields an error bound for the approximation of an appropriate solution of theequation by a truncated subnormal series--the bound holding in some subregion.The chapter closes with applications to Bessel functions.

Chapter 8 covers the asymptotic analysis of sums and sequences--a veryclear derivation of the Euler-Maclaurin summation formula, many applications,error analysis, summation by parts, Darboux’s method, Haar’s method. Theauthor doesn’t attempt to present all the important extensions of Darboux’smethod that writers have discovered in the last 20 years, nor does he give anyspace to the problem of deriving asymptotic expansions directly from powerseries where coefficients are known in closed form. To adequately treatDarboux’s method would take another book. Material on power series is generallyavailable [5].

Chapter 9 treats further methods applicable to integrals--a lot of excitingmaterial by a number of different mathematicians which has never before appearedin book form.

In Chapters 11, 12 and 13, the author moves into his true m6tier: the asympto-tic analysis of differential equations containing a large parameter in the neighbor-hood of a turning point. I won’t present a detailed description of these chapters.Anyone familiar with the author’s research papers in this area knows how impor-tant and useful the material is. As have indiated before, this is mean analysis.It makes no apologies for its complexity. Its appeal, unfortunately, must be to thespecialist, though there can be no overestimating the significance of its practicalapplications. These three chapters alone, in my opinion, justify the hefty price ofthe book.

The book closes with a chapter on the numerical use of asymptotic expansionsand the estimation of remainder terms.

One of my colleagues has successfully used this book as a basis of a one-semester course on asymptotics. Also, Academic Press has issued the first sevenchapters of the book as a separate volume, Introduction to Asymptotics andSpecial Functions. It is clear the author has devoted a lot of effort to the selectionof the abundance of exercises that grace the book, and these are one of the book’sbest features--not just for the student, but also for the research worker who wantsto develop his skills.

The traditional book review closes with a recommendation that the readerbuy the book, or, occasionally, a caveat. Although I have written reviews doingsometimes one and sometimes the other, have come to see such importuning asa very banal commentary on a project which has consumed a major portion ofsomeone’s life. A mathematics book, after all, is just as much a work of art as anovel or poem. In each, the author, using a kind of creative insight, has molded andtransformed his experience into something new--something which must alwaysbe considered on its own terms. The most happy situation is when the richness ofthat experience and the effectiveness of the strategy for organizing it are ideally

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BOOK REVIEWS 575

matched--and then the result is a personal statement very much like the presentbook.

JET WIMPDrexel University

REFERENCES

[1] A. ERDLYI, Asymptotic Expansions, Dover, New York, 1965.[2] N. G. D BRtJIJN, Asymptotic Methods in Analysis, North-Holland, Amsterdam, 1961.[3] W. WAsow, Asymptotic Expansionsfor Ordinary Differential Equations, Interscience, New York

1965.[4] J. HF.ADIY(, An Introduction to Phase-Integral Methods, Methuen and Co., London, 1962.[5] M. A. EvGFov, Asymptotic Estimates and Entire Functions, Gordon and Breach, New York,

1961.[6] B. L. J. BRKSNA, Asymptotic Expansions and Analytic Continuations for a Class of Barnes-

Integrals, P. Noordhoff, Groningen, the Netherlands, 1963.[7] A. EID(iLI ,D M. WYM,y, The asymptotic evaluation of certain integrals, Arch. Rational

Mech. Anal., 14 (1963), pp. 217-260.[8] C. S. MEIJER, On the G-function. I-VIII, Proc. Ken. Ned. Ahad. Wet., 49 (1946).[9] H. JVlEYS AYD B. S. JEFVFYS, Methods of Mathematical Physics, Cambridge University Press,

New York, 1956.[10] G. N. WTSOY, Theory ofBessel-functions, Cambridge University Press, New York, 1922.

Markov Processes and Learning Models. BY M. FRANK NORMAN. Academic Press,New York, 1972. xiii + 267 pp., $15.00.This valuable monograph is a detailed and thoughtful presentation of

stochastic learning models and their associated mathematics. As one of the fore-most contributors to the unification and rigorous development of mathematicallearning theory, the author is able to write about it with clarity and elegance.Some of the results reported are published for the first time in this volume.

To quote from the Preface" "Those who are familiar with mathematicallearning theory will notice that the emphasis on ’continuous state models’ in thisbook, at the expense of ’finite state models,’ is the reverse of the emphasis in thepsychological literature. In particular, the book makes no contribution to theanalysis of models with very few states of learning. These models are quite wellunderstood mathematically, and they have been extremely fruitful psychologi-cally."

The author does not, in fact, attempt to establish the psychological utilityof the many state models which he treats, but provides references for readersinterested in this question.

In the Introduction, a general learning model is set forth as a discrete timerandom process. At the beginning of trial n, the subject is characterized by hisstate of learning X,, which may be, for example, the probability that he will make acertain response. On this trial, an event E, occurs in accordance with a probabilitydistribution depending on X,, and E, in turn effects a transformation

u(X,, E,) of the state. The sequence {X,} of states, belonging to a state space X,is then a Markov process with stationary transition probabilities. This theoreticalframework includes stimulus sampling models as introduced by Estes, linearmodels of Bush and Mosteller type, and additive aodels which generalize theso-called "beta" models.

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