Analysis I: Integral Representations and Asymptotic Methods

RX Gamkrelidze

(Ed.)

Analysis I Integral Representations and Asymptotic Methods

J

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Contents I. Series and Integral Representations M. A. Evgrafov 1 II. Asymptotic Methods in Analysis M. V. Fedoryuk 83 III. Integral Transforms M.V. Fedoryuk 193 Author Index 233 Subject Index 235

I. Series and Integral Representations M.A. Evgrafov Translated from the Russian by D. Newton

Contents Introduction ................................................... Chapter 1. The Evolution of the Concept of Convergence. ............ $1. NumericalSeries ............................................ $2. Improper Integrals .......................................... 6 3. Regular Methods of Summation. .............................. $4. Function Series ............................................. Q5. Convergence in Function Spaces .............................. 5 6. Regularization of Integrals ................................... 0 7. Formal Series and Asymptotic Series. .......................... Chapter 2. The Techniques of Operating with Series and Integrals ..... 0 1. Newton Polygons .......................................... 9 2. Finding the Coefficients of a Power Series. ..................... 0 3. Series of Partial Fractions ................................... $4. The Gamma-Function and the Euler-Maclaurin Formula ........ Q5. The Zeta-Function and the Dirichlet Series .................... 0 6. The Mellin Transform ...................................... 9 7. Integral Representations for the Sums of Power Series ........... $8. Laplace’s Method .......................................... § 9. Another Version of Laplace’s Method ......................... $10. The Hypergeometric Function ............................... $11. Theorems on the Singular Points of Power Series ............... References .....................................................

2 3 3 6 10 14 18 21 25 29 29 33 38 42 46 49 53 59 66 69 75 78

M.A.

Evgrafov

Introduction Infinite series, and their analogues-integral representations, became fundamental tools in mathematical analysis, starting in the second half of the seventeenth century. They have provided the means for introducing into analysis all of the so-called transcendental functions, including those which are now called elementary (the logarithm, exponential and trigonometric functions). With their help the solutions of many differential equations, both ordinary and partial, have been found. In fact the whole development of mathematical analysis from Newton up to the end of the nineteenth century was in the closest way connected with the development of the apparatus of series and integral representations. Moreover, many abstract divisions of mathematics (for example, functional analysis) arose and were developed in order to study series. In the development of the theory of series two basic directions can be singled out. One is the justification of operations with infinite series, the other is the creation of techniques for using series in the solution of mathematical and applied problems. Both directions have developed in parallel. Initially progress in the first direction was significantly smaller, but, in the end, progress in the second direction has always turned out to be of greater difficulty. It would be a mistake to think that the justification of operations with series interested our predecessors less than us, or that they valued techniques more highly than rigour. Newton’s proofs were completely rigorous, and he was reluctant to publish an insufficiently justified theory of fluxions. In my opinion, the small advances in the justification of operations with infinite series is explained by the absence of a suitable language in which to conveniently speak of these operations, and the creation of a language requires incomparably greater efforts than the proof of individual results. As a rule, the creation of a language is the work of several generations. In this respect we can refer to the example of Euler, whose research affected his contemporaries by its depth and non-triviality, but shocked them with its lack of rigour. To a modern reader the arguments of Euler do not seem to be so very non-rigorous. Simply, Euler already understood the principle of analytic continuation (for single-valued analytic functions), but the absence of a suitable language prevented him from transmitting this understanding to his contemporaries. In the mid nineteenth century there was already a completely modern understanding of a convergent series which allowed one to prove the required results with complete rigour and to distinguish valid arguments from invalid ones. However, left over from the seventeenth and eighteenth centuries were many puzzling unjustified arguments which, for all their lack ofjustification, led to true results by significantly briefer routes. The expansion of the main points of these arguments and the creation of new means of justifying operations with divergent series and integrals was one of the basic achievements of the last century. A short

I. Series and Integral

Representations

3

account of the stages in the development of the modern approach to these questions forms the content of the first chapter of this article. The second chapter is devoted to the second direction; techniques for using series and integral representations in mathematical analysis. The selection of the material for this chapter presented a most difficult problem, and the chosen solution is purely subjective. I have desisted from an attempt to list results, since this route would have required a much larger volume and would have ended only with the production of a reference book; completely useless for reading. A unique opportunity for me, it would appear, to give an exposition of fundamental methods. However, even this path has its own obstacles. The fact is that almost every method which has been used in analysis has generated, in its applications to different objects, extensive theories. Some of these theories have been successfully concluded, some are being rapidly developed and some have come to a dead end. In any of these cases a detailed story of these theories is inadvisable. I have decided to recount in this article only those analytic methods which have not yet been developed into a general theory. Almost all of it is around 100 years old (or more), but is familiar only to sophisticated analysts. To establish the authorship of these methods is most often impossible; they represent the birth of “mathematical folklore”. I have tried not to overburden the article with historical or bibliographical information (although the temptation in both directions was strong). In compiling the bibliography I have proceeded on the premise that its purpose is to assist the reader to quickly find the necessary sources (and not to display the erudition of the author). Therefore I have avoided references to obscure literature. If the reader wishes he can find exotic references in the bibliographies of the books quoted.

Chapter 1 The Evolution of the Concept of Convergence $1. Numerical Series The theory of convergence of numerical series assumed its completely modern form in the middle of the nineteenth century .l In the last 150 years there have been no new results and no new notations. We will now list the basic definitions and results. ‘It would be correct to say that at the beginning of the nineteenth century they began to speak of convergence of numerical series in a language close to the language of the textbooks of our time. The idea of convergence itself, apparently, was not that different from that contemporary with Ancient Greece, but to detach this notion from its method of expression is very difficult.

M.A.

4

Evgrafov

I. Series and Integral

A numerical series is an infinite sum Jtl 4 =

t41

+

u2

+

243 +

..*

(1.1)

where the u, are real or complex numbers. The number u, is called the general term of the series and the number s,=

f: u,=u,+

... + U”

k=l

is called a partial sum of the series. No real meaning,’ in general, is imposed on the infinite sum (1.1). If there is a finite limit s = lim s, “+Zl then the symbolic notation (1.1) gains a meaning. In this case the series is called convergent and s is called the sum of the series. The Cauchy criterion. A numerical series (1.1) is convergent if and only if for each E > 0 there is a number N(E) such that for all n > N(E) and for all m 2 0 the inequality Iu, + ..-u,+,I < E

is satisfied. A series is called absolutely convergent if the series

From the Cauchy criterion it is clear that: Each absolutely convergent series is convergent. If a series converges then its general term tends to zero. Although the idea of convergence of a series was precisely formulated only at the beginning of the nineteenth century the majority of the tests for convergence were found somewhat earlier. We list the basic convergence tests, beginning with tests for absolute convergence. All the tests for absolute convergence rest on the so-called comparison test: Leta, +a, + * *. be a convergent series with non-negative terms. If the general term of (1.1) satisfies n= 1,2,..., Id S a,, then the series (1.1) converges absolutely. *A characteristic example of the slowness of change in the language of mathematicians. Even in modern terminology there are still traces of lost beliefs. The definition given implicitly endows any series (regardless of its convergence) with some value. In the seventeenth century it was firmly believed that each series had a definite sum, although we might not know a method of finding it.

5

The simplest infinite series, whose convergence was well-known even in antiquity, is the geometric progression with a multiplier less than one. Comparison with a geometric progression gives us Cauchy’s test: If limsup (lu,l)“” < 1, “‘03 then the series (1.1) converges absolutely. D’Alemberts test is obtained by the same route: If lim u,+1 < 1, “-+a31 u, I then the series (1.1) converges absolutely. Cauchy’s test is essentially stronger than D’Alembert’s test, but the latter is rather more convenient to apply when the general term of the series is in the form of products and quotients of factorials. In order to obtain more precise absolute convergence tests by means of the comparison test it is necessary to have a larger stock of convergent series. This stock has been obtained via the integral calculus. With its help the following test has been obtained, known as the Cauchy integral test: Let a positive function f(x) be continuous for x 2 a and monotonely tending tozeroasx+ +oo.If lim x f(t) dt < al, x++ca s II

z1 Iu?iI is convergent.

Representations

then the series f(a) + f(a + 1) + f(a + 2) + ...

(1.2)

converges absolutely, and if lim y-(t)dt x’+m s a

= +co,

then the series (1.2) diverges. For f(x) = X-a,

f(x) = (1ogV x

f(x)

= ~~o!idloi%xl)-”

xlogx

’

“‘.

the integrals can be calculated, and the Cauchy integral test gives us a scale of comparisons sufficient for the majority of problems. Namely: the series

zl n-“,

f Oog@x W’ -, n I2(logn)-” nlogn “**

n=2

converge for a > 1 and diverge for u S 1.

n=3

6

M.A.

I. Series and Integral

Evgrafov

In the nineteenth century many other tests for absolute convergence were devised but at present have been forgotten as unnecessary.3 Convergent but not absolutely convergent series are called conditionally convergent. If absolutely convergent series are practically no different to finite sums, then conditionally convergent series require a very much more careful treatment, as is shown, for example, by the following result. Riemann’s theorem. By varying the enumeration of the terms of a conditionally convergent series (with real terms) it is possible to obtain a series which converges to a preassigned sum, or even a divergent series. A similar result holds for series with complex terms but the sum of the new series may either be any point of the complex plane, or any point on some line in the complex plane.4 There are comparatively few tests for conditional convergence of series which do not reduce to tests for its absolute convergence. The most general is Abel’s test: Let a, and b, (n = 1,2,. . .) be two sequences of complex numbers, having the properties

I / iIak

SM
n=l,2

,...,

and

F lb,+,- bnl< *-

PI=1

Then the series

7

more about this later, but first we will give the definition of the concept and list the basic tests for convergence. Let a complex-valued function f(x) be Riemann integrable in each interval (a, b’), where a < b’ < b, but not (Riemann) integrable in (a, b). Then we may speak of an improper integral of f(x) on (a, b).5 If the limit of the integrals of f(x) over (a, b’) exists as b’ + b, b’ < b, then we say that the improper integral converges at the point b. Similarly we define an improper integral convergent (or not) at the lower limit of integration. An improper integral of a function f(x) over a given interval is called absolutely convergent if the improper integral of 1f (x )I is convergent over the same interval. Tests for convergence of improper integrals are not very different to the tests for convergence of numerical series. As for series the basic test is the comparison test: Let g(x) be non-negative on (a, b) and let the integral of g(x) over (a, b) converge. If 1f(x)1 5 g(x) holds on (a, b) and f (x ) is integrable on each interior interval, then the improper integral of f(x) on (a, b) is absolutely convergent. A scale of convergent and divergent integrals is constructed even more easily than for series since integrals are more easily evaluated. We will give a scale of convergence in two versions; one for a = 0 the other for b = +oo. We introduce the notation log, x = logx;

log, x = log(log,-,

If a function f(x), which is integrable sufficiently large x the inequality

x),

k = 2, 3, . . .

on each interval

(a, c), satisfies for

Px, C” IfWl 5 M x log, x . . . log,-, x

n=l

converges (generally speaking, conditionally). The best known is Leibniz’s test, which is obtained from Abel’s test when a, = (- 1)” and b,, is a positive sequence monotonely converging to zero as n-co.

0 2. Improper

Representations

Integrals

The concept of an improper integral was formulated in the nineteenth century and to this day is used in elementary textbooks on mathematical analysis, however, from the modern point of view, it has lost its significance. We will say 3A reader wishing to familiarise himself with the ancient tests for convergence may see [9,30]. 4This analogue of Riemann’s theorem for series with complex terms is set as a problem in Polya and Szego [44] without reference to authorship.

for some positive integer k and some c1> 1, then the improper integral of f(x) over (a, +cc) is absolutely convergent (at infinity). If a function f(x), which is integrable on each interval (c, b) with 0 < c < b, satisfies for sufficiently small x the inequality

(1%,(w)-” IfWl 5 M xlog,(l/x)...log,-,(1/x) for some positive integer k and some tl < 1, then the improper integral of f(x) over (0, b) is absolutely convergent (at zero). For conditional convergence of improper integrals also there is a test similar to Abel’s test for series. Let the functions g(x) and h(x) be given on (a, +co), moreover let g(x) be positive and monotonely tending to zero as x -+ +cc and let h(x) satisfy the 5 For b = co Hardy

named

these integrals

“infinite”.

M.A.

8

I. Series and Integral

Evgrafov

Representations

where ,u(t) is the Lebesgue measure of the set of points of D for which

condition

If@ l,...,X,)I 2 t.

x 2 a.

Then the improper integral of the function f(x) = g(x)&) on (a, +co) converges (at infinity). The essential difference between series and improper integrals is the absence of a simple necessary condition for convergence of the integral (similar to the general term of a convergent series tending to zero). In particular, in an integral absolutely convergent at infinity, the integrand need not tend to zero. As an example take the function f(x) = fJ n” exp[ - 22”(x - n)2},

m > 0.

n=l

It is easily verified that the integral of this function over the whole line converges (and, moreover, absolutely since f(x) is positive). At the same time it is easy to see that n= 1,2,... f(n) > nm, We have already mentioned at the beginning of this section that from the modern viewpoint the idea of an improper integral has lost its significance. Here we must distinguish between absolute and conditional convergence. First we will discuss the question of absolute convergence of improper integrals. In modern mathematics the Riemann integral has for a long time given way to the Lebesgue integral. If we consider an improper Riemann integral, then its absolute convergence is a corollary of the existence of the integral as a Lebsgue integral. However, in defence of the classical heritage it is worth saying that we deal rather with a terminological improvement. In fact, the question of existence of the Lebesgue integral of a measurable function easily reduces to a question of absolute convergence of an improper Riemann integral. Namely, suppose we have a Lebesgue integral I =

f(q,...,x,)do sD

(1.3)

where D is a domain in R” and do is the volume element in R”. According to the definition, the integral (1.3) exists if and only if the integral I* =

If(xl,...,xn)ldu

(1.3*)

sD

exists (also in the Lebesgue sense). According to a well-known formula from Lebesgue integration theory

I*=smw(t) dt 0

(1.4)

The function p(t) is non-negative and non-increasing. Therefore the function tp(t) is Riemann integrable in each (a, b) with a > 0 and b < +co, and (1.4) is then an improper Riemann integral. Its convergence (at both limits) is equivalent to the existence of the integral (1.3*) and hence (1.3). Thus, although in passing to the Lebesgue integral the notion of absolute convergence loses its significance, the tests for absolute convergence remain useful for research into the question of existence of the integral. The replacement of the concept of an absolutely convergent Riemann integral by the concept of existence of a Lebesgue integral is particularly convenient when the question concerns multiple integrals. The fact is that the definition given by us at the beginning of this section does not generalise well to the manydimensional case. The position with conditional convergence of integrals is noticeably more complex. The discussion of many-dimensional integrals suggests that conditional convergence is far from the best means to attach a meaning to a non-existent integral. In fact, for many-dimensional integrals, the significantly more convenient concept of the principal value of an integral is widely used. We will give the definition of this concept in one of the simplest cases. Let the function f (x1,. . . , x,) be continuous on the closure of a domain D c R", with the exception of one point (x7,. . . , x,“). Denote by D, the domain obtained from D by removing the ball Ix - xy

+ ... + Ix - x,“I’ < E2.

If the limit ;i

s4

f(xl,...,x,)du

exists, then it is called the principal value6 integral of f(xl, principal value integral is usually denoted

. . . ,x,) over D. The

P c f(xl,...,x,)du. JD

Both conditionally convergent integrals and principal value integrals are rather feeble attempts to attach a definite value to a non-existent integral. In the following sections we will speak of much more drastic measures taken in this direction. ‘The concept of principal value integral was introduced by Cauchy in the first half of the nineteenth century but only became widely used in the twentieth century. The basic works on this concept are [26] and [46]. For a detailed survey and bibliography see [8].

M.A. Evgrafov

10

I. Series and Integral Representations

0 3. Regular Methods of Summation In the seventeenth and even the eighteenth centuries mathematicians believed that each series (convergent or divergent) had a well-defined sum; for convergent series it was possible to find it simply by successively adding the terms, whereas for divergent series more complicated methods were needed. Almost nobody doubted the correctness of the formulae 1 - 1 + 1 - 1 + *a. = l/2 1 - 2 + 3 - 4 + **. = l/4, since the sums of these series, when computed by various methods were the same. The sum of the series 1 - l! + 2! - 3! + .*. was calculated by Euler to three decimal places using a method which we now call the Euler summation method. One of the most widely applied methods of evaluating sums of series is the following. First a certain number of partial sums of the series are calculated. If the sums start to coincide (to the desired accuracy) after some index, then the value obtained is regarded as the sum of the series. If the values of the partial sums continue to noticeably diverge, then their arithmetic means are calculated and if they start to coincide after some index, then the sum of the series is taken to be the value obtained. Often the arithmetic means of the arithmetic means are taken. The practical utility of the method described is unquestionable. In fact, the calculation of each successive term of a series is a fairly complicated problem. Managing without these calculations at the expense of calculating the arithmetic means of values already found meant containing a great saving in time. Euler’s method is close in spirit to that just described but is more relined. To wit, given a series a1 + u2 + uJ + . .. Euler constructed a new series v1 + u2 + v3 + . . . as follows 1 u1 =-IA

2 l’

where

0

v,=-

1 2” (

u,+

n are the binomial k

n(

1

‘)u~+-.-+(~)u~),

n=2,3 ,...

coefficients. Having written down this calculation

scheme it is not difficult to see that the partial sums of the new series are not much more difficult to calculate than in the method of arithmetic means. At the same time Euler’s method, in many cases, gave significantly greater acceleration of the convergence. Both the method of arithmetic means and Euler’s method can be applied with equal success to convergent and divergent series. It is easy to see that if the

11

original series converges then both the methods lead to the sum. However, these methods give determinate values even for the sums of many divergent series. In the nineteenth century a great many methods of summation were proposed (the invention of new methods of summation stopped with the advent of the twentieth century).7 The first fundamentally new step in the theory of summation methods was made in 1911 by Toeplitz.8 He proved a theorem describing all the regular methods of summation-summation methods having the natural properties of linearity and taking convergent series to convergent series. It is more convenient to formulate Toeplitz’s theorem in terms of sequences rather than series. To apply the theorem to series the sequence of partial sums is taken as the sequence. Let n,k=

A = (%I,),

1,2 ,...,

k 5 n,

be an infinite triangular matrix. The matrix A assoeiates with each sequence {s,} the sequence {st}, where n St

=

1 k=l

ankSk?

n= 1,2,...

We will say that {s,} is summablewith sum s by the method defined by A, if lim s,” = s. a+m The summationmethod defined by A is called regular ifs, + s implies s,” + s. Toeplitz’s theorem is as follows. A summation method defined by a matrix A is regular if and only if the following two conditions are satisfied,

and lim ank = 0, k= 1,2,... n+m Summation methods defined by Toeplitz matrices include most of the known summation methods. For example, the method of arithmetic meanscorresponds to the matrix A with ank

=

1/ny

n, k = 1, 2, . . . ,

k 5 n,

and Euler’s method corresponds to the matrix with ‘A detailed survey of summation methods, containing the proofs of almost all the results and a great deal of interesting historical information, is given in [27]. ‘Toeplitz’s original paper was published in an inaccessible Polish journal, but his results are presented in detail in many books, for example, [27] or [SS].

M.A.

12

1 k ank

=

Zji

0

n

?

I. Series and Integral

Evgrafov

n,k=l,2

,...,

k 5 n.

Nevertheless, certain summation methods widely used in analysis do not fall within the Toeplitz scheme; for example, the Abel-Poisson method. This method attaches to the series ui + u2 + u3 . . . the sum

Representations

13

It is worth stressing that the majority of summation methods have been developed, not out of an abstract desire to find a sum for any divergent series, but for the summation of divergent series which arise in concrete problems. We will now give two results of this kind. Let the power series f(z) = nzl fnz”

s = lim F u,?. x-1-0 n=l This defect was eliminated almost immediately by Steinhaus.g The simplicity of Toeplitz’s criterion made it easy to construct regular summation methods with fairly unusual properties. For example, it turned out to be possible to construct regular methods which summed the series 1 - 1 + 1 1 + * *. to any preassigned number. It could be said that Toeplitz’s theorem struck a decisive blow at the naive belief in the existence of a definite sum for each series. A very interesting subclass of Toeplitz summation methods are the Hausdorff means, considered by Hausdorff. lo These methods are defined by matrices A of the form A = 6~6, where p is a diagonal matrix with positive diagonal elements, and 6=((-1)x(c)),

n,k=1,2

,...,

nzk.

For Hausdorlf means interesting answers to the question of comparative strengths of methods and their consistency have been obtained. The research into Hausdorff means completed the destruction of the naive belief mentioned above. It appears that Euler already understood the necessity to be aware of the “parentage” of a numerical series in order to sum it properly. In fact, Euler almost always dealt with power series, and the methods of summation he applied were by analytic continuation of power series. Thus, for example, the above-mentioned Euler method of summation reduces to regarding the series ai + up + u3 + **. as the value of the power series

converge in some neighbourhood of z = 0. The Mittag-Leffler star of the series f(z) is the domain DJ- consisting of points to which the series can be continued along line segments passing through z = 0. DI is a simply-connected domain and the result of continuing f(z) to D, is a single-valued function which we denote by F(z). The limit LIZ” lim F a++0 n=~ r(l + n6) exists for each z E Df and is equal to F(z). This result is due to Mittag-Leffler. l1 As a second result we quote Fejer’s theorem.” The Fourier series of any piecewise continuous function 4(x) sums by the method of arithmetic means to *{4(x + 0) + 4(x - O)}. Summation methods have been discussed not just for series but also for improper integrals. The generalizations of the formulae present no special problems. For example, the summation formula for a series ul + u2 + *** by the method of arithmetic means takes the form s = lim t n-tm

for summation

of the integral

o* fWk

5 u,xn n=l

x at x = 1; to calculate this value the series is expanded in powers of y = ~ 1+x and the value of the new series at y = 4 is taken. Many of the manipulations carried out by Euler stopped being mysterious after the creation, by Riemann and Weierstrass, of analytic function theory. gSteinhaus’ result, published in the same journal issue as Toeplitz’s result, was obtained on the basis of Toeplitz’s result and using the same method. Therefore the more general Steinhaus result is often called Toeplitz’s theorem. “The summation methods named Hausdorff means were introduced not by Hausdorff but by Hurwitz. They are named after HausdortTbecause he studied them in depth in a series of papers [28] and [29]. One of the chapters of [27] is devoted to an account of Hausdotlf’s theory.

s

k=l

it takes the form

a>0

The fundamental results on summation of integrals are concerned with the theory of Fourier integrals, where theorems analogous to the above theorem of Fejer have been proved.13 “There is an account of this result in [27]. i2 See the original article by Fejer [20]. An account of Fejer’s theorem and its various can be found in any book on trigonometric series. i3Many results on the summation of Fourier integrals are given in [SO] and [56].

generalizations

14

M.A.

Evgrafov

I. Series and Integral

Toeplitz’s theorem has also been generalized to summation methods for integrals. In this case the role of the matrix is played by a function of two variables.i4

At the foundation of infinitesimal calculus is the systematic utilization, not of numerical series, but of function series. However, although the concept of convergence of a numerical series had taken a completely modern form by the turn of the nineteenth century, the development of the modern concepts of convergence of function series required another complete century. We will briefly tell the story of the formation of these ideas. The meaning of convergence for a numerical series is fairly obvious-a convergent numerical series must have a sum which can be calculated to any degree of accuracy from its sequence of partial sums. A function series can, certainly, be considered as the collection of numerical series associated with all possible values of the variables and we can say that the function series is convergent if all of these numerical series converge. Such an approach has turned out to be unsatisfactory in many respects. The fundamental deficiency is that it severely restricts the role of the function series in analysis. The basic advantage of function series (widely used from the very beginnings of infinitesimal calculus) is that it is easy to perform many formal operations with these seriesthey can be added, multiplied, integrated, etc. The convergence of the series which occur in the intermediate steps is of no consequence. All that counts is the validity of the final result. Thus the notion of convergence of a function series must be aimed at providing a foundation for formal operations with function series. The first serious results on function series were obtained by Weierstrass in the mid nineteenth century. He introduced a notion of convergence for function series which was stricter than convergence at each point. This notion, which was named uniform conoergence, rapidly gained universal recognition.’ 5 It has come down to the elementary analysis courses of our time with practically no change. The fundamental theorems on uniformly convergent series are. a) The sum of a uniformly convergent series of continuous functions is a continuous function. b) A uniformly convergent series of continuous functions can be integrated term by term. r4These results are also presented in [27]. ‘sAlthough Weierstrass had used the idea of uniform convergence by 1841, formally he the author of the idea. The definition was published in 1847 independently by Seidel and The notion, nevertheless, has been associated with Weierstrass who willingly explained it in and other verbal communications, but, in the words of Klein, he had an aversion to printers did not like to publish his work.

was not Stokes. lectures ink and

15

These theorems now occupy an honourable place in analysis textbooks and are called the Weierstrass theorems. We must mention also a simple test for uniform convergence called the Weierstrass

$4. Function Series

Representations

test.

c) If functions f.(x), which are continuous in a domain D, satisfy the inequalities If.(x)1 5 u, in this domain and the numerical series with general term u, converges, then the function series with general term f,(x) converges uniformly in D. Using these theorems it became possible to accurately justify the foundations of operations with uniformly convergent function series. However, the solution was by no means completely satisfactory-the requirement of uniform convergence of all intermediate series was clearly excessive. We pass over the struggle with the severity of this requirement. In the nineteenth century mathematicians studied mainly two types of function series-the power series and the Fourier series (including Fourier integrals). These series are essentially different, and we discuss them separately. In questions concerning power series more or less complete answers are given by the following two theorems of AbeLl 1. If a power series

(1.5) converges at a point z = z. and z. # 0, then it converges uniformly

in the disc

IzI 5 r, for any r < IzoI.

2. Under the assumptions of the preceding theorem, the power series (1.5) converges uniformly on the line segment joining z = 0 to z = zo. The first Abel theorem implies the existence of a number R, R 2 0, such that (1.5) converges for lzl < R and diverges for IzI > R. This number is called the radius of convergence of the series (1.5) and the disc 1zI < R is called the disc of convergence (the latter name is used only when R > 0). Yet Euler, who carried out a variety of operations with power series, often operated with them outside the disc of convergence. From the point of view of uniform convergence such operations were totally illegitimate, but Euler obtained correct results with their help (unfortunately, operations almost the same as Euler’s led other mathematicians to incorrect results). In clarifying his astonishing operations with divergent series Euler said that he was working not on the series, but on the functions which could be expanded into these series. The possibility of free transition from series to functions and conversely was guaranteed by the uniqueness of the expansion of a function into a power series.” i6Abel’s formulation has been somewhat modernised. In 1826, when he published his theorems, the notion of uniform convergence did not exist. “Books on the history of mathematics do not mention who first noted the uniqueness of the expansion of an analytic function into a power series. This fact was considered selfevident long before the origin of a clear notion of a function.

16

M.A.

I. Series and Integral

Evgrafov

The theory of analytic functions created by Weierstrass allowed many of Euler’s arguments to be justified. We briefly explain the essence of the idea of analytic continuation which lies at the foundation of the Weierstrass theory. A function expanded in a power series in a neighborhood of each point at which it is defined, was for a long time the principal object of study in mathematical analysis. It was thought at first that other functions simply did not occur; but when the existence of others was recognised, these functions came to be called analytic. It is easy to deduce the following result from the uniqueness of the power series expansion. The principle of analytic continuation. Let f(z) be a function analytic in a domain D, and let G be a domain containing D. Then there is at most one function F(z) analytic in G and coinciding with f(z) in D. The function F(z) is called the analytic continuation of f(z) from D to G. From the principle of analytic continuation it follows, in particular, that an analytic function is completely determined by its values in an arbitrarily small neighbourhood of a point at which it is analytic. Therefore it is natural to speak of the analytic continuation of an analytic function from one point to another. Unfortunately the definition says nothing about the properties of the continuation. Weierstrass carried out a deep analysis of the notion of analytic continuation. He showed that analytic continuation from one point to another (if it was possible) reduced to a finite number of elementary operations, consisting of the expansion of a power series -f C”(Z - a)

(1.6)

into powers of (z - b), where b is a point inside the disc of convergence of (1.6). The result of continuation depends, in general, on the choice of intermediate points or, what is the same thing, on the choice of curve along which we proceed from one point to another. Weierstrass’ research allows us to understand the essence of the phenomenon of multivalency of analytic functions, which was a fundamental source of error in the clumsy attempts at analytic continuation. Weiestrass’ research on analytic function theory was developed further by Riemann. After Weierstrass and Riemann analytic continuation became a standard tool in the hands of analysts. Thus, research into power series in the nineteenth century led to the development of analytic function theory and the uncovering of the mystery of analytic continuation. Essentially different problems arose in the analysis of Fourier series and Fourier integrals. The trigonometric series (1.7)

Representations

17

is called the Fourier series of a function f(x), integrable on (- rc,K), if the formulae f, = & j:

f(x)eminxdx,

n=O,

?I

+l,

+2 ,...,

(1.8)

known as the Fourier formulae, hold. For Fourier integrals slightly different terminology is currently accepted. Let the function f(x) be defined on R”. Then the integral f(l)

= (27p

R” f(x)e”@”

dx,

s

(1.9)

where (4

dx = dxI...dx,,

= x151 + ... + %Ll,

is called the Fourier transform of f(x). The formula f(x) = (27p

R” f(<)e-i’cxt;)

d<,

(1.10)

s

is called the inversion formula for the Fourier transform. Formulae (1.9) and (1.10) are also called Fourier formulae. Trigonometric series occurred in mathematics long before Fourier, but only after his work did they attract the serious attention of mathematicians. This was because in his research into heat diffusion Fourier showed that trigonometric series could be a powerful tool for the solution of partial differential equations (the Fourier method). Fourier’s work provided a major stimulus for the study of divergent series and integrals. Fourier’s operations with series and integrals were, from the viewpoint of the Weierstrass theory of uniform convergence, totally illegitimate, but the results obtained using these operations aroused no doubt whatsoever. As a typical example we quote the derivation of Pursevul’s formula. Let f(x)

=

2 fneinx, n=-cc

g(x) =

2 gneinx. “=-a

Then &

; f(x)s(x)dx s II

=

? f,sw “=--a

To derive this formula the series for f(x) and g(x) are multiplied integrated term by term, after which the elementary equalities 1 = ei(m-n)xdx = a,,,, K s -x are used, where a,,,, =

0

m#n,

1 m=n.

m,n=O,l,...,

out and

M.A.

18

I. Series and Integral

Evgrafov

This derivation is justified by the Weierstrass theory if the Fourier series of f(x) and g(x) converge uniformly on the whole interval [-‘it, rr]. However, as was established quite a long time ago, the demand for uniform convergence of Fourier series imposes an excessively high degree of smoothness on the functions f(x) and g(x). To present this requirement more clearly we will now give the basic tests for convergence of Fourier series.” Dirichlet’s

R s

is the Fourier seriesof a function f(x) satisfying the condition

-II

if and only if

(1.12)

dx < ~0

lL12< 00.

In this connection the relation

If(xo + t) + f(x0 - t) - Zf(x&

cot(t/2)dt

s0 is finite, then the Fourier series of f(x) at x = x0 converges to f(xo). These tests give sufficient conditions for convergence of a Fourier series, but these sufficient conditions are quite close to necessary. In the first half of the twentieth century many different examples were constructed in which the Fourier series converges as badly as possible. Amongst these an example due to A.N. Kolmogorov was particularly remarkable: the Fourier series of an integrable function which diverged at all points.ig The justification of operations on Fourier integrals was in an even more parlous state than the justification of operations on Fourier series. In the theory of Fourier integrals were circulating formulae such as cw”

r

ei’(xe;)dx = S(l),

JR”

(where S(5) is the so-called Dirac delta function). Many attempts were made to justify operations on Fourier series and Fourier integrals by using summation methods. However, the simplest and most effective route to this end lay in another direction.

6 5. Convergence in Function Spaces The first new step in the development of the modern trends in convergence of function series could be regarded as the following result.”

are [21]

If(

f

R

“See [ll] or [58]. “See [11] or [58]. “‘The original papers series.

(1.11)

“Em fneinX

f(x + 0) + f(x - 0) 2 Dini’s test. If the integral

19

Riesz-Fischer theorem. The trigonometric series

test. If f (x) is of bounded variation on [ - rr, n] (that is is a difference

of two monotone functions), then its Fourier series converges at each point to

Representations

and [45].

An account

can be found

in any book

on trigonometric

lim N-a,

n f(x) - $,

fnei”“~2clx = 0

s -R I

(1.13)

holds.

In modern terminology, instead of (1.12), we would say that f(x) belongs to L2(-rc,~), and instead of (1.13) we would say that the series (1.11) converges to f(x) in the topology of L2( - rr, rr) (or in the metric of this space). The Riesz-Fischer theorem was obtained independently by both of these mathematicians in 1907. It is interesting to note that as long ago as 1876 such a result had been published:21 The minimum value of the quantity

z,=

f(x) -

9

c,einx ‘dx,

II=-N

with respect to all possible values of the coefficients c,, occurs when the coefficients are equal to the Fourier coefficients of f(x). Now it is not easy to understand why the transition from this result to the Riesz-Fischer theorem took more than thirty years. Evidently a role was played here, not so much by objective difficulties, but rather by a definite inertia of thought. In order that the reader may himself estimate the worth of these two factors we will enumerate the stages in the transition from one result to the other.22 The first stage, apparently, was the transfer of the minimal property to series with respect to any orthogonal system (the transfer itself is trivial, but the idea of an orthogonal system of functions had to be introduced). The second stage was establishing the analogy between orthogonal systems of functions and *I See [52]. “The history of the development of the notion in one of the essays in the book [S].

of infinite-dimensional

spaces is dealt with in detail

M.A.

20

Evgrafov

I. Series and Integral

orthogonal systems of vectors in Euclidean space. In this analogy a function expanded as a series plays the role of an arbitrary vector, and the Fourier coefficients play the role of the coordinates of the vector. At this stage Hilbert created the theory of the infinite-dimensional sequence space which we now call the Hilbert space 12. For the later stages the creation of Lebesgue integration theory was essential. Only with the Lebesgue theory did it become possible to establish an isomorphism between I2 and the space of square-integrable functions L2. After all this has been said the Riesz-Fischer theorem now reduces to the technical problem of proving the completeness of the system of functions {einx}, n=o, fl, f2,... in the function space L2( - n, 7~). It has long been customary to say that if (1.13) holds then the series (1.11) converges in the mean square to f(x). It is clear that a series converging in the mean square is by no means sure to converge in the usual sense. This situation opened the way to new methods of justifying operations with divergent series and integrals. In fact, the theory of Fourier series and Fourier integrals in L2 is based on it. The most significant consequence of the discovery of mean square convergence was probably the very idea itself that there are many different modes of convergence. This idea was exploited in many ways. Firstly, many new notions of convergence (some quite eccentric) were suggested. Secondly, a logical analysis of the idea was carried out and the axiomatic theory of topological spaces was born. Thirdly, the already established notion of mean convergence penetrated more and more deeply into the classical domains of analysis. With the creation of the beautiful theory of Banach spaces it became clear that the Weierstrassian theory of uniform convergence could by no means be regarded as somehow better than convergence in the mean. Both are particular cases of convergence in the topology of some Banach space.23 In fact, let A be a Banach space whose elements will be denoted by f, and let A’ be its dual space whose elements will be denoted by 4. We use the notation (~$,f) for the value of the functional 4 on the element f, thus stressing the symmetry between A and A’. We will denote the norm off E A by 11f 11.Then the following results hold: a) if a series

Representations

21

c) if elements f, of A satisfy 11f, 11S u,, and the numerical series with general term U, converges, then the series (1.14) converges in the topology on A. If we take as A the space of functions f(x), continuous on a set D, with the norm

then from a) and b) we obtain Weierstrass’ theorem on uniformly convergent series and the Weierstrass test for uniform convergence. Thus, with the creation of the theory of Banach spaces, the Weierstrassian theory of uniform convergence lost its exclusive position. For some time it was thought that analysis had no need of topological spaces other than Banach spaces. However, it soon turned out that this opinion was mistaken. In various questions in analytic function theory and Fourier transform theory it turned out to be a great deal more convenient to deai with topological vector spaces outside the confines of Banach spaces.24 We mention two basic reasons for this, restricting ourselves to Fourier transform theory. The first reason is that in many problems it is desirable to have a simple test for membership of a given space, both for a function and for its Fourier transform. In other words, both the space itself, and the Fourier transformation of it, must be able to be described fairly simply. Among Banach spaces the space L2, and very few others (all easily expressed in terms of L2), has these properties. Outside the confines of Banach spaces the possibilities are greatly enhanced. The second reason is that the condition for convergence in a specified Banach space imposes certain limitations on the divergence of series (or integrals) which converge in the topology of the space. Outside Banach spaces, even here, the possibilities are enhanced. Yet another reason for the denial of Banach spaces is that Banach spaces turned out to be far from simple and convenient even from the viewpoint of functional analysis. Roughly speaking, Banach spaces differ too much from the simplest spaces-the finite-dimensional ones.

9 6. Regularization of Integrals

(1.14) n=1

converges in the topology on A, then it converges to some element of A; b) if (1.14) converges in A and 4 is any element of A’, then the numerical series J

<&ti>

converges; 23 Any information analysis.

required

regarding

Banach

spaces can be found

in any textbook on functional

In connection with the problem of giving a meaning to divergent Fourier integrals it is worth mentioning yet another method of attaching a meaning to an integral, the unique value of which, at an uneducated glance, could only be infinity. We begin with an account of the similar attempts applied to series. The question of the possibility of attaching a finite sum to a divergent series with positive terms was fairly keenly discussed in the eighteenth century. Euler regarded it as perfectly possible, but his opinions on this had more opponents 24The theory of Fourier transforms,based on a wide Banachspaces,is presentedin the books [23] and [47]

use of topological vector spaces different (and also in many later books).

to

M.A.

22

I. Series and Integral

Evgrafov

of integrals.

The method of regularization of integrals is fundamentally very little different to the method which Euler used to attach a meaning to the sum of a divergent series with positive terms. Namely, for the regularization of a divergent integral an additional parameter z is introduced. This parameter must be chosen so that the integral converges for some domain of its values and is an analytic function of z. After this the integral is analytically continued to the value of z for which we obtain our original integral. The value of the integral obtained in this way is called a regularized value. There are also more primitive methods of regularization, again borrowed from the arsenal of eighteenth century mathematicians. The question of how the regularized value of the integral depends on the way in which the additional parameter is introduced was not raised for some time. An important new feature of the modern method of regularization of integrals is the concern that the analytic continuation of the integral with respect to the parameter should be single-valued. This allows one to avoid the problems demonstrated by the example at the beginning of this section. In fact, each result on regularization of integrals is a result on single-valued analytic continuation of some class or other of integrals (and conversely). The simplest result on single-valued analytic continuation of integrals was probably already known to Euler (the continuation of the integrals, defining the gamma-function and the beta-function). The following result has also been known for some time. Let a function a(x), defined on the positive semi-axis, converge to zero, as x + co, more rapidly than any power of x, and close to x = 0 let it have an asymptotic series

of the power series

loi% j&=zlf from the disc Izl < 1 to the point z = 2 puts before us two perfectly equivalent candidates for the role of sum: rci and -xi. Nevertheless, the history of the summation of divergent series with positive terms did not end with the attempts by Euler. One original method was applied in his papers by the Indian mathematician Ramanujan, who was alive at the beginning of the twentieth century. 25 His method was related to the EulerMaclaurin

formula kgl f(k)

= J’ f(x)dx a

+ c + f$

- f

(- l)*&f’“L’Vn)

k=l

and consists of this; the series

kzl“f(k) is assigned a sum equal to the constant C in the formula. The constant C = C(j,a) depends on a in a fairly simple way, but its dependence on f(x) is far from simple. When f(x) is analytic in the halfplane Re z 2 a, z = x + iy, and satisfies the condition If(x + iy)l 5 M If(x)Ie(2n-a)lyl,

a(x) x f anxan+8, n=O

& > 0,

where CIis a positive number and p is an arbitrary complex number. Then the integral

fU +it) -f(l-it) dt If(l) Ii O” c(f,a) =_s’f(x)dx

then the expression for C(f, a) takes the form 2

a

Z5We are not talking here of published Ramanujan and to his notebooks.

work

s0

by Ramanujan.

A(z) =

eznr - 1

Hardy,

in [27]

refers

to letters

23

In the majority of the problems solved by Ramanujan the calculation of C was reduced to the analytic continuation of the Riemann zeta function in the halfplane Rez < 1. The problem of giving a meaning to divergent integrals with positive integrands aroused significantly more attention in the twentieth century. The fact is that such integrals often appear in the solution of partial differential equations. For their treatment a certain system was devised, named the method of regularization

than proponents. In the nineteenth century there were practically no proponents remaining of these paradoxes. It was even suggested that a divergent series with positive terms should be regarded as convergent with sum +co. A significant role here was played by the fact that Euler’s ideas on analytic continuation were only understood sometime later, and when they were understood, by Weierstrass, then the serious obstacles which prevented the sum of a series being defined in a reasonable manner, using analytic continuation, were also understood. The fact is that the result of analytic continuation of a series outside its disc of convergence, in general, is not single-valued. Therefore, for example, an attempt to define the sum of the series

using analytic continuation

Representations

” of

al s a(x)x*-l

dx

0

converges in the halfplane Re(z + /I) > 0, and may be analytically continued to ’ the whole complex z plane except for the points z, = -(cm + /I), n = 0, 1, . . . , at which it has simple poles with residues a,.

M.A.

24

Evgrafov

I. Series and Integral

On the basis of this result it is possible speak of the regularized values of A(z) for any z # z,. Sometimes one goes even further and speaks of the regularized value of A(z) at z,, meaning by this the value lim a(z)-* z-+z, 1

a(r, z) = s

n1

[G(x)]‘-‘a(x)

25

s CQWI -Vb4 do, n

(S, the unit sphere in R” and do the differential volume form on S,). The function a(r, z) has compact support with respect to I and is analytic in z throughout the complex plane. Therefore the simplest of the above results on analytic continuation can be applied to (1.16) (a small modification is needed in view of the dependence of a(x) on z; but since this dependence is analytic, the basic part of the result remains true). By analytically continuing the integral (1.16) to the point z = - 1, we arrive at a regularized value of the original integral (1.15). If the homogeneous function Q(x) can become infinite on 1x1 = 1, then a(r,z) will no longer be analytic in the whole z plane. In this case, in order to investigate a(r, z), we must apply the second result given on analytic continuation. Using it we can show that a(r, z) is a mermorphic function of ‘z.

--the finite part of A(z) at the pole z, (the concept of finite part is easily generalized to poles of higher order). For the purpose of regularization of integrals this result has been appreciably generalized. One of the strongest generalizations takes the following formz6 Let the functions a(x) and G(x) be defined on the whole of R”, where a(x) has compact support and G(x) is the product of a polynomial and a smooth function, positive throughout R”. Then the integral A(z) =

Representations

dx

JG>O

0 7. Formal Series and Asymptotic Series

(dx the volume element in R”), can be analytically continued to the whole complex z plane as a meromorphic function. If desired this result can also be formulated for integrals of functions on smooth manifolds. The regularization of a fairly broad class of integrals can be reduced to these theorems.” As an example we will discuss how to regularize the integral I= s R”

Q(xM(x)dx,

Now we pass on to the kinds of series with which, usually, there is connected no notion of convergence. The scornful attitude, left over from the nineteenth century, towards the term “formal series” is still fairly widespread. Related to this attitude is not a precise definition of terms, but rather a vague feeling that there is nothing to say about the convergence of such series. However, the notion of a formal series is quite interesting and we now define its precise meaning. In what follows we will discuss only formal power series.28 Let m be a multi-index in Z:, that is, m = (m,, . . . ,m,), where the mk are non-negative integers, and let z = (z 1,. . . ,z.) be a vector of symbolic variables. We will use the notations

(1.15)

where Q(x) is a function, homogeneous of order p, p > 0, smooth and positive on the sphere 1x1 = 1, and d(x) is a function of compact support. We introduce a parameter z by putting

Z(z)= 1

CQW-24(x) dx.

JR=

Iml =m,

Z(z) obviously converges for Re z > 0 and is an analytic function of z. To obtain (1.15) we must analytically continue Z(z) to the point z = - 1. Changing to polar coordinates I = 1x1and o = x/1x1, we obtain a, (1.16) r’“+“-‘a@, z) dr Z(z) = s0

p = z-1 . ..z. %I

+.**+m,,

Denote by P” the set of polynomials in z with complex coefficients. Obviously P” is a vector space. We define a topology on this space by giving the following notion of convergence. A sequence {PJz)}, where PJZ) = C&)z”,

where

is . called convergent to zero in P” if for any m there is an N = N(m), such that g p’,“) = 0 for all k > N. 26The assertion of a “very strong generalization” is not completely correct. In other generalizations stronger restrictions are imposed on the function G(x), but in return stronger results on the poles of the integral A(z) are given. The result given here was conjectured in [22] and proved in [2]. “A large number of interesting examples are discussed in [22].

$ “‘There is nothing functions.

to prevent

the definition

of a formal

series over any system of linearly

independent

26

M.A.

’

Evgrafov

I. Series and Integral

The space P”, with the topology just described, is not complete. By completing it we arrive at a complete topological vector space P” which is called the space

T(P) = P2 -

of formal power series.

Representations

q’(x)

+

z$

We will consider the action of this operator on polynomials depending on x. We will show that a formal series

The topology described above is called the topology of formal series.29 In the topology of formal series each power series is convergent. Suppose we have an operator T, in general nonlinear, which is defined on the set of all polynomials from P”, and suppose that this operator takes each such polynomial to a function T(P(z)) which is analytic with respect to z in some neighbourhood of z = 0. We will say that the formal series

P(z) = F p,(x)z”

P(z) with coefficients

(1.17)

E Fl,

m=O

whose coefficients p,(x) satisfy the recurrence relations

P(z) = &lzm PO(X) = 4(X)>

satisfies the equation T(P) = 0 if the sequence

converges to zero in the topology of formal series. This condition means that T can be extended from P” to P” and is continuous at the point P(z) E P”. It would be possible to extend this definition by considering operators T defined, not on the whole space P”, but on some subspace containing all partial sums of the formal series P(z). Let us look at some examples. Example 1. The formal series

P(z) = z - 1!z2 +

2!z3

-

..*

E P’

satisfies the equation z2p’(z) + P(z) - z = 0. In fact, put PN(z) = 5 (-l)“-‘z”(m

- l)!

m=l

Then

.

+

Pin-lb)

=

03

(1.18)

satisfies T(P) = 0. An application of T to a polynomial P E P’ again gives us a polynomial in z. Further, by applying T to a partial sum of the formal series (1.17), we obtain a polynomial for which, in view of (1.18), the coefficients of the powers of z with exponents less than the index of the partial sum are zero. According to the definition, this sequence of polynomials converges to zero, as the index of the partial sum increases, in the topology of formal power series. Hence, it follows that the formal series (1.17) satisfies the equation. The topology of formal series is very closely connected with a classical method of solving equations; the method of undetermined coefficients. The above formulation of a formal series satisfying an operator equation corresponds exactly to the fact that the method of undetermined coefficients produces recurrence relations.30 Thus, we have shown that for formal series there is a quite practical, but weak, concept of convergence. The position with asymptotic series is different-the concept of convergence is quite irrelevant for them. Asymptotic series are objects of a completely different kind. To define them we must begin by defining the notion of an asymptotic formula. Let E be a set in a topological space and let 5 be a limit point of the set. If f(x) and 4(x) are two complex-valued functions defined on E, then the forrnulae31

QN(z) = z’P;v(z) + P,(z) - z = (- l)NN!zN+‘.

According to the definition the sequence (- l)NN!zN+’ converges to zero in the topology of formal series, therefore the series satisfies the equation.

and

Example 2. Let q(x) be an infinitely differentiable function on an interval (a, b) of the real axis, and let T(P) be defined by the formula

mean respectively that 30At the time of Newton

29 At present it is customary to introduce a topology on a topological vector space by the assignment of a system of open sets, and not via the notion of convergence (that is, the assignment of a system of closed sets), since it requires the verification of fewer conditions. However, the idea of convergence is more intuitive and I prefer to introduce the topology in that way. The credit for introducing a topology on the space of formal power series belongs, apparently, to Krull [31].

kio P&)Pm-k(X)

f(x) = 0(4(x)),

x+<,xEE

(1.19)

f(x) = W(x)),

x+<,x~E

(1.20)

in the sense of the topology of formal series was spoken of as contrasting it with “convergence in the arithmetic sense”, roughly comparable to our notion of convergence of a numerical series. 31The symbols o and 0 were introduced by Landau at the beginning of the twentieth century and they have become quite firmly rooted in analysis despite the logical defects noted in the text. The attempts undertaken by Bourbaki [6] to replace these symbols has not yet had any notable success.

“convergence

in the algebraic

convergence sense”,

; lllllllllllllllllllllllllllllllllllllllllllll1lllllllllllIIIIIiIIIIIIIIl

M.A. Evgrafov

28

f(x) -= cw

.!L

I. Series and Integral Representations

(1.19*) O

and f(x) (1.20*) I @j I < c0Relations of the form (1.19) and (1.20) are called asymptotic formulae. Included in asymptotic formulae are also relations like (1.21) x+&x~E f(x) - 4(x), :*y,:;

which is easily replaced by an equivalent formulae of the form (1.19) x + 5, x E E.

f(x) - d(x) = 4&4)~

f(x) = o(m)9

If (1.23) holds for a function f(x) and, what is more, N is equal to infinity, then we say that f(x) has an asymptotic series.32 The whole collection of formulae (1.23) are then usually written as one, f(x) = J

c,h(x),

g(x) = oU(4)

ought to imply (but by no means do imply!) the equality of f(x) and g(x). The modern style would correspond more to the use of the inclusion sign rather than the equality sign. Fairly often, for one function f(x), we can write not one, but several, asymptotic formulae, each more accurate than the preceding one. We will consider this possibility in somewhat more detail. In order to talk about the accuracy of asymptotic formulae we must have a scale of comparison functions h(X)? 42 (4, . . . >MX)~

x + 5, x E E.

(1.24)

The definition given above makes no references to the question of the convergence of the series on the right in (1.24). This series may, with equal success, be either convergent or divergent. And, what is more, it may be convergent and its sum differ from f(x). The simplest example of an asymptotic series is a convergent power series f(x) = f

It must be recognised that the traditional method of writing asymptotic formulae in the form (1.19) and (1.20) is an abuse of the equality sign. In fact, in view of the transitivity of equality the formulae

29

c,xn,

x + 0,

for a function f(x) analytic at x = 0. One of the early sources of divergent asymptotic series was the Euler-Maclaurin formula referred to earlier in 86 of this chapter. Asymptotic series (or, to be precise, asymptotic formulae) have long been used for the purposes of calculation. To this end they are far more convenient than convergent series. The merit of asymptotic series was unquestioned even at the time of the hard struggle with divergent series. At that time the only justification for the existence of a formal series for the solution of an equation was the hope, that the series would turn out to be an asymptotic series for a genuine solution.33

Chapter 2 The Techniques of Operating with Series and Integrals

satisfying the natural conditions A+1(-4 = 44&))9

9 1. Newton Polygons

n = 1, 2, . . . , N - 1.

The asymptotic formulae themselves must take the form f(x) -fib)

- ... - f.(x) = O(d”bN,

(1.22)

where s.+1(x) = 44n,(x))9

n=l,2

,..., N-l.

n=l,2

,..., N,

The simplest case is when f,(x) = d&4~ where the c, are constant multipliers. form

Formula (1.22) can then be written in the n= 1,2,...,N

(1.23)

One of the very old methods of expanding a function in a power series was found by Newton when he had to expand the solutions of algebraic equations a2The modern notion of an asymptotic series was introduced by Poincare in [40], where he showed that the formal series satisfying linear differential equations with analytic coefftcientswere asymptotic series for certain genuine solutions of these equations. A little earlier Stiehjes had suggested a similar idea, but his definition was less successful and was not pursued (the term “semi-convergent series” suggested by Stieltjes was also unsuccessful). 33At present, even in elementary textbooks, questions of convergence are very clearly separated from questions concerning asymptotics. It is all the more remarkable therefore to meet, in quite modern monographs, terminological confusion regarding formal series. For example, certain authors call the formal series satisfying an equation an “asymptotic solution” of the equation (with no attempt to establish whether or not the formal series is an asymptotic series for a genuine solution). These connotations grossly misinform the reader, even when an accurate definition is given.

M.A. Evgrafov

I. Series and Integral Representations

in series. The method is not used so much for obtaining the whole series but rather for a sequential search for any number of the leading terms of the series. We will demonstrate the power of the method on two examples. For the first we choose the one that Newton himself had to solve. Let

In fact, in the contrary case, the largest term in the expression for P(x, y(x)) will obviously be different to zero (the question concerns the size of the terms for small x). In every concrete case this result enables one to find admissible exponents y. However, writing down an exponent using a formula is not so simple. Newton found an intuitive geometric method for finding the exponents, useful both for concrete calculations and for general arguments. Newton’s idea lies at the foundation of the area of mathematics now known as convex analysis. We will describe Newton’s construction using the now customary rudiments of convex analysis. Consider the pairs (&, vk) as points of the plane. In the (CL,v) plane form the conuex hull of the set of points

30

P(x, y) = t AkX@kYVk

(2.1)

k=l

where all the coefficients A, are non-zero, and pLkand v, are non-negative integers satisfying min (pk + vk) > 0,

(2.2)

k

and min pLk= min v, = 0.

(2.3)

k

k

Condition (2.2) means that P(O,O) = 0, and condition (2.3) means that P(x, y) is not divisible by x or y. We will seek a solution of the equation P(x, Y(X)) = 0,

y(0) = 0.

(2.4)

If (0,l) occurs amongst the pairs (pk, vk), then the search for the required solution presents no problems. We write the solution y(x) as a series y(x) = c1x + +x2 + c3x3 + ... with undeterminedcoeficients ck. Substituting this series into (2.4) and equating to zero all the coefficients of the power series obtained for P(x, y(x)), we arrive at recurrence relations for the coefficients of the series, which allow ck to be found if c1, . . . , ck-l have already been found. In the general case this method does not work. The simple example of the equation y” - x = 0 explains why: the sought after solution may not be expandable in a power series in x. We try for a solution of (2.4) as a series with fractional powers y(x) = UIXU’ + azxa* + . ..)

(2.5)

where a, # 0,O c a, < at c ... First of all we try to find the leading term of the series, that is, the exponent a1 and the coefficient a,. For this we use the following simple reasoning. A. In order that there is a solution y(x) of (2.4) satisfying the condition Y(X) - cxv,

x + 0,

with some c # 0, it is necessary that the quantity pk

+

Yvk,

k = 1, 2, . . . , N,

take its least value for more than one value of k.

bk,

vk’k),

k= l,...,N,

31

(2.6)

that is, the smallest convex set containing all of these points. This convex hull lies in the first quadrant. Because of (2.2) and (2.3) it does not contain the origin but does have points on both coordinate axes. We will call the segment of the convex hull of (2.6), which is visible from the origin, a Newton polygon. Subsequently we will denote it by L,. From the definition of convex hull each vertex of L, is at one of the points (pk, vk). Let I be a link of L,, that is, a line segment joining two adjacent vertices (both vertices are regarded as being in the link). Denote by K, the set of indices k = 19 **a, N for which (&, vk) lies on 1 (from the definition of a link K, contains at least two indices). If k E K,, k’ E K,, and k # k’, then the quantity y,= -pk-pk’ v, - V,?

does not depend on k and k’, but depends only on 1. Further we associate with a link 1 a polynomial Pl(x, y) defined by P,(x, y) = 1 AkXPkYYk. keK,

This polynomial is the collection of terms of the original polynomial P(x, y) which have (one and the same) minimal degree when y is replaced by xv*. Using this notation the admissible exponents are expressed easily and equations for the coefficients obtained. Corresponding to each link 1 of L, there is one value of the index a and a number of coefficients a # 0, for which a solution y(x) of (2.4) can exist satisfying x + 0. y(x) - ax’, The index a is equal to JJ~,and the values of a are the roots of the equation c ksK,

AkaYk= 0.

(2.7)

M.A.

32

I. Series and Integral

Evgrafov

This result resolves the question of finding the first term in the expansion of y(x) in a series (2.5). We now discuss the problem of finding the subsequent terms in (2.5). If the first term a, ~‘1 in the expansion (2.5) for y(x) has been chosen, then we write CI~as an irreducible fraction p/4 and introduce new variables y, = y - ulxal*

x1 = x1/q,

The old variables are expressed in terms of the new ones by the formulae x = xf,

y = y, + qxp.

= 2 &$y;;.

If a, is a simple root of (2.7) then obviously (0,l) will occur among the pairs (,& vi). In this case, as we know, the solution vi(x) can be expanded in a series of integer powers of x1 and recurrence formulae obtained for the coefficients of the series. This completes the expansion of y(x) in a series (2.5). If a, is a multiple root of (2.7), then we must again apply the procedure of constructing a Newton polygon. This gives us the second term of (2.5). It can be shown that this process terminates after a finite number of steps.34 As a second example we consider the problem of solutions of power growth for the system of differential equations dx - = Hy(x, y), dt

dy z = - Kh,

Y),

(2.8)

where H(x, y) = 5 AkXpk+ly"k+l, k=l

(we place the same conditions on the numbers A,, pk and v, as in the first problem). We are interested in solutions {x(t), y(t)} of (2.8) for which x(t) - at=,

t-+0

y(t) - bt@,

t-0

for constants a, b, c( and /? with a # 0 and b # 0. Denote by L the Newton polygon constructed relative to the set of points (2.6). For each link I we define, as before, a set of indices K, and quantities y, and also introduce the polynomial s4This

problem

has been investigated

in detail

in [54].

AkXfl*+lyvk+l,

analogous to the polynomial s(x, y). Again if we apply A and use the Newton polygon technique we arrive at the result: Corresponding to each link 1 of the polygon L are a pair of exponents (CI,/?) and a certain number of coefficients (a, b), for which a solution {x(t), y(t)} of (2.8) may exist which satisfies the conditions (2.9). The indices ct and B are defined by 1 pk

-

p= Ylvk’

yl pk

-

Ylvk’

where (pk, vk) are any points on the link 1 of L, and the coefftcients a and b satisfy the system of equations H;,(u, b) = aa,

k=l

33

keK,

UC -

Substituting these expressions into P(x, y) = 0 and cancelling, if possible, a power of x1, we arrive at a new equation Pi (xi, y,) = 0, where P1(xl,yl)

I&(x, y) = c

Representations

H;,(u, b) = - /3b.

If desired it is possible to look for a solution {x(t), y(t)} and a series expansion analogous to (2.5). The successive terms of the series may be sought in approximately the same way (the arguments become more complicated) but it is not always possible to prove convergence of the series.35 Calculation A, in some form or other, is used in the asymptotic analysis of arbitrary nonlinear equations, but applying the Newton polygon technique is comparatively rarely successful. In those cases when it does apply, the results obtained are deeper and far more interesting.36

$j2. Finding the Coefficients of a Power Series In the previous section the topic in question was the possibility of successive determination of the terms of a series, whereas here we will talk about a statement of a problem which is different in principle-the search for the general form of the coefficients of a series. Ignoring the geometric progression, the first example of a power series whose coefficients were found in general form was the binomial series due to Newton a(a - 1) (1 + ux)a = 1 + tlux + 12 uZxZ +

a@

-

w

-

2) u3x3

+

. . .

1~2.3

(a an arbitrary complex number). So long as it was only necessary in mathematics to deal with functions, obtained from an independent variable by the four arithmetic operations and the extraction of roots, the binomial series was the “The convergence question for this problem is far from simple. 36There are a number of interesting results in [32] and the work 32 (1976), 1-31.

of the same author

in Invent.

Math

M.A.

34

I. Series and Integral

Evgrafov

unique function for which the coefficients could be expressed in an explicit and fairly compact form depending on the index (of course, one does not count functions represented as a sum of binomial series, which also includes any rational function since they can be decomposed into a sum of partial fractions).37 The position changed significantly with the appearance of the so-called transcendental functions

log x, eX, sin x, cos x.

Representations

35

available; its use at the time reduced to a tautology-the derivative was defined by means of the power series). The possibilities of this method were quickly exhausted. The most effective method proved to be one found, apparently, by Euler. It consists of this, a differential equation is found that is satisfied by the function to be expanded and this equation is then solved by means of a series. We give two examples of this method. Example 1. Let us expand the function f(x) = eX into a series. It is not difficult to see that our function satisfies the equation

For the first of these was found the series38 log(1 + x) = 2 (-l)+ PI=1

f(0) = 1.

f’(x) = f(x), By substituting

the series

and shortly afterwards there followed the series

m

f(x) = “ZO hx” into this equation and equating the coefficients of like powers of x, we obtain the relations

and Xzn

x2n+l

cosx = z (-l)“p (2n)! ’ n=O

sinx = 2 (-1) n=O

12= 0, 1,2, . . .

The initial condition f(0) = 1 gives f. = 1, and then

Next the series arctanx

(n + l).L+l = f,,

(2n + l)!

f,=

= f (-1)“2, n=O

1.2.‘....n

=A.

Thus we have obtained the familiar series for eX.

and arcsinx = F (--lym.g II=0

(2n)!

were found. Even at the time of Euler the question of expanding a function in a power series was considered to be precisely the problem of finding explicit formulae for the coefficients of the series (of course, these explicit formulae might contain the gamma-function; otherwise it would have been necessary to reject many of the series written above, including the binomial series itself). The first method used to solve this problem was the integration of already known series (Taylor’s formula for the coefficients of a power series was not a7 At first glance it may seem obscure as to why seventeenth and eighteenth century mathematicians regarded the coefftcients of a product of two binomial series as much more complicated than the coefficients of one binomial series. There is a clear foundation for such an opinion: the coefficients of the binomial series are easily calculated successively by a simple recurrence formula containing only two terms; the number of terms in the recurrence formula for the coefftcients of the product of two binomial series increases with the index of the coefficient and this seriously complicates the calculation. 38This series was found by Mercator in 1668.

Example 2. Let us expand the function f(x) = (arc sin x)’ into a power series in x. The function f(x) is even and f(0) = 0. Therefore its power series takes the form: f(x) = $I %X2”. In addition, the formula sin x N x implies that a, = 1. By writing down the first and second derivatives of f(x) it is not difficult to discover that (1 - X2)f”(X) = xf’(x) + 2. Substituting the power series for f(x) into this relation and equating the coefhcients of like powers of x we find (2n + 2)(2n + l)a,+, - 2n(2n - l)a, = 2na,,

n=

or 1 7

4n-

a “+l = (2n + 1)(2n + 2)‘“’

n = 1,2,....

1,2,...

M.A.

36

Evgrafov

I. Series and Integral

22”-‘[(n - 1)!]2 (2n)! ’

n= 1,2,...

Consequently

f(0) = 0.

f(x) = xF(f(x)),

(arcsinx)*

m 2*“-‘[(n = 1

- 1)!]2

(2n)!

a=1

Then

X2n

By comparing the derivatives of a given function many differential equations can be written for it. Naturally there arises the question of exactly which differential equations enable us to obtain explicit formulae for the coefficients of the power series. A fairly complete answer to this question is given by the following result. Let f(x), analytic at x = 0, satisfy the differential equation ~A,k~mf’k’(~)

= c B,,,,“’

(both sums finite and all A,, non-zero). If there are only two different numbers among the differences m - k, formed with respect to the indices occurring in the sum on the left, then the coefficients of the series

f(x) = “.l Lx”,

where fi = F(0) and

iI= l-1) 2ni(n s,fl=r Cw-w~~,

f(x) = F f,X”T

Example 3. Let us expand into a power series the function f(x) satisfying the equation

can be expressed explicitly in terms of the gamma-function. The differential equations described in this result attracted a great deal of attention in the seventeenth and, particularly, the eighteenth centuries. They include the hypergeometric equation and the generalized hypergeometric equation. The general solution of these equations, in general, cannot be expressed in terms of the elementary functions (although they have coefficients of a simple kind). At the beginning of the nineteenth century Cauchy gave a new formula for the coefficients of a power series = “go fnx”*

of f(x) in a neighbourhood

of x = 0 and takes

(2.10) where I satisfies just one condition: f(x) must be analytic in Ix 15 r. The Cauchy and has turned out to be very useful in the theoretical investigations of power series, but in the search for the explicit form of the coefficients it gives comparatively little. True, with its aid the expansion in series

formula has great flexibility

f(0) = 0.

According to the result given above we have for the coefficients f, the expressions

n=O

This formula uses the analyticity the form

n = 2,3,...

Although this formula does not promise an explicit formula in the general case, in individual cases the integral can be explicitly calculated.

f(x) = xef’“‘,

f(x)

37

of rational functions and certain transformations has become somewhat simpler. In particular, with its use it is very simple to obtain the following result39, which gives several more cases of explicit formulae for the coefficients. Let F(t) be analytic at t = 0 with F(0) # 0 and let f(x) satisfy the equation

Hence, since a, = 1, we obtain a, =

Representations

iI=l s

nn-2

C&=-=27ci(n - 1) Ifl=, t”-’

Consequently

(n - l)!

n”-’

n! ’

f(x) = F qxn. n=1

.

In conclusion we mention one further aspect of the problem in question. In many cases, for those functions for which it is possible to find explicit expressions for the coefficients of the series, it is possible to find other, more cumbersome, formulae for these coefficients. Such cases are a rich source of highly non-trivial identities. Example 4. Let us prove the identity n (2k)!(2n - 2k)! 1 24”+‘(n!)2 c [k!(n - k)!]* ‘(2k + 1)(2n - 2k + 1) = (2n + 2)! k=O 39Thi~ result (although with the formula for the coefficients expressed in terms of derivative rather than integrals) was known long before Cauchy. In the literature it is called the Lagrange series or Bthann-Lagrange series. Many cases of the formula were known to Euler. The basic formula was published by Lagrange in 1770 and a generalization of it was published by Btirmann in 1799.

M.A.

38

I. Series and Integral

Evgrafov

Square the series for arc sin x, given at the beginning of this section. Cancelling similar terms we obtain as the coefficient of x2”+2 the left hand side of the identity over 22”. From the series for the function (arc sin x)~, obtained in example 2, we get as the coefficient of x~“+~ the right hand side of the identity over 22”.

0 3. Series of Partial Fractions Another problem which attracted a great deal of the attention of eighteenth century mathematicians was the expansion of meromorphic functions into series of partial fractions and into infinite products. Euler succeeded in finding the expansion of sine as an infinite product

Representations

function f(x), we can at once write its corresponding

39

series

?A. “Cl x - A,

(2.11)

Here 1, are the poles of f(x), that is, the points at which f(x) becomes infinite. (For simplicity we will subsequently always assume that the poles of f(x) are simple; however, the generalization to the multiple pole case presents no particular difficulty.) The lirst problem is that the series (2.11), in general, diverges. The second problem is that, when the series does converge in some manner, it is still necessary to find the difference between the sum and the function f(x). Both of these problems can be overcome comparatively easily if there is a number p such that the series (2.12)

and the cotangent as a series cotx=;+

converges, and if there is a sequence (r,>, r, -+ +co, such that 2x f n=l x2 - n27c2

Euler used these expansions widely and obtained many significant identities from them. In particular, expanding cot x - l/x in a series of powers of x in two ways, he found that $l;=gy

$lf=f.

Here we will discuss only the expansion of a meromorphic function f(x) into a series of partial fractions, since the expansion into an infinite product easily reduces to the expansion of its logarithmic derivative f’(x)/f(x) into a series of partial fractions followed by integration of the series obtained. The idea behind obtaining an expansion into a series of partial fractions is very simple. For a rational function f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and the degree of P(x) is less than the degree of Q(x), there is the so-called Lagrange formula

m=cg&.

n

Q(x) = Wx - A,)

(we are assuming that the denominator has simple zeros). The expansion of a meromorphic function into a series of partial fractions is obtained by transferring Lagrange’s formula from rational functions to meromorphic functions, We note the basic problems which arise in this translation. The representation of a meromorphic function f(x) as a ratio of two entire functions, as a rule, causes no problems in each concrete case, and so we will not dwell on this question. Let us assume therefore that in giving a meromorphic

n-tco. (2.13) max IfWl = W,P), 1x1 =rll An accurate and suitable method of solution was given by Cauchy, based on his theory of residues. The method consisted of calculating the integral

1

f(x)

dx

(I4 < m)

4hi I~I=~, xqx - z) (p an integer not less than the p of formula (2.12)) as a sum of residues, and passing to the limit as n + ~0.~’ The result obtained by this method takes the form: Suppose that (2.12) and (2.13) hold, and p is an integer larger than p. Then f(x)

=

;g

;f(k)(o)

+

“J

A,

(&

+

t

+

$

+ n

*** +

g n

>

The method used by Cauchy is easily applied to other similar problems.41 We give two examples. Example 1. Let us find the sum

s= F Ai2 II=1 where il, are the positive roots of the equation tan x = x. 40The proof that the integral tends to zero was obtained by a simple direct estimate. 41 Many other examples of the use of Cauchy’s method are given in textbooks on analytic functions theory. A large number of examples of this type can be found in [18], [41], [49] and [SO].

M.A.

40

Evgrafov

I. Series and Integral

Representations

41

pass to the limit as m + co, we obtain

Consider the function g(x) =

sinx - xcosx

1 jt--;s(A(x)

X3

It is not difficult to show that this function has only real zeros and that they are of the form f A,,, where A, are the roots of interest to us. Therefore the function f(x) = !g

= -

sin x 3 - -3 cos x - (l/x) sin x x

From which the required formula follows. The function cot rrx has an interesting property which enables us to extend somewhat the possibilities of the Cauchy method. On the circle 1x1 = m + 4, where m is a positive integer, the following estimates hold for cot XX

has poles only at the points x = +A,, and the residues at these poles are equal to 1. On the circles 1x1= mlr, for large values of the integer m, it is easy to obtain the estimate max If(x)1 = O(l), m + co.

cot nx + i = O(e-l’mXI),

ImxLO,

cot 7cx- i = O(e-l’mxl),

ImxSO.

Using these estimates, and the Cauchy method, we can prove the following result. Let f(x) be analytic in the whole complex plane except for a finite number of poles a, , . . . , a,, not lying on the real axis, and for sufficiently large 1x1 suppose that

1x1 =mn

Therefore, evaluating the integral 1

rcxg 271i IfI=nm 4

where E(X) + 0 as x + fco. If the integral

f’(0) + 2s = 0.

m f(t) dt s -a,

We calculate f’(O) and find that S = 0.1. converges then the series

Example 2. Let us prove the formula

where A(x) is an arbitrary rational function whose denominator has zeros at the points x = ak and for which the degree of the numerator is at least two less than the degree of the denominator (in order that the series make sense there must be no integers among the ak). For the function cot rcx it is easy to obtain the estimate max lcotrrx) = O(1) 1.X=m+1/2 (m a positive integer), and from the condition

IA(x)1 = O(mm2),

2ni -4

1x1 =m+1/2

on A(x) we have m+c9.

f(t)dt

- n 1 Imak>O

-

71 1 Imak
res {f(x)(cot rcx + i)}

x=ak

res {f(x)(cotrrx

x=ak

- i)}.

Example 3. Let A(x) be a rational function not having poles at the integers, and let 8 be a real number in the interval (0,274. We will prove that

A(x) cot rcxdx

as the sum of the residues at the points x = ak and x = 0, + 1,.

converges, moreover

As an example we mention a special case of this theorem when the integral can also be calculated using residues, and the sum of the series is expressed in finite form.

Ixl=m+1/2

Therefore, if we calculate the integral 1

G?< 275

If(x + iy)l -c c(x)ea’YI,

as the sum of the residues at x = 0 and x = + 2, (for all n for which /&I < rrm), and then passing to the limit as m + co we get

max

71x)+ ; =E ‘4(n) = 0. ” 03

“Em A(n)eine = -n 1 ,‘=“,” (A(x)eiXe(cot 7tx - i)>, k

f m, and then

where the ok are the poles of A(x).

M.A.

42

Evgrafov

I. Series and Integral

We calculate the integral

43

value to 10 places:

1 4271i

Representations

y = 0.5772156649.. .

A(x)eiXe(cot XX - i) dx ItI =m+

l/Z

From (2.14) it is easy to obtain

as the sum of the residues at x = ak and x = 0, f 1,. . . , + m, and pass to the limit as m + co. The proof that the integral tends to zero is no more complicated than in the previous cases. In the upper half-plane the boundedness of the cotangent is used, and in the lower half-plane, the second of the above mentioned estimates is used. In addition, it is necessary to apply Jordan’s lemma.42 The rest of the proof is as in example 2.

0 4. The Gamma-Function

and the Euler-Maclaurin

Formula

T’(x) p=-y-;-ea4 and

$logT(x)=n=O f -!(n + x)’ From the same formula (2.14) it is easy to obtain the identity T(x)T(l

Binomial coefficients and factorials were known to mathematician long before the advent of differential calculus. However, the extension of these functions from an integer variable to a real (and even complex) variable required the development of the apparatus of analyticity. This problem was solved by Euler who created the gamma-function T(x), for which r(n + 1) = n!, and the beta function B(x, y), related to the gamma-function by the relation

“4 n(n + x)

- x) = --L sin 71x’

from which it follows, in particular, that r(3) = A. Afterwards Euler obtained an integral representation T(x)

(2.15)

for the gamma-function

=

which easily reduces to the formula 00

The first formula that Euler gave for the gamma-function r(x)=;g{(l

+:)x(1

T(x)

took the form43

s 0

+$l}.

At present it is preferred to write the formula a little differently: (2.14) where y= lim 1 +i+k+*** n+m (

1 +;-logn

tX-’ e-‘dt.

=

>

The constant y is called the Euler constant. Euler himself evaluated its value to 20 decimal places, the well-known nineteenth century English calculator Adams found 200 places; at present approximately 2000 places are known. We give its 4ZA fairly elementary estimate of the integral, presented in all the textbooks, concerned with the calculation of integrals by the method of residues. 43 The history of the investigation of the gamma-function is presented in many books on the history of mathematics. For the majority of the references the information in [55] is sufficient.

It is this formula which is now taken as the definition of the gamma-function. Euler also obtained several integral representations of the beta-function. Nowadays the following formula for the beta-function is regarded as fundamental 1 t”-‘(1 - t)Y-’ dt. Hx, Y) = s0 Arising in connection with the gamma-function was a problem whose solution (given by Euler) was a topic of study for many analysts during the last century. The problem initially was to find a simple formula which was a good approximation to the gamma-function for large values of the variable. Its solution was the so-called Stirling’s formula T(x) - $G

xxe112emx,

x+

+co.

To reline this formula the series log~(x)=xlogx-r-flogx+;log2n-&+--... named Stirling’s series, was used.

1 720x3

M.A.

44

I. Series and Integral

Evgrafov

Euler posed and solved the more general problem of finding a formula which was a good approximation to the sum

0)

-=

z

1 +;-

1 - eP

Representations

45

2 (-l)k&P. k=l

Therefore the expression obtained for F’(x) can be written as

= i f(k) k=l

for large values of n. This formula, named the Euler-Maclaurin Euler-Maclaurin series, takes the form44

formula

F’(x) = f(x) + if’(x)

or

F(x) =sxf(t)dt +c+if(x) -kzl (-ljk&fc2k-1Yx) (2.16)

0

where the Bk are the Bernoulli numbers (we will give their definition later). Stirling’s series is the particular case of the Euler-Maclaurin series when f(x) = log x. Eulers arguments (completely non-rigorous, both from the viewpoint of his contemporaries and the modern viewpoint) in the derivation of (2.16) were fairly simple. We reproduce them here because they served as a model for many generations of mathematicians, who were not troubled by lack of rigour in proofs. Assume that f(x) is defined for any values of the variable and not just for the integers. The question of finding

- kg (- 1)’ &f@*‘(x).

Integrating we arrive at (2.16). The constant C in the Euler-Maclaurin formula is not determined in the above argument. Euler devoted a great deal of effort to finding this constant for various function f(x). In particular, he found many interesting methods of accelerating its calculation. An analytical expression for C was found significantly later. The divergence of the Euler-Maclaurin series (and, in particular, Stirling’s series) was known to Euler and his contemporaries. However, the series was often used in calculations because of it asymptotic properties. The validity of the asymptotic formulae obtained from the Euler-Maclaurin series was proved not just for individual functions f(x), but also for many classes of functions. It is interesting to note that for the Stirling’s series itself a modification has been found which contains only convergent series: ck f k=r(x+l)...(x+k)

log~(x)=xlogx-x-;1ogx+

0) = k f(k)

(2.18)

k=l

where

will be solved exactly if we find a function F(x) which satisfies F(x) - F(x - 1) = f(x). In the solution

of this equation,

(2.17)

for brevity, we will denote the differential

d operator - by D. By Taylor’s formula dx

F(x - 1) = 2 y m=O

1

l

Ck=s

o

x(x + l)...(x

In particular, it is clear from (2.18) that Stirling’s formula T(x) - ~*x”-1’2e-”

.

D”F(x) = emDF(x).

Therefore (2.17) can be written as

is valid not only for x -+ +co, but also for x + co in any sector Iarg XI 5 8,8 < rr. Replacing x by x + iy, we easily obtain an asymptotic formula for the behaviour of the gamma-function on lines parallel to the imaginary axis: Ir(x

J+yx)

= f(x)

which gives us

+ iy)l N JGly(x-1’2e-(n’2)~y~,

F’(x)= & The Bernoulli numbers Bk are defined by the Euler-Maclaurin

formula,

see [27].

f CG

1

T(x) = 271i s L

(2.19)

Y-+*00

Of the other formulae concerning the gamma-function worthy of mention: 1

44 Regarding

+ k - 1)(2x - 1)dx.

s

Hankel’s

formula is

ezzmx dz.

Here L is the boundary of the half-strip IIm zI < E, Re z < E (E an arbitrary positive number) traversed so that the half-strip remains on the left; the function zsX, under the integral, is taken to be exp( - x log z), with the principal value for log z.

M.A.

46

Evgrafov

Hankel’s formula gives a representation plane.

5 5. The Zeta-Function

I. Series and Integral

for (l/T(x))

and the Dirichlet

in the whole complex

Series

In addition to the gamma-function and the beta-function, Euler introduced yet another function, the analysis of which occupied not only him, but also a number of other outstanding mathematicians right up to the present day. The function in question is (2.20) later called the Riemann zeta-jiinction.45 Closely related to the study of the zeta-function is the analysis of series of the form

called Dirichlet series. The Dirichlet series (2.20), which defines the zeta-function, converges in the half-plane Rex > 1. One of the problems which attracted Euler was that of the analytic continuation of the zeta-function to the whole complex plane. The formula

(1-

21-*),3x)

Using (2.21) we get the series

where ,u(n) is the Mijbius function, defined as follows: 1. ,LL(l) = 1; 2. p(n) = 0, if n is divisible by the square of any integer other than 1; 3. p(n) = (- l)k, where k is the number of prime divisors of n, if n does not fit into either of the first two categories. This series also provides no opportunity for analytic continuation of the zeta-function. Riemann found several simple integral representations of the zeta-function, from which he was able to accomplish the desired analytic continuation. The most commonly used of these representations is

w = & J: g,

Using this it is no longer difficult to show (see for example the simplest result on regularization of integrals in Ch. 1, 6 6) that the zeta-function is analytic in the whole complex x plane, except for the point x = 1 where c(x) has a simple pole. Another method of analytic continuation of the zeta-function makes use of the functional equation [(l - x) = i(x)21-Xnxr(x)cos7.

(2.21)

- P-T,

P

where the index p is taken over all the prime numbers. This formula played a major role in many questions, but did not help with analytic continuation since the product (2.21) also only converges in the half-plane Rex > 1.

l(x) = kx(x - 1)71-‘“‘2’r

5 C(x). 0 It is clear from (2.21) that the zeta-function is never zero in Rex > 1. From (2.22) it is easy to deduce that in the half-plane Re x < 0 the zeta-function is zero only for x = -2, -4, -6,... (th ese are called the trivial zeros of the zetafunction). Apart from the trivial zeros c(x) has zeros only in the strip 0 < Rex < 1, called the critical strip. The famous Riemann hypothesis is the following. All zeros of the zeta-function which lie in the critical strip lie on the line Rex=*. Up to now the Riemann hypothesis has been neither proved nor disproved. The asymptotic behaviour of the zeta-function along lines parallel to the imaginary axis can be analysed fairly easily if the lines do not lie near the critical strip. Let us introduce the function k(x) = lim sup log ‘i:

full expositibn

of results concerning

the zeta-function

(2.22)

,I

gives a continuation of the zeta-function to the half-plane Rex > 0, since the Dirichlet series in this formula converges uniformly for Rex > 0. Although the sum of the latter series, as we have seen, is an entire function Euler failed to obtain an analytic expression for it which could be used in the left half-plane. This was done only much later. Euler did succeed in finding an expansion of the zeta-function as an infinite product

OA

Rex > 1.

This equation is equivalent to the evenness of 5(x + i), where

= 2 0”’ PI=1

lxx) = flu

Representations

can be found

in [Sl].

y++m

l iy)‘,

-co<x
M.A.

48

Evgrafov

I. Series and Integral

It is easy to deduce directly from (2.22) or from (2.21) that x > 1.

k(x) = 0,

From the functional equation (2.22) and using (2.19), it is easy to deduce that x < 0.

k(x) = $ - x,

Ojxil.

F akemAkX k=O

Initially only real exponents 1, were discussed, but subsequently series with complex & came to be studied.

The LindelSf hypothesis is that 44

=

3 o 7

x,

these theorems

see [49],

[Sl]

or [19].

$6. The Mellin Transform

x < 3; x 2 3.

Although the Lindeliif hypothesis is weaker than the Riemann hypothesis it also, up to now, has been neither proved nor disproved. The analysis of the zeta-function plays a major role in number theory, since many number theoretic concepts obtain an analytic interpretation via the zetafunction. In conclusion, we will say a few words on the convergence theory for Dirichlet series. The analysis of these series greatly influenced the formation of the notion of convergence of function series in the nineteenth century, since in them, for the first time, were noted certain interesting properties absent in power series. The fundamental result on convergence of Dirichlet series is stated as follows. If the series 1 c,neX converges (absolutely converges) at a point x = c, then it converges (absolutely converges) in the half-plane Re x > Re 4’. It follows from this that there are numbers c and c* with the properties: for Rex > c (Rex > c*) the series 1 c,n-’ converges (absolutely converges) and for Rex < c (Rex c c*) diverges (does not converge absolutely). The numbers c and c* are called, respectively, the abscissa of convergence and the abscissa of absolute convergence for the Dirichlet series. The notions of abscissa of convergence and abscissa of absolute convergence are very similar to the notion of radius of convergence of a power series. An essential distinction between Dirichlet series and power series is that the abscissa of convergence of a Dirichlet series, in general, does not coincide with the abscissa of absolute convergence. It turns out that the inequality c 5 c* 5 c + 1 holds, moreover c* = c + 1 is possible, as is shown by the series c( - l)“ndX. Another essential difference between Dirichlet series and power series is that a power series must have at least one singular point on the boundary of the disc of convergence, whereas a Dirichlet series with a finite abscissa of convergence 46Regarding

49

4 may represent an entire analytic function. An example is again provided by the ; series C (- l)“VX. At the beginning of the twentieth century the theory of generalized Dirichlet +’ series began to be vigorously developed.47 These series have the form

By theorem46 from the theory of entire functions k(x) is convex downwards. This gives us the inequality k(x) 5 G,

Representations

At the beginning of the nineteenth century series ceased to be the fundamental tools of analysis. Their position began to be occupied by the integral transforms of Laplace and Fourier, to which, at the end of the nineteenth century was added ,the Mellin transform. The Laplace transform of a function f(t) is the integral co F(z) =

f(t)ePdt.

s -* The Fourier transform is obtained from the Laplace transform by the simple replacement of z by - it. The Mellin transform of a function 4(x) is the integral @J(s)=

om4(x)x’-’ dx. (2.23) s It is easy to see that the Mellin transform of +4(x) is the Laplace transform of f(t) = +(e-[). Clearly all information referring to any one of these transforms can be easily translated to refer to the other two. The separation of the Laplace, Fourier and Mellin transform is purely traditional. The Mellin transform is used traditionally in work with analytic functions and, therefore, in the theory of this transform the possibility of deforming the contour of integration is used widely (based on Cauchy’s theorem on the independence of the integral on the path). Mellin transform theory is usually regarded as part of the theory of residues. The originality of Mellin transform theory starts with the inversion formula. The classical result on the inverse Mellin transform is most conveniently formulated as follows: 47There

is a modern

survey

of Dirichlet

series in [33]. Of the earlier

books

[34] is of interest.

M.A.

50

Evgrafov

I. Series and Integral

<, to denote the integral

let 4(x) be analytic for 0 < x < 00, and satisfy the conditions K4 = o(x-a),

x + 0,

w

x-+00,

= 0(x-?,

{‘+im

@(s)x-“ds,

u
(2.24)

a-im

The main disadvantage of this theorem is that it does not establish a one-toone mapping of one function space to the other. Therefore we introduce immediately function spaces which are more suitable for our purposes. Denote by A4& the vector space of functions d(x) which are analytic in some sector V,, = {x: largxl < y’}, Y’ ’ Y9 and satisfy the conditions $4(x) = 0(x-“‘), (25(x)= 0(x-Q

exp(z log( --4) dx, sf(x)

taken over the boundary 8D of any simply-connected domain D containing the real axis chosen so that f(x) is analytic within D and on its boundary. The boundary 8D of D is assumed to be traversed in such a way that D lies on the left. The value log( - x) is taken to be the principal value. The version of the Mellin transform of interest to us is associated with the formulae

and

Y(s) =i2711 4(x)( -4-l dx s,=‘o,

(2.23*)

(2.24*)

X+O,XE

v,.,

x + co, x E v,.,

for any u’ > a and p’ < B. If required it would not be difficult to define a topology in this space, but for us that will not be necessary. Further, denote by A?& the vector space of functions Q(s) analytic in the strip CI< Res < /? and satisfying the condition I@(s)1 < C(a’, P,)e-y”‘ms’,

51

dD

where CI< j. Then the function Q(s), defined by (2.23), is analytic in the strip u
Representations

a’s Res 5 p’,

for some constants C(a’, p’) < co and y’ > y, for any a’ > a and r < j? (the constant y’ does not depend on u’ and p’). With these notations we have: If 4(x) E M& then a(s) E n&i, and the inversion formula (2.24) holds. If @p(s)E fi&, then the function d(x), defined by (2.24), belongs to M&, and 0(s) is expressed in terms of d(x) by (2.23). These two results mean that the Mellin transformation is a one-to-one mapping of M& onto A?&. In working with the Mellin transform we most often have to deal with functions 4(x) analytic not only for x > 0, but also at the point x = 0. In these cases it is sometimes more convenient to use a somewhat modified Mellin transform. In order to write down the formula for this transform we have to introduce some new notation. Let f(x) be analytic for x 2 0. We will agree to use the symbol.

mj-(x)( -x)‘dx sC-0)

In order to formulate the inversion theorem for this version of the Mellin transform it is more convenient to use slightly different spaces. Denote by M; the vector space of functions 4(x), analytic in some sector v = {x: larg(x + &)I < y’},

22> 0, Y’ > Y,

and satisfying qqx) = 0(x-@‘),

X+O,XE

vy!,

for any p’ < p. Denote by &$ the vector space of functions Y(s), analytic in the half-plane Re s < B and satisfying the conditions Y(k) = 0,

k=

1,2 ,..., p,

where the integer p is defined by p + 1 2 /I and p < /I, and 1y+)l < ~(~)~~lResl+(~-y’)l~msl, Ressp’, for some constants a 2 0, y’ > y and C(F) < co, for any p’ < j3 (the constants a and y’ do not depend on p). Under these condition we have the following result. If d(x) E MJ with b > 0, then Y(s) E fii and (2.24*) holds with any crsatisfying o 0, then the function 4(x) defined by (2.24*) does not depend on 0 for 0 c B < /I and belongs to Ms. Then Y(s) can be expressed in terms of d(x) by (2.23*). This version of the Mellin transform can be extended to the case /I S 0, but (2.24*) then takes a slightly different form. Namely

M.A.

52

I. Series and Integral

Evgrafov

<(x)x-‘-Ydx = ‘-‘-‘. som

(2.24**)

hereq=[-P],a-q-l 0. Then the integral in (2.23*) converges for s 5 0, in particular, when --s is a non-negative integer n. In this case the integrand in (2.23*) is a single-valued analytic function with a pole at x = 0. By Cauchy’s theorem the contour of integration can be replaced by a sufficiently small circle 1x1 = p to obtain

tl)“+‘w dx = (l)n+l P’(O) V-n)=& Xn+l s

T(y + 1)smlry’

Making

the substitution

s

Thus (2.23*) enables us to extend the coefficients of the analytic function b(x), defined as functions of an integer variable (the index), to complex values of the variable. The same formula, if desired, can be used to define the notion of a fractional (even complex) order derivative of an analytic function. The need for such derivatives often occurs in various branches of analysis. We give an example of the use of the Mellin transform in the calculation of definite integrals.48 Example 1. Denote by l(x) the unique positive solution of the equation <er = x,

53

O
x = <et in the latter integral we arrive at the formula

om(< + l)t-Ye-‘-Yc&

=

1 yy+‘ZJy + l)sin7cy’

O
The method used in this example enables us to obtain highly non-trivial integrals starting from any power series expansion with explicit formulae for the coefficients. Even such a simple series as the geometric progression

n!

1x1 =P

Representations

-=

1

x+a

F t-1)” --

”

n=o a n+1

a>0

3

gives the well known, but not entirely trivial, formula

s

* xsel dx = 7caS

O<s
sin xs ’

0 x+a

In conclusion we mention one more version of the Mellin useful in the study of Dirichlet series. Let

l)(s) = F l&n-,

x > 0.

transform which is

Res>a

n=l

From example 3 of Chapter 2, Q2, it follows that t(x) has the expansion

and

g-1 5(x)

=

g1

Rex > 0.

t-l)“-‘~c

valid for 1x 1 < l/e. Using Stirling’s formula (see Ch. 2 6 5) it is not difficult to verify that the function y(s) = t-v-l r(1 - s)

Then we have the formulae (2.26)

and

belongs to iii 0””. Therefore, by (2.24*) with q = 0, we can write

(-s)-s-l 1 +m -x)s-l dx, s 5(x)( r(1 - s) = 5Z

d(x) = &

On the other hand, for Re s > - 1, we have _1

+* <(x)(-x)s-l

271i s t-oj

dx = -F

+m <(x)x’-1

!P(s)r(s)x-Sds,

0 > a.

(2.27)

a-im

Res < 0.

(-0)

s”im

(2.25)

Putting $(s) = 1 in (2.26), we arrive at the well known (see Ch. 2 $4) integral representation of the gamma-function. Putting I&) = c(s) we arrive at the integral representation of the zeta function mentioned earlier (Ch. 2 0 5).

dx.

s0

0 7. Integral

Representations

for

the Sums of Power Series

Therefore 48 I have not come across this example,

or the use of this method,

in the literature.

Among the basic reasons why power series came to lose their popularity in analysis were the difficulty in analytically continuing them outside their disc of

54

M.A.

I. Series and Integral

Evgrafov

convergence, and the analysis of their singular points. Both of these problems are considerably simpler to resolve when the function in question is represented, not as a series but, as the integral of an analytic function with respect to a parameter, and the dependence on the parameter is also analytic. Therefore, in the nineteenth century, began the search for the possibility of representing the sum of a power series as an integral with respect to a parameter. We will now expound some of the methods used to this end. To start we will discuss a simple example. Example 1. Let us find an analytic continuation

for the sum of the series

outside its disc of convergence 1zI < 1. From the integral representation of the gamma-function the formula 1

1

na - - m

O”

tam’ e-“’ dt,

gives us an analytic continuation of f,(z) to the whole complex plane with a cut along the ray (1, toe”) (for 181> 7c/2this continuation leads us back into the circle Izj < 1 after passing round z = 1). If desired we could analyse the behaviour of f(z) in a neighbourhood of z = 1, where it has the expansion f,(z) = (z - 1)“~‘(a, + ar(z - 1) + . ..) + b, + b,(z - 1) + 6 The following considerably generalizes the method used in the example. : a4 Let p(x) belong to A4$ (Ch. 2,§ 6), where 0 :::I a = 0, y > 742. P > 0, 8 P; in Then for the sum of the power series (2.28)

) I there is the integral representation

a > 0. I

x-SdS g(x) =LsO+icc -@(s)

where

27ci a-im

which transforms to

It is immediately clear that the latter formula gives an analytic continuation of f,(z) to the whole complex z plane with a cut along the ray (1, +co). In fact, the integrand depends analytically on z, if z # 5 and 5 lies on the mentioned ray. Furthermore, this formula permits us easily to analytically continue f,(z) even across the cut (1, +co) (but not through the point z = 1). In fact, by Cauchy’s theorem, the path of integration may be changed to any other with the same endpoints provided that on deforming one curve into the other we do not pass through singular points of the integrand. 49 In particular, taking z to lie on the interval (0, l), we may replace the cut (1, +co) by the cut (1, coeie), where - rc < 0 c rc. The formula

of the question in [18].

of analytic

55

it is easy to derive

s0 Substituting this expression for na into the series for f,(z) and interchanging the order of integration and summation, we arrive at the formula

49 There is a detailed account of the contour of integration,

Representations

continuation

taking

account

of deformation

T(S)

’

acoc/3,

and G(s) is the Mellin transform of p(x). Sometimes a somewhat simpler, but less effective, method can be successfully applied. Let D be a bounded, simply-connected domain lying in the strip 1Im [ 1< rc, and let h(c) be analytic in the complement of D with respect to the extended complex plane. If p(x) takes the form

then, for the sum of the power series (2.28), there is the integral representation (2.29)

It follows from (2.29) that the singular points of (2.28) are fairly simply related to the singular points of the function h(c) by which p(x) is expressed. Roughly speaking, this connection is that each singular point 1 of h(l) corresponds to the singular point e” off(z), and the singular points of both functions have approximately the same character.50 “There

are various

theorems

of this kind

in [3].

M.A.

56

I. Series and Integral

Evgrafov

Representations

57

Integral representations of the sums of power series are not used solely for analytic continuation. Research into the behaviour at infinity of an entire analytic function, represented by an everywhere convergent power series, also presents serious problems which are noticeably simplified by the use of integral representations. One of the simplest representations used to this end takes the form

More complicated, but also more effective, is the integral representation based on the Abel-Plana formula.‘* Let D be a domain in the complex s plane containing the ray (a, +co), where 0 < LX< 1, and the half-strip

(2.30) -1
Suppose that D is bounded by two simple curves C+ and C-, going from a to infinity and not intersecting the real axis (C’ goes above the axis, C- below). Then the following is true. If a function p(s), analytic in D and on its boundary curves, satisfies the condition

“$O(-l)“B(n)z”=l

Ip(

< Mexp(aIResl

The representation

+ (rr - s)IImsl},

E > 0.

(2.3 1)

(2.30) works fairly well when the function

Res > CI~,

Ip(

bsl

< MeaRes,

< &.

s E D,

for some constants M and a, then

f(z) = f (- l)“An)z”

zl p(n)z” = jrn p(x)z”dx a

is bounded as z + cc along the positive real axis. However, for functions which grow strongly in this direction (2.30) does not work.51

+ s

‘(‘)”

2nis c+

_

1

ds

e

Example 2. We will investigate the behaviour as z + +cc of the function

where l/2 < p < 1. It is clear from Stirling’s formula that (2.31) is satisfied for p > l/2. As (r we can take any number greater than - p. A direct estimate of the integral gives us

If(

= w7?

z+ +co.

To obtain a better estimate we must move the line of integration as far as possible to the left. Since there is a pole of the integrand at z = -p we cannot move the contour to the left of this point without changing the value of the integral. However, translation is possible if we take into account the residue at this pole. Taking account of the residue at s = - p, we obtain the formula z+ +co. (2.32) + 0 ; ) 0 This formula can be made more precise if we take into account the residues at subsequent points of the integrand. We remark that the same considerations permit us to prove (2.32) not only for z + +co, but also for z + 00 in the sector largzl < x - 42~. Outside this sector f(z) grows, and (2.29) becomes unusable because of the divergence of the integral. f(z) =-71z-p p sin RP

51 There are many

examples

of the use of this representation

in [24].

In (2.33) the integrals over C+ and C- comprise the remainder term. Its estimate becomes more precise the more to the left we move the contours C+ and C-, without disturbing the convergence of the integrals. A completely acceptable (power growth in z) estimate is obtained if we can take the curves C+ and C- to be the rays (cr,cr+ ice) and (~1,CI- ice) parallel to the imaginary axis. This becomes possible if we impose more restrictive conditions on p(s). If p(s) is analytic in the half-plane Re s 2 CI,where -cc < a < 0, and satisfies therein the condition Ip(s)I < M.exp(alResl

+ (27~- .s)lImsl),

E > 0,

then

za An)z” = O3d4z”dx + Wzl”), sa

(2.33*)

Z-CO.

The first integral on the right in (2.33) and (2.33*) is analysed by the usual steepest descent method. The integrals over C+ and C- in (2.33) can also be analysed by the steepest descent method, if (2.33) cannot be changed into (2.33*). In the example given of the use of the Abel-Plana formula we will not carry out the analysis of the integral by the steepest descent method but will make use of a ready-made result.53 “For more detail on the Abel-Plana S3This integral has been investigated, there are a few analogues.

formula see [19], [27]. for example, in [17]. In [19] this example

is omitted,

however,

M.A.

58

Evgrafov

I. Series and Integral

ngo P(W = g Ia

Example 3. Let us examine the behaviour as z + co of the function F,(z) = i

n-z”,

m=--4) 0

f < y c 4

(the restriction y > 2/3 is not essential; it is imposed only to simplify the writing down of the final result). The function p(s) = e -YS’OgSis analytic in the half-plane Re s 2 a for any a > 0, and from the easily verified inequality 5 :[Irns]

- Res(log(Res)

- 1)

it follows that (2.33*) is applicable if y c 4. The integral is easily analysed by the steepest descent method and there is the asymptotic formula

m-bXZXdX =&&z1,2Y exp(izlly)(l s01x

+ C(i))

21’2yexp {:zl/Y} + 0 ( [z/(1/2yf-1exp(izlly))

+ 27cm)) dx.

0 8. Laplace’s Method

+ O(lzl”), One effective method of solving linear functional

equations is the Laplace

method. It consists of this, a solution is looked for in the form

which is valid as z + co, largzl 5 rc.Therefore (2.33*) gives

F,(z) = $i+.

p(x)lzl”exp{ix(argz

59

The Poisson formula does not represent the sum of a series as a single integral; it contains an infinite series of integrals. However, as a rule, these integrals are far from equal in value. Usually one (or two) of them is the leading term of an asymptotic formula, and the rest a remainder. The Abel-Plana formula consists of amalgamating the integrals in the Poisson formula into three groups: corresponding to m = 0, values of m < 0 and values of m > 0. That said, the Poisson formula is valid only for functions p(x) which do not tend to zero too quickly (more slowly than exp( - ax2)). For a function p(x) which does tend to zero more quickly the picture changes drastically.5s

n=l

Re(-slogs)

Representations

y(x) =

+ O(lzl”).

$(z)e”“dz,

(2.34)

sC

where the contour of integration is not assumed to be known in advance, and is to be found in the process of solution (the search for the contour is usually a fundamental part of the process of solution in the Laplace method). Concerning C it is only assumed a priori that it satisfies the following condition: A. At the endpoints of C the integrand and a sufficient number of its derivatives are zero. To save space we will write (2.34) as

It is necessary to keep both remainder terms since the value exp y-z l/Y (e > may converge either to zero, or to infinity, depending on the argument of z. Had we not made the assumption y > 213, then it would have been necessary to discard the second remainder term in the asymptotic formula for argz greater than 3rcy/2. The function F,(z) can also be investigated for y > 4, but in that case it is necessary to use (2.33). The leading term in the asymptotic formula remains the same, but there are other exponential terms in the remainder. In conclusion we mention the Poisson formula.54

Y(X) 6 d(z)

If condition

(Cl A is satisfied on a contour C, then (2.34) implies y’k’(X) 1 zkqqz)

(2.35)

(Cl

xky(x) 1 (- l)k#k)(z) valid for any function $(x) satisfying the condition m -a, IW)I dx < a~. f In its applications to power series this formula takes the form s4 For more details

on the Poisson

formula,

see [19],

[SO].

(2.36)

z for any positive integer k. From Laplace’s method the so-called operational calculus was developed, in 2 which the contour C was taken in advance to be the whole real axis, or its positive : part. The operational calculus is now a widely developed branch of mathematics, :~ provided for in textbooks and reference books. We are only interested in that G

$ ‘sThe asymptotic behaviour i,-5. comparatively little attention. 1

of series of functions p(n) which tend to zero very Some examples are discussed in [24].

quickly

has received

60

M.A.

I. Series and Integral

Evgrafov

part of Laplace’s method which cannot be put within the framework of operational calculus.56 We consider first of all one of the best known examples of Laplace’s method -the construction of a fundamental system of solutions for the differential equation (2.37)

where uk and bk are constants.57 It is easy to deduce from (2.35) and (2.36) that a function y(x) represented by (2.34) satisfies (2.37) if d(z) satisfies (2.38)

NW(z) = Cm) - A’(mw, where A(z) = i

ukzk,

B(Z)

k=l

= i

bkZk.

k=l

Equation (2.38) has a unique (up to a constant factor) solution, which can be written in the form

d(z)= “nl(z- z,)- ‘=P

.

(2.39)

Here zl,. . . , z, are all the different zeros of the polynomial A(z), and P(c) and Qk([) denote certain polynomials of [ equal to zero for [ = 0. P(z) is non-zero only when the degree of A(z) is no greater than the degree of B(z), and the degree of P(z) is one more than difference of the degrees of B(z) and A(z). Qk(c) is non-zero when zk is a multiple zero of A(z), and the degree of Qk([) is one less than the multiplicity of the zero. The following result holds. Let la,,1 + 1b,l # 0. Then there are n linearly independent solutions y,(x), . . . , y,(x), of (2.37), having the form Y,(X) =

cj(z)exZ dz,

s Gl

where 4(z) is the solution of (2.38). We next present a general principle for choosing the contours C,. A fundamental role in the choice of contour is played by the points zk and the directions in the z plane along which the function 56 The operational calculus was orientated mainly towards the solution of the Cauchy problem for differential equations with constant coefficients, however, in serious texts, say [41], nonstandard problems have been discussed. “This problem has been discussed in a number of books on differential equations.

eXp

XZ

+

p(Z)

+

Representations

-f k=l

Qk

61

(2.40)

tends to zero as z -+ co, or as z + zk. We discuss the simplest case first; when P(l) and Qk([) are absent, that is, all the zeros of A(z) are simple and n in number. We will construct successively a COntOur Ck for each zk and, by the same token, the solutions yk(x). The contours will depend on the values of x for which the solutions are to be constructed. For definiteness, we will construct them for x > 0. For x > 0 the function (2.40), in our case exr, tends to zero as z + co in the left half-plane and, moreover, more quickly than any power of z. In order to satisfy condition A we must choose a contour C, having both endpoints at infinity in the left half-plane. Choose any curve L,, going from infinity (in the left half-plane) to zk and not passing through any other z,. We form C, from three parts: 1) The part of L, from infinity to the first point of intersection with the circle lz - zkl = p, where p is any sufficiently small positive number. 2) One complete counter clockwise circuit of the circle lz - zkl = p. 3) Part 1 traversed in the opposite direction. It can be shown that the solutions yk(x), constructed with respect to these contours, are linearly independent if there are no positive integers among the 1,. If ,l, is a positive integer, then the corresponding solution yk(x) is identically zero. We will show how to rectify the situation in this case. If Re I, > 0, but 1, is not an integer, then we can replace the integral over Ck by the integral over L,. These integrals will differ only by a constant factor. In fact, for Re I, > 0 it is possible to pass to the limit as p + 0, and the integrand on the first and second L, differs only by the factor e2ni’k. Therefore the integral over C, differs from the integral over L, by the factor e’“‘“k - 1. This factor is irrelevant in the construction of yk(x), provided it is not zero. That is precisely what happens when ;1, is an integer. The situation is completely rectified if for an integer i, we take the integral over L, instead of the integral over C,. We move on to the case when all the zeros of A(z) are simple but are n - p in number. In this case P(z) is not zero, and has degree p + 1. When P(z) has degree p + 1, the function (2.40) tends to zero as z + cc in p + 1 sectors. Having chosen one of these sectors we can construct n - p contours C, by the same means as in the first case (but now the integral for the solution yk(x) will converge for all x since the convergence to zero of (2.40) does not depend on X; it is completely determined by P(z)). The missing contours are obtained by taking curves coming from infinity in the chosen sector and going out to infinity in another of the sectors where the function in (2.40) tends to zero. If there are positive integers among the 2, then, of course, we must make the same rectification as in the first case. It remains to consider the case of multiple roots.

62

M.A.

Evgrafov

I. Series and Integral

If a zero zk of A(z) has multiplicity 4 + 1, then Qk([) has degree q. In order to obtain the necessary number of solutions we must connect exactly q + 1 contours to the point zk. For a polynomial Qk(c) of degree q > 0 there are q sectors with vertex at zk, in which the function in (2.40) tends to zero, as z + zk, faster than any power of (z - zk). Choose one of these sectors and join it: 1. To infinity. 2. To itself after a complete circuit around zk. 3. To the remaining sectors of the same kind. This gives us the required q + 1 contours. The integrals with respect to these contours (except, perhaps, the first) converge for all x. When these solutions are represented by integrals which do not converge for all x there arises the natural problem of their analytic continuation. We will not discuss this problem in the general case, but will consider the questions which arise in an example. Example 1. We will investigate the solutions of the equation xy” + 21y’(x) + xy(x) = 0.

(2.41)

From equation (2.38) we easily find that qqz) = (1 + 22)1-l. If x > 0 and 1 is not an integer, then as L, and L, we take the rays (-cc + i, f i) and obtain, following the general recommendation, solutions (i+O)

Ylb)

=

-m+i (22 + l)l-leX=dz, s

and Y2(4 =

c-+0) -co-i (z’ + l)‘-lexzdz, s

where (z2 + l)“-’ = exp{(L - l)log(z + i) + (A - 1) log(z - i)>, with the principal values taken for the logarithms. The brackets around the upper limits of integration in the solutions mean that the integration is taken over two sides of a cut with circuit of the upper limit by a small circle. The integrals for yr (x) and y2(x) converge for Rex > 0. For Re 1 > 0 these integrals can be simplified, and the simplified formulae take the form yl(x) = 2i sin 7cA

(i - z)“-‘(z

Representations

63

By discarding the constant factors we obtain, for Re L > 0, the solutions Y?(X) =

--m+i (i - z)“-‘(z

+ i)‘-‘exzdz,

s’

and Y%4 =

-lmi (z - i)“-‘(-i

- z)l-lexZdz.

s. These solutions are suitable even for positive integral 2. It is easy to see that for those 1 the solutions can be expressed in terms of elementary functions. These solutions, in general, are many-valued analytic functions and to describe them completely it is necessary to be able to find their analytic continuations relative to any path along which continuation is possible. The problem of completely describing the analytic continuation of a manyvalued analytic function along any ray, in general, is extraordinarily difficult. It is made noticeably simpler if the function to be continued satisfies a linear differential equation with polynomial coefficients. The first simplification is that it is possible to immediately describe the paths along which analytic continuation is possible. Any solution y(x) of the differential equation P,(x)y(“)(x) + . . . + Pn(X)Y(X) = 0, where PJx) is a polynomial, can be analytically continued along any path which does not pass through points at which P,(x) is zero. Applying this result to the solutions yl(x) and y2(x) of (2.41), these solutions can be continued along any path which does not pass through x = 0. Further, suppose that we have a pair of solutions yi(x) and y2(x) of (2.41) which form a fundamental system, and suppose these solutions are defined in the half-plane Rex > 0. Analytically continue these solutions along any path going once round x = 0 in a counter clockwise direction. As a result of the continuation we obtain two new solutions y:(x) and y:(x). Since the original pair of solutions formed a fundamental system the new solutions can be expressed in terms of them

Introducing

Y:(x)

=

~llYlc4

+

%,Y,W,

YW

=

@2lYlb)

+

a22Y2w

A

=

the matrix

+ i)rl-lexZdz,

El1

@12

( a21

@-22 >

1 and putting the solution

and

Y(X)

y2(x) = 2i sin 712 -lmi (z - i)“-‘(

S’

- i - z)“-’ exZ dz.

I

=

QYlW

+

by,(x)

(2.42)

in correspondence with the vector (a, b), we obtain that as a result of one counter

M.A.

64

Evgrafov

I. Series and Integral

clockwise circuit about x = 0 the solution associated with (a, b) translates to the solution associated with

Continuation of a solution along any closed path reduces to some number of circuits of x = 0. Clearly the result of continuation of the solution (2.42) with respect to a path which goes m times round x = 0, is associated with the vector (a@),b’“‘) = A” * (a, b). (m is positive if the path is counterclockwise

round x = 0, otherwise it is negative). Thus, the problem of analytically continuing any solution of (2.41) along any path is solved if the matrix A can be found for the chosen fundamental system of solutions. The analytic continuation of the integrals representing the solutions yi(x) and yz(x) is accomplished by rotating the contour of integration5’ with a simultaneous change in arg k. It can be shown that the matrix A for these solutions is 1 _

1 (

e2niJ.

_

1

>.

Laplace’s method can be applied not only to the solution of differential equations, but in the solution of other equations some new problems arise. For example, applying Laplace’s method to the equation

ukeakz,

-:--yz ( The function (2.40), in our case, takes the form

exp -f (

>

+ (x - y)z . >

For x - y > 0 this function tends to zero as z + 00 in the left half-plane and also as z + co in the right half-plane along the strip (2m - +)7c < Im z < (2m + .+)n,

m = 0, fl,

A suitable complete set of contours of integration Cm = (z: Imz = 27tm},

f2,...

are the lines

m = 0, fl,

f2,...

For m = 0 we obtain the solution

1 Making the substitution er = at in the integral and using the fundamental i integral representation of the gamma-function, we obtain the formula i j It is easy to verify also that

y,(x) = u-r(x

- y).

+ Uk) = 0,

exp -$ + (X - Y)~ dz = ux-ye2nim(x-y)r(X _ y). s cm i I 1 Since our equation is linear, then any function of the form Y,(X) =

we can again obtain equation (2.38) for i(z), where now A(z) = i

From (2.38) we find

e2nil

1

z1 @kX + bk)Yb

65

+(z)=exp

(a’, b’) = A * (a, b).

A=

Representations

B(z) = f

k=l

bkeb@.

k=l

Equation (2.38) is known to be solvable in quadratures but, in general, it will no longer be solvable in elementary functions and this rather complicates the analysis. Another complication in the analysis is that there are infinitely many linearly independent solutions of the initial equation. We will discuss one example of this kind. Example 2. Let us investigate the solutions of the equations5’ Y(X + 1) = 4x - Y)Y(X)F where a and y are constants and, for simplicity,

(2.43)

we will suppose a to be positive.

Y(X) =

f CmY,(X), in= -cc where the c, are arbitrary constants, will also be a solution provided the series converges. Slightly more convenient is another description of the solution. A solution of (2.43) takes the form Y(X) = uX@ - Y)@(X), where Q(x) is an arbitrary periodic function of period 1. Among the simplest special solutions of (2.43), apart from y(x) = uXr(x - y), we mention ,*nix

58 Analytic continuation of integrals 59The monograph [37] is devoted domain.

using rotation to the study

of the contour is explained in some detail in [18]. of difference equations in the real and complex

y(x)

=

ux

T(y + 1 - x)

It is clear that these results remain valid for complex values of a.

66

M.A.

Evgrafov

I. Series and Integral

6 9. Another Version of Laplace’s Method

(2.44)

qqs)xSds.

sC In this version two conditions are imposed on the contour C: A. At the endpoints of the contour the integrand and a sufficient number of its derivatives are zero. B. The contour C, obtained by shifting C a distance 1 (parallel to the real axis), is equivalent to C, that is, both go round the same singular points of d(s). For economy of space we will write (2.44) as Y(X) s 4(s).

d(s) = as

&Y(X) I Sk&)

Imbk # ImbY,

k #vi

and Im ak # Im/$?

(2.45)

((3

(2.49)

1) WY

+ l)...(T(s-&+

Imak # Imc&

P = 4;

In the new version of Laplace’s method (2.44) implies the equalities

T(s - al). . . (T(s - cl&J

qs-p1

where Q(s) is an arbitrary periodic function of period 1. Therefore, in the construction of the solutions of (2.47) we must choose not only the contour of integration C, but also the solution d(s). The possible choices of contour are severely restricted by condition B, according to which if C goes round one of the poles of T(s - tlk), then it must go round the remaining poles of this function. We will first describe the construction of a fundamental system of solutions of (2.47) under strong restrictions on the values ak and fik, and then we will show how to remove these restriction. We begin with the simplest case when

(C)

(x$>

67

In contrast to the basic version of Laplace’s method the solution of this equation is by no means unique. Its general solution can be written in the form

Laplace’s method can also be applied in other ways when the solution of the equation is sought in the form6’ y(x) =

Representations

k, v = 1, 2,. . . ,p.

Consider the half-strips

and

L, = (s: IIm(s - olk)l < E, Re(s - ak) < E}, xy(x) I&

- 1)

(2.46)

(Cl

(condition B on the contour is essential for (2.46)). The new version of Laplace’s method can be applied to other questions and the problems which arise are also different. We will discuss its application to a typical problem in differential equations. Suppose we have the differential equatio#’ x,(x&),(x)

- Q(X~)Y(x)

and Ljf = {s: IIm(s - /Ik)l < E, Re(B, - s) < E}, where E > 0 is small enough so that L, and L, (and also L: and L,*) have no points in common for k # v. The boundary aL, of L, satisfies condition B, and if 1~x1 > 1 it also satisfies condition A. Therefore the functions

= 0,

T(s Yk(X)

=

&

where

P(z) = t$, (z - a,), k=l

Q(z) =

kfi

(Z

-

-

s d&r(Se&

cQ)...T(s

-

LIP)

+ l)...T(s-p,+

1)

(ax)s ds,

(2.50)

for laxI > 1, are solutions of (2.47). The integral in (2.50) is the sum of the residues at the poles of the function r(s - ak). Evaluating these residues we obtain the expansion

pk).

It is easy to deduce from (2.45) and (2.46) that a function y(x), represented in the form (2.44), will satisfy (2.47) if $(s) satisfies the equation Wd(s)

= Q(s + MS

+ 1).

(2.48)

60Thi~ version of Laplace’s method is not usually even mentioned in textbooks. However, it was fairly widely used in the early investigations of the hypergeometric equation. 61 This equation is called the generalized hypergeometric equation. The overwhelming majority of special functions satisfy equations of this type. In Q2 of Chapter 2, in the investigation of power series expansions, we also arrived at equations of precisely this type having polynomial right hand sides.

where A,=

sin

sin n(a, - ‘$)

a”*

. n(&

-

@k)

n v#k

sin

d/$

-

pk)’

It can be shown that yl(x), . . . , y,(x) are linearly independent

a fundamental system of solutions to equation (2.47).

and hence form

M.A.

68

Evgrafov

I. Series and Integral

Let us construct a fundamental system of solutions for the same equation (2.47), defined now for 1~x1 < 1. To this end we remark that the function b(s) = as

m

qa,

- 4. * * wp

- 4

- s + l)...T(a,

- s + 1)

aL~r(ul --s+

l)...T(a,-s+

m-k zk(x)

1)

The functions y:(x), . . . , y,*(x) are also linearly independent, and form a fundamental system of solutions of (2.47). We move on now to the removal of the restrictions imposed. Note that the series expansions of yE(x) and y:(x) are still quite usable if we assume that the differences bk

-

b,

tk

+

v,

(k, v = 1,2,. . . ,p)

bv?

“r(s-/?l

+ l)...T(s-p,+

1)

k=

3

1,2,...

are solutions of (2.48) and therefore the functions T(s zk(x)

=

& s

-

c(l).

. . T(s

-

up)

&, T(s - B1 + 1). . . T(s - a, + 1)

(pits-W

_

l)k-l(ax)S&,

where the fk,(x) are functions analytic for /axI > 1 (their coefficients can be expressed in terms of the gamma-function and its derivatives of order up to v). Solutions z:(x) can be constructed in a completely analogous way when there are integers among the /& - /I,. We now move on to the case when there are integers among the differences (2.52). This case requires no additional constructions, however, some clarification is necessary. The fact is that in this case some of the solutions yk(x) or y:(x) are missing, that is, are identically zero. However, if yk(x) is missing then y:(x) turns out to be defined for 0 < 1x1< co and not just for laxI < 1. Therefore, in the construction of a fundamental system of solutions of (2.47) for /axI > 1, yk(x) can be replaced by y:(x). When & = elk + 1, the formulae for both solutions y,‘(x) and y:(x) give zero. However, it is not difficult to verify that there is a solution

(2.52)

where nk are non-negative integers and the differences elk - c1i, for k > m, are not integers. The functions (e2ni(s-gl) _ l)k-1

fk,v(x)(logx)“.

A similar situation holds in the case when p # q. In this case again there are not enough solutions either for laxI > 1 or for laxI < 1. On the other hand, the insufficiency is always compensated by solutions which turn out to be usable for all x.

1,2 ,..., m,

k=

c v=o

y,‘(x) = y:(x) = x”*.

are not integers. Under the same assumption it is easy to rectify the integral representations. Namely, instead of the rectilinear half-strips L, and Lz we must take certain curvilinear (snakelike) half-strips. The greatest complications arise when there are integers amongst the numbers (2.51) or (2.52). We first consider the case when the integers only occur in the differences (2.5 1). Suppose the elk enumerated so that uk = a, - nk,

xix’

(2.5 1)

and elk -

=

(2.50*)

(ax)s ds

are solutions of (2.47) for 1~~x1< 1. The integral in (2.50*) is equal to the sum of the residues at the poles of the functions r(& - s) and, evaluating these residues, we obtain the expansion

uk - @,,

69

are solutions of (2.47). The functions zi (x), . . . , z,(x) can be used to replace the solutions (2.50) with the same indices. The integral in the solution zk(x) can also be calculated using residues, but the formula is significantly more cumbersome because the integrand has multiple poles. These formulae can be written in the form

is also a solution of (2.48). Therefore the functions

Yx4 =&s Wl-4...wp-4

Representations

0 10. The Hypergeometric Function The adjective “hypergeometric” is traditionally used in connection with three [ different objects. I The first of these objects is the hypergeometric series62 F(a, b, c; z) =

r(c) ca r (a + n)T(b

WW4

[ The second object is the hypergeometric

n=o

T(c+n)

+ n) z” n!

(2.53)

equation

x(1 - x)y”(x) + [c - (a + b + l)x]y’(x)

- aby

= 0

(2.54)

s2 At the beginning of the study of the hypergeometric series Euler obtained many relationships and integral representations for it. A thorough investigation of the hypergeometric series was carried out by Gauss (for some time the hypergeometric series was called, for this reason, the Gauss series). The fundamental investigation of the hypergeometric equation was carried out by Riemann.

70

M.A.

Evgrafov

I. Series and Integral

The coincidence of names is explained by the fact that the hypergeometric series is one of the solutions of the hypergeometric equation. The third object is the hypergeometric function. This notion has not been defined very clearly, and different authors use it in different senses. For us the hypergeometric function will be any solution of (2.54) considered as a manyvalued analytic function. In particular, an analytic continuation of the hypergeometric series is, in our understanding, a hypergeometric function. It follows from the general theorems on the existence of solutions of linear differential equations that a hypergeometric function can be analytically continued along any path not passing through the points 0, 1, co, which we call the singular points of the hypergeometric equation. The singular points of the equation need not be singular points for each branch of any solution of it. For example, the hypergeometric series does not have z = 0 as a singular point. There are a great variety of formulae connected with hypergeometric functions (by no means all of them are contained even in the detailed reference texts on analytic special functions). 63 We discuss here just one classical problem-the continuation of the hypergeometric series-and so will give only those formulae which bear directly on this problem. It is not difficult to verify that the hypergeometric equation can be written in the form

F(a, c,c;z)=WV(c r(c)- b) sol F’(l

- ty-l(l

-

zt)-dt,

63 Each reference text on hypergeometric functions, for example Cl], contains a great many formulae concerning the hypergeometric series and the hypergeometric equation. There is a brief exposition of the theory of hypergeometric functions in [55]. Of the more modern books we mention [35] and [38]. Some non trivial formulae can be found in [53].

71

although this integral representation is valid only under the condition parameters b and c satisfy the inequalities Re c > Re b > 0. The Mellin-Barnes integral representation gives another solution:

m) 1 im r(a F(a, c,c;z)=T(a)T(b) 2k s-im

that the

+ s)T(b + s)T(-s) (-z)“ds, r(c + s)

where the contour of integration is a somewhat deformed imaginary axis (it is deformed so that the poles of r(a + s) and T(b + s) lie to the left, and the poles of r( - s) lie to the right, of the contour). The Mellin-Barnes formula is valid for all values of the parameters. In the Mellin-Barnes formula the function (- z)s must be considered in the z plane with a cut along (0, +co) (on which the value of -z is negative) and is taken to be equal to exp(slog( -z)), where the principal value is taken for the logarithm. This same agreement will hold in all later formulae which contain expressions of the form LYE(unless otherwise stated). In order to analytically continue the hypergeometric series to other sheets of a Riemann surface we will use the following method. We evaluate the integral in the Mellin-Barnes formula as the sum of the residues of the poles lying to the left of the contour of integration. This gives us F(a,b,c;z)=A(-z)-“F

x~(c-l+n~)y(r)=x(o+x~)(b+x~)y(n)(2.55) so that its solutions can be found by the method described in Chapter 2, Q9. An important property of the hypergeometric equation is that a change of variable from x to 1 - x or l/x again leads to an equation of the same type (but with other constants a, b and c). These changes are the sources of a great many formulae. In all that follows we will assume for simplicity that the parameters a, b and c satisfy the conditions: 1. --a, -b and -c are not integers; 2. a - b is not an integer. The simplest problem of analytic continuation is the analytic continuation of the hypergeometric series (2.53) from the disc 1zI < 1, where it converges, to the z plane with a cut along the line (1, +a). This problem is solved, for example, by the Euler integral representation

Representations

a,a-c+

+ B(-z)-~F

La-b+

1;’ z>

b,b - c + 1,b -a

+ l;i

z> ’

(2.56)

where A = r(W(b

- 4

T(b)T(c - a)’

B = rGW(a

r(a)r(c

- 4

- b)

This relation, and the relation obtained from it by changing z into l/z, permit us to analytically continue the hypergeometric series onto any sheet of a Riemann surface. In fact in order to reach some sheet or other, we perform a sequence of circuits around z = 0 and the segment (0,l) (equivalent to a circuit around 00). A circuit of z = 0 does not change the hypergeometric series. To go around the segment (0,l) we use (2.56) and then revert to the hypergeometric series in z (but with other parameters). The inconvenience of this method is that each change of variable increases the number of terms. Therefore, if there are many circuits it is better to calculate the monodromy group of the equation (2.54). It will be more convenient for us to define the monodromy group in the general case64 and not just for equation (2.54). 64The notion of a monodromy differential equations.

group

is expounded

in any course

on the analytic

theory

of

M.A.

72

Evgrafov

I. Series and Integral

Suppose we have a linear differential equation y’“‘(x) + c,(x)y’“-1’ (x) + ... + c,(x)y(x) = 0

(2.57)

with coefficients c,Jx) which are single-valued and analytic in a given domain D, and let (2.58) {Yl(X)~...~Yrl(X)~ system of solutions of (2.57) defined in a neighbourhood of

be any fundamental a given point x0 E D. The solutions of (2.57), in general, need not be single-valued in D, but they can be analytically continued along any path in D. We will discuss the analytic continuation of solutions of (2.57) along a curve in D starting and ending at x,,. We recall certain facts about these curves. The symbol C, C, denotes the curve obtained by first traversing C, and then C,. The symbol C-’ denotes the curve C traversed backwards. Curves C, and C, are called homotopic if one can be translated into the other by a continuous deformation not leaving D, and without removing from them the point x0. The set of all curves homotopic to a curve C is called the homotopy class of C and is denoted [C]. A multiplication can be introduced for homotopy classes:

cc11cc21 = cc, Gl,

[cl-’ = [c-l]

The set of homotopy classes forms a group with respect to this multiplication, denoted by ~(0, x0) and called the fundamental group of D. One of the basic theorems in analytic function theory reads? If a function, defined and analytic in some neighbourhood of a point x0 E D, can be analytically continued along any curve C lying in D, then the result of the continuation only depends on the homotopy class of C. Let us analytically continue all the solutions in (2.58) along a curve with homotopy class 1. We then arrive at some other system of solutions

The set of matrices T(A), ,J E ~$0; x,,) is a group, and this group is called the monodromy group of equation (2.57). It is clear that the monodromy group of (2.57) depends on the choice of fundamental system (2.58). The hypergeometric equation is one of the few equations with a non-trivial monodromy group which can be calculated. The domain D for the hypergeometric equation is the whole complex plane with puncture points 0 and 1. The fundamental group of this domain is the free group on two generators, which can be taken to be one counter clockwise circuit of x = 0 and one counter clockwise circuit of the segment (0,l). We denote their homotopy classes by 1, and 1,. Any homotopy class A can be expressed uniquely in the form il = 12, A$. . . 12r;p,,

TV) = (Wmp))EP.. . mAnl))E1, and the problem of calculating the monodromy group of the hypergeometric equation reduces to the representation of 1 in the form (2.59) and finding T(&) and T(&,) (after choosing some initial fundamental system of solutions). As a fundamental system of solutions of the hypergeometric equation one of the following two pairs is most suitable: Y,, l(x) = F(a, b, c; 4

and y,,,(x) = (-x)l-“F(u

- c + 1,b - c + 1,2 - c;x),

or

Yi34

= “il %(4Y”(X)T

These equalities associate a matrix T(1) = (a,,); with A E n(D; x,), where it is easy to verify that qil-‘) = (T(A))-’ Wl4) = Wl)W,)~ and the unit matrix is associated with the unit element of the fundamental 65 This theorem

is usually

called the monodromy

theorem.

u,u-c+

group.

&u-b+

l;!

X

(

>

and y,,*(x) = (-x)-~F

b,b - c + 1,b - a + l;L

(

k = 1, 2,.,., n.

(2.59)

where the index mktakes the values 0 and co and the exponent sk takes the values 1 and - 1. Then

y,,i(x)=(-x)-OF defined in a neighbourhood of the same point x0. Since the system (2.58) was fundamental, then the new system can be expressed in terms of the old one and we can write

73

Representations

X

. )

The first pair is defined initially for 1x1< 1 and the second pair for 1x1 > 1 (the hypergeometric series in their description converge in these regions), but, as we have seen, these solutions can be analytically continued, by the Mellin-Barnes formula, to the whole complex plane with a cut along the positive part of the real axis. We denote the matrices T(i) associated with the first pair by To@) and those with the second pair by T,(I). Since the hypergeometric series is analytic at z = 0 it follows that

M.A.

74

Evgrafov

I. Series and Integral

WA,) =

Representations

75

In conclusion we will say a few words about the analysis of the hypergeometric function in the neighbourhood of x = 1. For this it is convenient to use the formula

and

F(u,b,c;x)=A.F(u,b,u+b-c+l;l-x) +B.(l-x)‘-“-bF(c-u,c-b,c-u-b+l;l-x), Finding To(&) and T,(&) is rather more complicated. For this we must express the first pair of solutions in terms of the second pair and vice versa. Formula (2.56) gives us (2.60)

Y,,,(X) = In a similar way =

where A = f(c)f(c - a - b) f(c - u)f(c - b)’

B = Wf(a

+ b- 4 WW4 ’

In particular, it follows from (2.62) that for Re(c - a - b) > 0 F(u, b, c; 1) =

f(b - a)r(2

Y,,,(X)

(2.62)

f(c)f(c - a - b) f(c - u)f(c - b) ’

- c)

r(1 - a)T(b - c + 1) Ym,lb) +

T;F

$2:

;f’

1J Y,,~(x).

(2.61)

It is not difficult, using (2.60) and (2.61) to obtain formulae for the matrices and T(A.,) of interest to us. They may be written more conveniently if we introduce the matrix

0 11. Theorems on the Singular Points of Power Series

To(&)

f(c)f(b f(b)f(c

A=

f(c)f(a f(u)f(c

- a) - a)

f(b - u)T(2 - c)

- b) - b)

f(u - b)r(2

- c)

r r(1 - u)T(b - c + 1) r(1 - b)T(u - c + 1) / Then we obtain T,(L)

= ~Tm&J,

T,(I,)

We formulate a final result on the continuation Let y(x) be the solution of the hypergeometric y(x) = crF(u,b,c;x) + /?(-xy-T(u

= /1-’ To@,).

of the hypergeometric function. equation taking the form

- c + 1,b - c + 1,2 - c;x)

and defined on the disc 1x1 < 1 with a cut (0,l). After analytic continuation of this solution along a closed curve of homotopy class (2.59) we arrive at the solution y*(x) = cr*F(u, b, c; x) + /!I*( -x)‘-“F(u

- c + 1, b - c + 1,2 - c; x),

where (a*, 8*) = G(&,))EP

. . . G%An,N%

0

In particular, putting c1= 1 and /I = 0, we obtain a formula for the analytic continuation of the hypergeometric series.

The first of the theorems to be discussed was formulated at the beginning of the nineteenth century. It reads as follows. On the circumference of the disc of convergence of a power series there is at least one singular point of the sum of the series. This result quickly became very popular, but for a fairly long time was unique of its kind. It was known that other series (for example, the Dirichlet series) did not have this remarkable property, but the reason for this remained elusive. The first serious attempts at investigation began only in the twentieth century. They led to the creation of a fairly extensive division of mathematics lying on the boundary of the theory of entire functions and functional analysis. The results of interest to us are in some sense “waste products”. To formulate the results we introduce some ideas. Let {A,} be an increasing sequence of positive numbers. The values lim sup f, n+m ”

lim inf t +m n

are called, respectively, the upper and lower density of {A,}. If the upper and lower densities coincide, then their common value is called the density of {A,}, and the sequence itself is called measurable. The maximal density of a sequence {A,} is the lower bound of the densities of measurable sequences which contain {A,} as a subsequence. Similarly, the minimal density is the upper bound of the densities of all measurable subsequences of {Al>.

I. Series and Integral Representations

M.A. Evgrafov

16

The fundamental result, generalizing the classical result quoted above, is:66 Let the sequence (2”) have maximal density A and satisfy An+l

-

A,

>

6 >

0,

n = 0, l,...,

and let the sequence of complex numbers (a,} satisfy limsup,-= logla,l

then the point z = A is a singular point for the sum of the series

o.

Then the series f

If

a,e-“n”

n=O

converges for Re z > 0, and on each segment of the imaginary axis of length 27cA the sum of the series has at least one singular point. The classical theorem can be obtained as a corollary by putting {A,} equal to w We mention a special case:67 Let {A,} be a sequence of integers with zero density. Then each point of the circumference of the disc of convergence of the series

f a,z” n=o

This result, as is not difficult to see, does not contain the previous result. Subsequently a theorem was proved which contained both results, but its formulation was somewhat artificial. We will now give a strong generalization of the first result without trying to include Fabry’s theorem.70 Let the series (2.63) converge for Izj < R and lim sup IRea,l’l” n+m

= k

Denote by {nk) the sequence of indices n for which Re a, # 0, and by {v,,,} the subsequence of {n,}, for which

Rea%+l.Rea+

< 0.

If the maximal density of the sequence {vm}, is equal to A, then on the arc is a singular point of the sum. An analytic function with an entire line of singular points aroused a great deal of interest among the mathematicians of the nineteenth century. Such functions were first noted in the theory of elliptic functions (modular functions), later examples were constructed using infinite series of partial fractions, and then using power series. The special case mentioned above (proved noticeably earlier than the general theorem) was a final step in a whole series of investigations. Even at the time of Euler much interest was aroused by the question of whether a given point on the circumference of the disc of convergence of a given power series was a singular point of its sum. The simplest result had been known for quite a long time.68 If all the coefficients of a power series are positive and its disc of convergence is the disc 1z 1< R, then the point z = R is a singular point of the sum of the series. In the same direction there is a difficult and elegant result called Fabry’s theorem.69 66The result given is a variant of Polya’s theorem, presented in detail in [3] and [lo]. The exposition in the original article [42] has its own merits. “This result was proved by Faber. The exposition in [3] is significantly better than in the original article. . 6* A simplified version of Pringsheim’s theorem. See [3] and [lo]. 69 See [3], where this theorem is given a great deal of attention.

IA = R

Iargzl 5 4

there is at least one singular point of the sum of the series (2.63). We mention some results concerning the so-called composition of series. The Hadamard composition of the series

A(z) = F c&z”, n=O

B(z) = f b,z”, n=O

is the series (A*@(z) = f a,b,,z”. n=o

The Hurwitz composition of the series A(z) = F a,z-n-1, n=O

B(z) = f b,z-“-l, n=O

is the series

(A*B)(z) =nzoc,z-, where “A version of Polya’s theorem.

See [3].

M.A.

78

I. Series and Integral

Evgrafov

n! T akL. k=O k!(n - k)!

c,=

In order to formulate the theorems on composition of series we will require still more notation. Let E and E’ be two sets in the complex plane. We will denote the set of points of the form z E E,

ZZ’,

z’ E E’,

by the symbol E 0 E’. Similarly, we will denote the set of all points of the form z + z’,

z E E,

z’ E E’,

by the symbol E 63 E’. The Hadamard and Hurwitz compositions of series are mainly of interest because of the following, in general false, assertions. Let E, and E, be the sets of singular points of the sums of the series A(z) and B(z) respectively. Then the set of singular points of the Hadamard composition of the series is contained in the set EA @ E,, and the set of singular points of the Hurwitz composition of the series A(z) and B(z) is contained in the set EA @ E,. The mistakenness of these assertions is due to the fact that they were formulated at a time when the modern notion of a singular point was still not clearly formulated. The assertions are true if the sums of the series have single-valued analytic continuations to the whole complex plane, except for a countable set of isolated singular points. They are also true when one of the series is a rational function. There are completely accurate formulations of the theorems on composition of series, but they are noticeably more complicated and not so romantic!71

References*

1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16.

18. 19. 20. 21. 22. 23.

“Theorems on composition of series are discussed in detail in [3]. *For the convenience of the reader, references to reviews in Zentralblatt fiir Mathematik (Zbl.), compiled using the MATH database, and Jahrbuch iiber die Fortschritte der Mathematik (Jrb.) have, as far as possible, been included in this bibliography.

79

transform: [39], [SO], [56], 1571. Operational calculus: [41]. Generalized functions and the Fourier transform: 1231, [47]. Theory of residues: 1173, [18], [19], 1241, [49], 1551. Analytic continuation: [3], [lo], 1341. Dirichlet series: [33], [34], [Sl]. Special functions: [l], [35],[38], [53], 1551. History of mathematics: [S], [36], [48]. I have included in the list of references several more books which would be difficult to place in any definite topic, but to which I feel strong sympathy (for example, [25] and 1441). Interspersed in the list of references are a small number of original articles, the contents of which are not reflected in later monographs, or which have some special merit. In this category are the works of Euler [12]-1161. The reader who makes himself familiar with them will certainly not regret the time spent.

17. The list of references given below consists mainly of textbooks and monographs devoted to the questions broached in this article. From the bibliographic point of view the list is far from complete -it includes only those books which I personally regard as most important and interesting. Amongst them are books remarkable in the completeness of their bibliographies. In spite of my efforts to contain the list of references it has nevertheless become fairly large. To help the reader in orientation I will give a brief clarification of the topics covered in the books listed. General theory of series: 171, [9], [30]. Summation of divergent series: [4], [27]. Trigonometric series: [ll], [SS]. Classical Fourier

Representations

Abramowitz, M., Stegun, I. (Eds): Handbook of mathematical functions with formulas, graphs and mathematical tables. Washington: Nat. Bur. Standards. Appl. Math. Ser. No. 55 XIV (1964), 1046 p. Table Errata. Math. Comput. 21, 747 (1967). Zbl. 171, 385 Bernshtein, I.N., Gel’fand, S.I.: Meromorphicity of the function P”. Funkts. Anal. Prilozh. 3, No. 1, 84-85 (1969). English transl.: Funct. Anal. Appl. 3, No. I, 68-69 (1969). Zbl. 208, 152 Bieberbach, L.: Analytische Fortsetzung. Berlin: Springer-Verlag, 1955. Zbl. 64, 69 Borel, E.: LeGons sur les sCries divergentes. Paris: Gauthier-Villars, 1928. Jrb. 54,223 Bourbaki, N.: Elements d’histoire des mathtmatiques. Paris: Hermann, 1960. Zbl. 129,245 Bourbaki, N.: Fonctions d’une variable rkelle. Paris: Hermann 1949/1951. Zbl. 36, 168, Zbl. 42, 92. Nouv. ed. 1976. Zbl. 346.26003 Bromwich, T.J.: An introduction to the theory of infinite series. London: Macmillan, 1931. Zbl. 4, 7 Bureau, F.: Divergent integrals and partial differential equations. Commun. Pure Appl. Math. 8, 143-202 (1955). Zbl. 64,92 CesBro, E.: Elementares Lehrbuch der algebraischen Analysis und der Inlinitesimalrechnung. Band I, II. Berlin-Leipzig: Teubner, 1904. Jrb. 35,294 Dienes, P.: The Taylor series. New York: Dover Publ., 1957. Zbl. 78, 59 Edwards, R.E.: Fourier series. A modern introduction. Vol. I, II Berlin: Springer-Verlag, 1979, 1982. Zbl. 424.42001, Zbl. 599.42001 Euler, L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive sol&o problematis isoperimetrici latissimo sense accepti. Opera omnia Ser. I, Vol. 25, 1952. Zbl. 49, 195 Euler, L.: Institutiones calculi differentialis. Opera Omnia, Ser. I, Vol. IO, 1913 Euler, L.: Institutiones calculi integralis. Opera Omnia Ser. 1, Vol. 1 I, 12, 1913, 1914 Euler, L.: Introductio in analysin inlinitorum. Opera Omnia Ser. I, Vol. 8, 9, 1922, 1945. Jrb. 48, 7; Zbl. 60, 11 Euler, L.: Letter to scholars. (Russian) Izdanie in-ta istorii estestvoznaniya i tekhniki Akad. Nauk SSSR, Moscow-Leningrad, 1963. Zbl. 127,243 Evgrafov, M.A.: Asymptotic estimates and entire functions. 1st ed. Moscow: Gostekh. 1957. Zbl. 114,277 English transl.: New York: Gordon & Breach 1961. Evgrafov, M.A.: Analytic functions. 2nd ed. Moscow: Nauka, 1968. Zbl. 157,393 English transl.: (of 1st ed.) Philadelphia-London: Saunders, 1966. Zbl. 147, 326 (Zbl. 144,70) Evgrafov, M.A.: Asymptotic estimates and entire functions 3rd ed. Moscow: Nauka, 1979. Zbl. 447.30016 Fejtr, L.: Untersuchungen iiber Fouriersche Reihen. Math. Ann. 58,501-569 (1904) Fischer, E.: Sur la convergence en moyenne. C.R. Acad. Sci. Paris 144, 1022-1024 (1907) Gel’fand, I.M., Shapiro, Z.Ya.: Homogeneous functions and their applications. Usp. Mat. Nauk 10, No. 3, 3-70 (1955). Zbl. 65, 101. English transl.: Am. Math. Sot. Transl., II. Ser. 8,21-85 (1958) Gcl’fand, I.M., Shilov, G.E.: Generalized functions Vol. I. Properties and operations. Moscow: Fizmatgiz, 1958 Zbl. 91, 111 English transl.: New York and London: Academic Press, 1964. Zbl. 115, 331 Generalized functions, Vol 2. Spaces of fundamental and generalized functions. Moscow: Fizmatgiz 1958. English transl.: New York and London Academic Press, 261 p. 1968. Zbl. 159, 183

80 24. 25. 26. 21. 28. 29. 30. 31. 32. 33. 34. 35. 36. 31. 38. 39. 40. 41. 42. 43. 44.

45. 46. 41. 48.

49. 50. 51. 52. 53.

M.A.

Evgrafov

Gel’fond, A.O.: Residues and their applications. Moscow: Nauka, 1966. Zbl. 152,59 Gel’fond, A.O.: Calculus of finite differences 3rd ed. Moscow: Nauka, 376 p. 1967. Zbl. 152, 80 (1st ed. 1952. Zbl. 47, 332) Hadamard, J.: Le probltme de Cauchy et les equations aux derivees partielles lineaires hyperboliques. Paris: Hermann, 1932. Zbl. 6,205 Hardy, G.H.: Divergent series. Oxford: Clarendon Press, 1949. Zbl. 32, 58 Hausdorff, F.: Summationsmethoden und Momentfolgen. Math. Z. 9,269-277 (1921) Hausdorff, F.: Momentprobleme fur ein endliches intervall. Math. Z. 16, 220-248 (1923). Jrb. 49, 193 Knopp, K.: Theorie und Anwendung der unendlichen Reihen. Berlin: Springer-Verlag, 1924. Jrb. 50, 150 Krull, W.: Allgemeine Bewertungstheorie. J. Reine Angew. Math. 267,160-196(1932). Zbl. 4,98 Kushnirenko, A.G.: Newton polyhedron and Milnor numbers. Funkts. Anal. Prilozh. 9, No. 1, 74475 (1975). English transl.: Funct. Anal. Appl. 9, 71-72 (1975). Zbl. 328.32008 Leont’ev, A.F.: Exponential series. Moscow: Nauka, 1976. Zbl. 433.30002 Levinson, N.: Gap and density theorems. New York: American Math. Sot., 1941. Zbl. 26,216 Luke, Y.: Mathematical functions and their approximations. New York-San FranciscoLondon: Academic Press, 1975. Zbl. 318.33001 Markushevich, AI.: An historical sketch of the theory of analytic functions. Moscow: Gostekh., 1951. Zbl. 45. 346 Nbrlund, N.E.: Differenzenrechnung. Berlin: Springer-Verlag. 1924. Jrb. 50, 3 I8 Olver, F.W.J.: Asymptotics and special functions. New York-London-Academic Press, 1974. Zbl. 303.41035 Paley, R.E.A.C., Wiener, N.: Fourier transforms in the complex domain. New York: American Math. Sot., 1934. Zbl. 11, 16 Poincare, H.: Sur les integrales irregulitres des equations lineaires. Acta Math. 8, 295-344 (1886). Jrb. 18,273 Pol, B. van der, Bremmer, H.: Operational calculus based on the two-sided Laplace integral. Cambridge: University Press, 1950. Zbl. 40,204 Polya, G.: Untersuchungen iiber Liicken und Singularitaten von Potenzreihen. Math. Z. 29, 549-640 (1929). Jrb. 55, 186 Polya, G.: Untersuchungen iiber Liicken und Singularitiiten von Potenzreihen. II. Ann Math., II. Ser. %,73 l-777 (1933). Zbl. 862 Polya, G., Szego, G.: Aufgaben und Lehrsltze aus der Analysis. Band I, II. Berlin: SpringerVerlag, 1925. English transl.: Problems and theorems in analysis. I, II, Berlin: Springer-Verlag. 1972. Jrb. 51, 173 Zbl. 236.00003 Riesz, F.: tiber orthogonale Funktionensysteme. Giitt. Nachr., 116-122 (1907) Riesz, M.: Integrale de Riemann-Liouville et le probltme de Cauchy. Acta Math. 81, l-223 (1949). Zbl. 33,276 Schwartz, L.: Thitorie des distributions. I, II. Paris: Hermann, 1950, 1951. Zbl. 37, 73, Zbl. 42. 114 Struik, D.: Abriss der Geschichte der Mathematik. Berlin: VEB, Deutscher Verlag der Wissenschaften(J961). Zbl. 93, 2 (Engl. orig. New York: Dover 1948 Zbl. 32,97) 6th ed. Berlin 1976 Titchmarsh, E.C.: The theory of functions. Oxford: University Press, 1932. Zbl. 5,210 (2nd ed. 1939) Titchmarsh, E.C.: Introduction to the theory of Fourier integrals Oxford: Clarendon Press 1937. Zbl. 17,404 Titchmarsh, E.C.: The theory of the Riemann zeta-function. Oxford: Clarendon Press, 1951. Zbl. 42, 79 (2nd ed. 1986; Zbl. 601.10026) Toepler, A.: Bermerkenswerte Eigenschaften der periodischen Reihen. Wiener Anz. 13.205-209 (1876). Jrb. 8, 133 Vilenkin, N.Ya.: Special functions and the theory of group representations. Moscow: Nauka,

I. Series and Integral

54. 55. 56. 57. 58.

Representations

81

1965. Zbl. 144,380 English transl.: Transl. Math. Monographs 22 (Am. Math. Sot. Providence 1968) Paris: Dunod 1969. Walker, R.J.: Algebraic curves. Princeton: University Press, 1950. Zbl. 39, 377 Whittaker, E.T., Watson, G.N.: A course ofmodern analysis. Cambridge: University Press, 1927. Jrb. 53, 180 Widder, D.V.: The Laplace transform. Princeton: University Press, 1941 Wiener, N.: The Fourier integral and certain of its applications. Cambridge University Press, 1933. Zbl. 6, 54 Zygmund, A.: Trigonometric series. Vol. I, II. Cambridge: University Press, 1959. Zbl. 85, 56

II. Asymptotic Methods in Analysis M.V. Fedoryuk Translated from the Russian by D. Newton

Contents Preface ....................................................... Chapter 1. Integrals and Series .................................. Q1. Introduction .............................................. 1.1. Simplest Examples ..................................... 1.2. Integration by Parts .................................... 1.3. Stationary Points ...................................... 1.4. An Example of a Non-Local Asymptotic Formula ........... $2. Laplace’s Method .......................................... 2.1. The Principle of Localization ............................ 2.2. The Asymptotic Form of the Contributions (One-Dimensional Case) ................................................. 2.3. The Many-Dimensional Laplace Method .................. 2.4. The Logarithmic Asymptotic Form of Laplace Integrals . . 0 3. The Method of Stationary Phase ............... 3.1. The Principle of Localization .............. 3.2. The One-Dimensional Case ................ . .. . . .. .. . 3.3. The Many-Dimensional Case .............. . .. . .. .. . 3.4. The Fourier Transform and the Legendre Transform . . . $4. The Method of Steepest Descent. . . . . . . . . . . . . . . . . . . . . . . 4.1. Heuristic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The Local Structure of the Level Lines of Harmonic Functions....................................... .. 4.3. Asymptotic Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 4.4. Examples of the Application of the Method of Steepest Descent......................................... .. 4.5. The Many-Dimensional Method of Steepest Descent . . .

84 86 86 86 90 92 94 97 97

.. .

. . .. . .. . . *. . . . .. . . .. .. .. .. . .. . .

.. . ..

98 102 106 106 106 108 110 112 113 113 115 116 117 124

84

M.V.

II. Asymptotic

Fedoryuk

$5. Supplement. Sums and Series ................................. 5.1. Merging of Singularities .................................. 5.2. Integrals with. Weak Singularities .......................... 5.3. Sums and Series ......................................... Chapter 2. Linear Ordinary Differential Equations .................. 5 1. Statement of the Problem. Regular Dependence on a Parameter .... 1.1. Statement of the Problem ................................ 1.2. Regular Dependence on a Parameter ....................... 1.3. Singular Dependence on a Parameter ...................... 9 2. Equations of Second Order Without Turning Points .............. 2.1. Formal Asymptotic Solutions ............................. 2.2. Asymptotic Diagonalization of Systems. .................... 2.3. WKB-Estimates ........................................ 2.4. The Asymptotic Form of the Solutions of (2.8) ............... 2.5. Higher Approximations. Additional Parameters .............. 0 3. Equations of n-th Order and Systems Without Turning Points ..... 3.1. Systems of Equations on a Finite Interval ................... 3.2. Equations of n-th Order on a Finite Interval. ................ 3.3. Large Values of the Argument ............................. $4. Equations in the Complex Domain ............................ 4.1. WKB-Asymptotic Formulae .............................. 4.2. Stokes Lines and the Domains Bounded by them. ............ 4.3. Boundary Conditions for Solutions and Domains of Applicability of the WKB-Asymptotic Formulae ............. 4.4. The Global Asymptotic Form of the Solutions of Equation (2.54). ................................................. 4.5. Equations of n-th Order and Systems. ...................... 4 5. Turning Points ............................................. 5.1. The Problem of Connection Formulae ...................... 5.2. Turning Points of Second Order Equations ................. 5.3. Turning Points of Equations of n-th Order and Systems ....... 5.4. Connection Formulae for the Second Order Order Painleve Equation .............................................. References .....................................................

129 129 134 136 140 140 140 141 142 143 143 145 148 150 153 155 155 156 158 160 160 162 165 168 175 176 176 177 183 187 189

Preface An asymptotic formula or asymptotic form for a function f(x) is the name usually given to an approximate formula f(x) z g(x) in some domain of values of x, where g(x) is ‘simpler’ then f(x). For example, if f(x) is an integral, then g(x) must either be given in terms of the values of the integrand and its derivatives at a finite number of points, or in terms of some simpler integral. Iff(x) is a solution

Methods

in Analysis

85

of an ordinary differential equation, then g(x) must either be expressed in quadratures or be the solution of a ‘simpler’ differential equation. This list can be extended-there is an unwritten heirarchy of asymptotic formulae. Of course all these definitions are very blurred. “‘What is asymptotics?” This question is about as difficult to answer as the question “What is mathematics?“’ [S]. Particular cases of asymptotic expansions were found and used as long ago as the eighteenth century by Stirling, Maclaurin and Euler. The rigorous definition of an asymptotic expansion is due to Poincare. Asymptotic methods have been applied in the investigation of integrals, sums, series, solutions of linear and nonlinear ordinary differential equations and systems, differential-difference equations, linear and nonlinear partial differential equations and systems, and integral and integro-differential equations. It is difficult to name a branch of mathematics, or the natural sciences, in which asymptotic methods could not be used. Their value most of all lies in the fact that they allow us to obtain a quantitative description of a phenomenon. Moreover, in many cases, it is not possible to obtain numerical results even with the help of modern computers. At present asymptotic methods are undergoing a period of blossoming. Even in classical problems, such as the calculation of the asymptotic form of rapidly oscillating integrals and the construction of asymptotic formulae in the large for the solutions of linear partial differential equations, fundamental results have been obtained only in recent years [2], [23]. This part is an introduction to asymptotic methods. In it we give the basic asymptotic methods for integrals, series and the solutions of ordinary differential equations and systems. Asymptotic methods in other areas have been discussed in other volumes in this series. In 0 1 of Chapter 1 we give the simplest examples of asymptotic expansions, and the basic ideas behind Laplace’s method and the method of stationary phase. In 4 2 we describe Laplace’s method, in 5 3 the method of stationary phase, and in 0 4 the method of steepest descent for one-dimensional and many-dimensional integrals. 5 5 contains some asymptotic estimates for sums and series and we also consider the merging of the singularities of the integrand (a stationary point close to the boundary, two nearby saddle points, a pole close to a stationary point). In Chapter 2 we describe asymptotic methods for the solutions of ordinary differential equations and systems. In 4 1 we briefly look at regular dependence of the solutions on a parameter, in 0 2 we discuss the classical WKB-method for second order equations and in 9 3 we discuss it for equations of higher order and for systems. In 8 4 we construct the asymptotic form in the large for the solutions of second order equations in the complex domain. In § 5 we study the behaviour of the solutions of equations and systems near to turning points. Proofs are not given, but their basic ideas are explained. A large number of illustrative examples are discussed.

M.V.

Fedoryuk

II. Asymptotic

Methods

in Analysis

k+ 1 =(xI-

Chapter 1 Integrals and Series

87

k+co.

Nevertheless, it can be used to calculate F(x) approximately Let us estimate the remainder:

$1. Introduction

for large values of x.

IR,(x) 5k&$ coe-X’& _@+‘I! X

1.1. Simplest Examples. We begin with the simplest examples.

x”+2

’

s 0

therefore

Example 1. Consider the integral

IF(x) - S,(x)15 3

(1.1) where x > 0. If we need to calculate F(x,) to a given degree of accuracy, then it can only be done by numerical integration. However, if x > 0 is large, then we can obtain a simple approximate (or asymptotic) formula for F(x). First we estimate the integral in (1.1) IF(x)\ 5

m CX’dt = ;, s0

therefore F(x) = 0(1/x) as x + +co. Integrating estimate: 1 F(x) = ;-; Repeated integration

by parts gives us a more precise

1 1 O” e?’ o (1 + t)2 dt =; + 0 s

x+

+a.

l)“n! fyx) = ; _ $ + $ + . . . + (-~ xn+l + R,(x), O” eeX’ dt o s (1 + t)n+z .

x -+ a.

This definition is equivalent to the following: there is a neighbourhood and constants C,, C,, . . . such that

f(x)- f. %hl(X)5 c,+,Ihv+1(x)I

F(x) = Ux) + R,(x),

U of a

(1.5)

(1.2)

for x E U and any N 2 0. The series ~~~oun~~(~) is called an asymptotic series for f(x) as x + a. We will use the notation

I (1.3)

This series diverges for all x since

whereas the actual value of the integral is F( 10) = 0.9156. The relative error made in replacing F( 10) by S,( 10) is less than 0.2%. The divergent series (1.3) is asymptotic to F(x) as x -+ +oo. Let us formulate a rigorous definition. Let (&(x)}, n = 0, 1,2,. . . be a sequence of functions defined in some neighbourhood of a point x = a. The sequence {dn} is called asymptotic as x + a, if for each n q$,+1(x) = o(&( x )) as x -+ a. The point a may be either finite or infinite.

f(x) - n$o %4l(x) = 0(4N,.l(X))*

We shall write this formula as

where S,(x) is a partial sum of the series

S,(lO) = 0.0914,

Examples. 1) &(x) = xn, x -+ 0. 2) #,Jx) = x-“, x -+ co. Let f(x) be defined in a neighbourhood of x = a and let a,, a,, . . . be constants. The formal series ~~zo u,&(x) is called an asymptotic expansion off(x), as x + a, ifforanyNz0

by parts leads to an expansion

R (x) = (- l)“+‘(n + l)! n Xfl+l

If we fix n and let x be large, then the right hand side of (1.4) will be small, and we will have an approximation for F(x) with a small error. For example, with n=3,x=lOwehave

f(x) = “go %&w)> which has become conventional recent decades. The notations

in mathematics

x + a, and mathematical

physics in

M.V.

88

Fedoryuk

II. Asymptotic

are also used. It is always clear from the context whether a given series is convergent or asymptotic, and the use of the equals sign in writing down asymptotic expansions will not give rise to any misunderstanding. An asymptotic series may be either divergent or convergent. For example, let f(x) be infinitely differentiable for small 1x1.Then its Taylor series is asymptotic as x --t 0: a,

f@)lCI\

x + 0,

f(x) = n=cJ c J+Xn, n:

Methods

Example 2.1. Let us consider the integral m

x+o. f(x)- n=O 2 fqxn . = O(XN+l), As is well known from courses on mathematical analysis the Taylor series need not converge, and, if it does, it may converge not to f(x) but to some other function. The term “asymptotic series ” is usually used to refer either to those series which diverge or to those whose convergence cannot be established, but indirect reasoning suggests that the series diverges. The possibility that an asymptotic series could diverge is included in its definition, in which the behaviour of the remainder for N fixed x + a is mentioned, but nothing is said about the behaviour of the remainder for x fixed N + co. In fact the constants C, in (1.5) may grow at any speed as N -+ 00. Suppose f(x) is defined not in a complete neighbourhood of a, but in a set M with a as a limit point. The definitions of asymptotic sequence and asymptotic series go over without change, except that we must write x E M, x -+ a instead of x -+ a. A typical example is: a = 0, M the interval 0 < x 5 b. In this case we write

x-+ +o.

f(x) = “z. a,hk4

:

g(z) = nf?o b,z-”

F,(x) =

eixt

---dt, s 0 l+t

(1.6)

where x > 0, and calculate its asymptotic form as x + +co. This is again done by using integrating by parts and leads to the formula 1 l! n! Fl(X) = ix + (ix)2 + ... + (ixy+l

+ R,(x),

R,(x) = -$$$

dt.

j-: (1 ;i;;n+2

Therefore we have the following representation for Fl (x) F,(x) = S,(x) + R,(x), k!

where S,(x) is a partial sum of the series f This series diverges for all x k=O (iX)k+‘. since it is the same as (1.3) with x replaced by -ix. The situation with the estimate of the remainder is rather more complicated than in example 1. In fact the estimate

lR.(x),~~jm

These definitions translate verbatim to functions of a complex variable z. We give some typical examples 1) a = 0, f&(z) = z”, M a sector of the form 0 < (zl < r, a 5 arg z 5 fl (j3 CI5 271). 2) a = co, d”(z) = Z-“, M a sector of the form R < Jz( < co, c( S argz j fl. The asymptotic series take the form

o (1 +

dtt)“+z=n! x”+l

shows that R,(x) = 0(x-“-l) as x + +co, but the last term n!(ix)-“-’ in S, has exactly the same order. To obtain a more precise estimate for the remainder we integrate by parts, use the previous estimate, and then we obtain R (x) = (y + l)! II

~-~ (zx)"+2

(n + 2)! O” eiXt (ix)“+2 s o (1 + t)"+2d4

(z E M,z + 00).

In example 1 we can see various basic features which are characteristic of all asymptotic methods. (J-6488jO599/K635/WSL/pp.86-88)

89

1”. The asymptotic formula is more precise the larger is the value of x. This follows from (1.4) with n fixed x -+ co. Laplace [21] referring to the method he used to calculate the asymptotic form of an integral said that “. . . this method is the more precise the more that it is necessary.. .“. 2”. An asymptotic formula does not allow us to calculate the value of F(x) with any assigned degree of accuracy. In fact, for x fixed, the moduli of the terms of (1.3) at first decrease and then, for n > x, begin to grow very rapidly. Thus an asymptotic formula has an insurmountable error from the point of view of calculation.

since for any N 2 0

f(z)= nzoa,z” (zEM,z+ 01,

in Analysis

[R,(x)1 5 2(;.t;21)!.

: By the same token we have proved that there is an asymptotic expansion

90

M.V.

II. Asymptotic

Methods

in Analysis

CD(L) =s(Ibf(x)e’“s’x’dx

Fedoryuk

91

(1.8)

This asymptotic series is also quite suitable for the approximate calculation of the integral for large values of x. Even the simplest formula, F,(x) = l/x (the series being replaced by the first term), for x = 1000 gives the value of F1(lOOO) with an absolute error of the order of 10e6 and a relative error of less than 0.1%. At the same time, the calculation of integrals of the type F,(x) for x >> 1 involves the following problems: the integrand is rapidly oscillating and therefore the number of nodes in a quadrature formula must increase as x increases. To calculate integrals of rapidly oscillating functions special quadrature formulae are used, whose construction use asymptotic methods. It is obvious that F,(x) = F( - ix), so in these examples we have analysed the asymptotic behaviour of the same function along different rays in the complex x plane. The asymptotic series obtained in both examples coincide; in particular F(x) - x-l both for x + +co and x = iy, y -+ +co. However, these integrals are completely different in character, and the fact that both converge to zero as x + +co, happens for quite different reasons. Let us write them in the form m 4(x) = j=O, 1, fj@, x) dt, F,(x) = F(x) s0 The integral F,(x) converges absolutely for all x > 0, the integral F,(x) converges only conditionally. The exponential emX’in F,(x) is equal to 1 for t = 0 and rapidly converges to zero as x -+ +co, if t > 0. Therefore all the area under the curve y = fo(t, x) is concentrated in a small neighbourhood of t = 0; in particular, this area tends to zero as x + +co. The integral F1 (x) is small for x >> 1, for other reasons. Let us consider Im F1 (x). The integrand (sin xt)/(l + t) oscillates rapidly for x >> 1 and the areas of two neighbouring ‘half-waves’ almost completely cancel each other. Therefore the area under the curve y = Imfi(t,x) is approximately equal to the area So of the first half-wave, which adjoins t = O-the endpoint of the interval of integration. In its turn this area is equal to n/x

sinxtdt s

which in order of magnitude proved above.

0

F(I) integral

=

b f(x)e”S(‘)dx s ll

(1.10) The leading term is

For definiteness suppose that S’(x) < 0 for x E I, so that max,., only at x = a. Then

S(x) is attained

(1.11) The leading term of the asymptotic formula is

Moreover, (1.10) and (1.11) are valid in certain sectors of the complex 2 plane. The expansion (1.11) is valid for lill -+ co, 2 E S,, where S, is the sector Iarg,4 5 ~42 - E. The number E > 0 may be chosen to be arbitrarily small but not depending on II. This expansion is uniform with respect to arg il, that is, &‘(A) = ens@) kio

ckn-k

+

RU)],

and for the remainder there is the estimate

X’

IMU

coincides with the asymptotic formula accurately

1.2. Integration by Parts. Consider t,he Laplace

and the Fourier

2 = -,

where I: a _I x 5 b is a finite interval. The trivial cases f(x) = 0 or S(x) = const, here and later, are not discussed. The function S(x) is real-valued, f(x) may be complex-valued. Let f(x), S(x) E P(Z) and let S(x) have no stationary points, that is, XEZ s’(x) # 0, (1.9) Then the asymptotic form of (1.7) and (1.8) is calculated using integration by parts. For @(A)there is the asymptotic expansion

integral

(1.7)

5 C,ITN-‘,

for A E S,, (II 2 1, > 0, where C, does not depend on arg 1,.The expansion (1.10) is valid as I il I + co uniformly in arg A, 0 5 arg II 5 X. As is well known, neither convergent nor asymptotic series have to be differentiable term-by-term. But (1.10) and (1.11) can be differentiated term-by-term any number of times by virtue of the following result. Let S be a sector a < arg 2 < p, l;il 2 R > 0, 0 < fi - LX5 272. A sector S’ of the form CI< a’ 5 arg,? 5 /Y < /?, ),?I 2 R, > R is called an interior closed subsector.

M.V.

92

Fedoryuk

II. Asymptotic

asJ.+

Let f(;l) be analytic in S and have the asymptotic series

Methods

in Analysis

93

+ccisgivenby h(x,, +co) = 1; h(x, +co) = 0 for x # x0.

uniform with respect to arg 1 in any closed subsector. Then for any integer n 2 0 there is the asymptotic expansion

Consequently, for I >> 1, h(x, A) has a sharp maximum at x0 and F(I) is approximately equal to the integral over a small neighbourhood of x0: x,+d F(1) x

f(x)exp(Wx))

s X,-6 In this small interval we can approximately and each of these is uniform with respect to arg 1 in any closed subsector. This is an analogue of Weierstrass’ theorem on term-by-term differentiation of a uniformly convergent series of analytic functions. Both the integrals (1.7) and (1.8) are entire functions of 1, since the asymptotic expansions (AE) (1.10) and (1.11) can be differentiated term-by-term any number of times. This remark holds for all the one-dimensional and many-dimensional Laplace and Fourier integrals which we will consider. Using integration by parts we can find the asymptotic form of Laplace and Fourier integrals when the limit of integration acts as a large parameter. Laplace had already used this method to obtain an asymptotic expansion of the error function

Erfx =

a0 s

x--t

+oo.

This AE is also valid for complex x: 1x1+ co, Iarg x 15 z/4 - E, uniformly respect to arg x. Other examples are-the incomplete gamma-function

= G4 y(a, 4=sxtamlee’dt T(a) -

e-xXa-k

f

k=Or(a-k+

1)

with

The method of integration

and then W)

Making

= evW(xo)}f(xo)

the substitution Jqj(x

- x0) = t

we obtain

f

(- l)k$k

+ 3))

f (t)ems@) dt,

m f(t)

Consequently, as 1+ x+

eiS@)dt,

(for more general conditions

- S(xo))3

+CO, F(A) z

+oo.

on f

1.3. Stationary Points. Consider the Laplace integral (1.7) and let max, EI (x) be attained only at the point x0, a < x0 < b. For’ simplicity let f(xo) # 0 and S”(x,) # 0. The limit of the function

4~~4 = f(4ew(Wx)

m s

emt2j2 dt = &.

JX

where, for example, f and S are polynomials and S see [ 111).

integral

-‘m

k=O

cc

e-‘2’2 dt. s +JS As I -+ +co the limits of integration tend to &cc and we obtain the Poisson

x--,+00

by parts applies to integrals of the form

Jx

f(x) 25 f(xo), S(x) = S(x0) + $S”(XO)(X - xfJ2,

’

and the Fresnel integral

ieix2 dt=2,,/& Q(x) =sxme@

put

aJTi?gj

x

0

dx.

The method given above, and this formula, are due to Laplace [12]. Of course these arguments are not totally rigorous, but they do lead to a correct result and converting them into a theorem is fairly simple (see 52 of Chapter 1). The Laplace method is important; essentially it is this. 1. If the integrand has a sharp maximum at x0, then the integral is approximately equal to the integral over a small neighbourhood of this point. 2. In a small neighbourhood of x0 the integrand can be made simpler by an appropriate change of variables. The substitution leads to a standard integral which, as a rule, can be evaluated. Both of the above situations are also applicable for n-dimensional integrals. The Fourier integral can be analysed similarly. Let S(x) have a unique stationary point x o, a < x0 < b, and let f(xo) # 0, S”(xo) # 0. This time @(A) is ap-

M.V.

94

II. Asymptotic

Fedoryuk

in Analysis

95

and its derivatives at one or more points. We will give an example of another kind. Consider the integral

proximately equal to the integral over a small interval [x0 - s”,x0 + s”] because the oscillation of the integrand near the stationary point is slowed down. For example, the function cos ;1x2does not change sign for 1x1 < rr/(Z$), and outside this domain the areas of the neighbouring half-waves almost completely cancel each other. Hence it follows immediately that the integral pa cos Ix2 dx has order const. 1-Y‘I2 . We make the same approximation as for the Laplace integral and in the last integral make the change of variable

JW(xJ

Methods

F(E) =

l ~f(x) 0 i X+E

dx

(1.12)

where E > 0 is a small parameter, fe C”(0, 1). If f(0) # 0, then the integral diverges for E = 0. An asymptotic formula for (1.12) as E + 0 can be calculated in many ways. For example, the leading term of the asymptotic formula is easy to find by integrating by parts:

(x - x0) = t,

then

F(E) = f( 1) log( 1 + E) - f(0) log E -

o1f’(x)log(x

+ E)dx,

s where 6 = sgnS”(x,). integral

meidt2/2 & =@ei”“/4, s-co

/

This method was developed by Kelvin

[19] and is called the method of

phase.

Earlier still the Laplace method was used by Riemann [28] in a memoir ‘An analysis of the possibility of representing a function by a trigonometric series with no special assumptions on the nature of the function’, which was published after his death in 1867. Riemann studied the asymptotic behaviour as n + co of an integral of the form

Q-4 - -f(O) log 6, 6 -+ + 0,

i i or, more precisely, i( /

and finally we obtain

stationary

so that

As 1 -+ +co the latter integral converges to the Fresnel

F(E) = -f(O)log&

s

o1f’(x)logxdx

+ o(1).

Is This asymptotic formula is non-local: it is given in terms of the values of f’(x) in 1 the whole subset of integration. Let us complete the example. Assume first of all that f(x) is analytic at x = 0, i and transform the integral: F(E) = f ( - e) log (!g

+ lo1 fw.f-s)dx,

The integrand is analytic in Efor small Is] and can be expanded in a series

‘(x;-+f;-E)= f. q(X)&“,

% 4(x) sin($(x) + nx) dx s having a stationary point t,V(x) + n = 0. But mathematicians have not given proper attention to this piece of work by Riemann. A more powerful method of finding the asymptotic form of integral of the form (1.7), with complex phase S(x), is the method of steepest descent (Chapter 1,94). This method was suggested by the celebrated English physicist and mathematician Peter Debye [9], whose role in the development of modern asymptotic methods is difficult to overestimate. In all the examples given above the asymptotic form of the integrals was local in nature: all the terms of the asymptotic expansion were given in terms of the values of the functions 1.4. An Example of a Non-Local

Asymptotic

Formula.

CJx)= (

[ ki, of’“’ +f(x)- f(O)].

X

Integrating term-by-term series, we obtain

and expanding the functions f( -E) and log( 1 + E) into

F(E) = log a f a,,&” + ‘f b,,P, E + +O, n=o

a = (- l)““f’“‘(O) ” n!

n=O

’

These series converge for small 1a].

(1.13)

b, = s0

’ G(x)dx

+ k$, icy;;;

M.V.

96

II. Asymptotic

Fedoryuk

+ Jo1f(x);f-E)dx

2.1. The Principle of Localization. one-dimensional

fN(-&)= f fT(-E)F n=O .

F(A) =

This leads to the formula

97

Consider the Laplace integrals:

* f(x)eAs’“‘dx sa

(1.15)

and many-dimensional

F(E) = logs f a,,~” + 5 b,,” + O(eN+l logs), n=O n=O

F(l) =

where the coefficients a,, b, are the same as before. However, in this case (1.13) generally speaking, will diverge. Integrals with non-local asymptotic form arise, for example, in problems in potential theory, in the study of integral transforms with singular kernels (Hilbert transform, Stieltjes transrorm), and in other problems. All the integrals considered above have the form

f (x)e”@) dx. (1.16) sD Here S(x) is a real-valued function, f(x) may be complex-valued and 1 > 0 is a large parameter. Further, I: a 5 x 5 b is a finite interval and D is a bounded domain in R” with a smooth boundary.

Let us give a rough estimate for the Laplace integrals; the domain of integration may be unbounded. If M = sup,, D S(x) < cc and the integral converges absolutely for some A0 > 0, then in the half-plane Re A >= A, IF(l)I

F(E) =

l f(x, E)dx,

(1.14)

s0 where E > 0 is a small parameter. We will normalise f by the condition max If@,&)1 = 1, xe[O,lI

0 < E 5 Eo,

and try to classify the types of dependence of the integral on the parameter. Let f E Cm for x E I, 0 < E 5 co; we can even assume that f is analytic in F in the ring 0 < I&l 5 Eo. 1. Regular dependence on the parameter. f E C” for x E I, 0 < E 5 Ed. TO obtain an asymptotic form it is sufficient to expand f in a Taylor series in powers Of

in Analysis

5 2. Laplace’s Method

If f(x) is not analytic, then (1.13) can be obtained as follows:

F(E)= fN(- E)log(q

Methods

E.

Any other dependence on the parameter is called singular. 2. Integrals with a weak singularity. The limit function f(x,O) E C” for 0 < x < 1, and at x = 0 either the function or one of its derivative becomes infinite. The integral (1.12) serves as an example. 3. The limit function f(x,O) is equal to 1 at a finite number of points x1,. . . , X, and is equal to 0 elsewhere. Typical examples are Laplace integrals. Here we must be more careful: the “principal part” off must be real near to Xl,..., X In’ Then, for small E > 0, F(E) will be approximately equal to the sum of the integrals over small neighbourhoods of the points xi,. . . , x,. 4. Oscillating integrals: If(x, E)I z const. for small E > 0 and lim,,+o f(X, E) does not exist. Typical examples are Fourier integrals.

The contribution

5 CeMReA

of a point x0, at which max,,n

K$) =s

taken over a small neighbourhood (1.15) and (1.16).

(1.17) S(x) is attained, is the integral

(1.18) f(x)e As(X)dx U(-%) U(x,) of x0. This definition applies for both

The principle of localization. The asymptotic form of F(a), as I -+ +co, is equal to the sum of the contributions of the points at which rnax,.. S(x) is attained. The precise meaning of this is as follows:

F(a)= ,$ Kj(4 + WxpMM - 4)) where c > 0. At xj we have lexp(lS(xj))l = leAMI, and the contributions of these points, as a rule, are of order O(XQ”), a > 0. Therefore the last term in (1.19) is exponentially small by comparison with contributions of this order. The principle of localization can be considered as an analogue of the residue theorem. In fact ni x (the sum of the residues),

the asymptotic form of F(1) = the sum of the contributions. This analogy can be extended. If z. is a pole of f(z), so that f(z) = (z - z,)-“‘g(z), g(z,) # 0, then res,,,O f(z) can be expressed using the values of f(z) and its derivatives at the

M.V.

98

Fedoryuk

II. Asymptotic

point zo. As will be shown below, the coefficients of the asymptotic expansion of the contribution V”O(n) can be expressed using the values of f, S and their derivatives at x0. Thus, the problem of calculating the asymptotic form of the Laplace integral F(A) reduces to a local problem-calculating the contributions of the maximum points. It also follows from the principle of localization that the differential properties of the functionsf and S are significant only near to the points at which max,.. S(x) is attained. We will assume that f, S E C” in neighbourhoods of these points. 2.2. The Asymptotic

Form of the Contributions

(One-Dimensional

Case). Let

x0 be a stationary point of order n of S(x), that is S’(x,) = ... = s’“-“(x,)

z.r0,

S’“‘(x,) # 0,

99

a0 = mf(o)

max S(x) = S(x,), xeI

s”(x,) # 0;

a<x,
be attained only at this point. Then for 2 E S,, /Al + cc [f(xo)

+ O(E~-‘)]e”S’“o)

(1.21)

There is the asymptotic expansion

Here d(t) E C” and d(O) = x0, qY(0) = 1, so the inverse function t = t(x) E C” in a neighbourhood of x0. If S(x) is analytic in a neighbourhood of x0, then 4(t) is analytic in a neighbourhood oft = 0 and the inverse function r(x) is analytic in a neighbourhood of x = x0. Let SEbe the sector larg AI $ 742 - E < 742 in the complex 2 plane. Watson’s lemma. Let f(x) E C” ([0, a]), a > 0 and CI > 0, j > 0. Then for i E SE, 11) -+ GO,there is the asymptotic expansion xfi-‘f(x)e-“”

f C,(logI))“, n=O

F(1) =

S’“‘(x,) S(x) = S(x,) + 7t”.

in Analysis

for ;1 E S,, [Al+ co. Here /? > 0, y is any real number and 0 < a < 1. Let us give the most ‘effective’ of all the formulae related with Laplace’s method. Let

and let the maximum

n 2 2.

Then, by a change of variable x = 4(t), S locally is a power:

CD(n) iisa

= P(log1)r

Methods

F(l) = eas(“o)“$” c,P-,

_

cn = 0 +t) d Pa (2n)! ( z >[ f(x)

2(S(x) - S(x,)) (x - x0)2

(1.22)

x=x”

Let us prove this formula: put x0 = 0, S(x,) = 0. If we make the substitution x = d(t), so that S(x) = - t2, we obtain f(d(t)M’(W”’

dx

1I --u2

dt.

0

(1.20)

Replace [ -a,, a2] by [ - 6, S] (the principle of localization) lemma, then

and apply Watson’s

A formal ‘proof’ is this: change a into +oo, expand f(x) into a power series in x and calculate the integrals obtained. A rigorous proof is several lines longer. It is perhaps worth noting that Watson’s lemma can be proved using integration by parts:

F(4 = e CfkW)M’@) + f(& - W’( - W-“‘” dt s

etc. Watson’s lemma easily carries over to the case when the integrand contains a logarithm:

Note that a,, is expressed in terms of a finite number of the derivatives off and S at x = 0. Therefore to calculate a 2n we can assume that these functions are analytic. We have (E > 0 small)

= $ f(n + 3)[(2n)!]-‘a2,A-“-“2;

M.V.

loo

a

II. Asymptotic

Fedoryuk

(2n)! f(qqt))qqt)t-2”-’ 2” = 427ci Jfl=E (2n)! f(x) 27ci IXI=E [-S(x)]““” d

dx

WN dx

[I++)],

x=x0(l)

(1.25)

U(4: Ix - XOWI 5 Il(~)IS~~(xo(~),~)l-“2

x+a.

We use the integral representation of the gamma-function T(x+

101

The simplest sufficient conditions are these. Let the integral be taken over a neighbourhood

[~~+1’2nx))~x&

As is clear from (1.22) the coefficients of the expansion are the values of linear differential operators on f(x) at stationary points (for a fixed function S(x)). The coefficients of all asymptotic expansions which arise in applications of the methods of Laplace, stationary phase and steepest descent, are constructed in exactly the same way. The contributions of degenerate maximum points can also be calculated using Watson’s lemma [ll], [6]. We will give a classical example-the proof of Stirling’s formula T(x + 1) = J&e-“x”

in Analysis

where f(x, 2) depends weakly on 1. Let S(x, A), for il 2 1, > 0, have a unique maximum point x0(A) which is, moreover, nondegenerate: S&(x,(A), 1) < 0. If the maximum is sufficiently sharp, then there is the asymptotic formula

dt

= ~~~,_~(x)[~~+1’2x-‘“-1 = (y(

Methods

l)=~~txe-‘dt=~~er”“‘dt.

Transform this to the form (1.15) by making the substitution

t + xt

O”ex(logl-t) & s0 The function S(t) = log t - t has a unique maximum point to = 1, S(t,) = - 1, S”(t,) = - 1, and from (1.21) and the principle of localization, (1.23) follows. In this example the integrand T(x + 1) = xx+l

f(t,x) = txe-’ has a maximum point t,(x) = x, which goes off to infinity as x + +co. The change of variable t -+ xt fixes the maximum point. This simple trick--fixing the maximum point-is very effective and allows us to calculate the asymptotic form of many special functions: for example, the Macdonald function K,(x) and the Weber function D-,-,(x) as x -+ +co [12], [7]. Consider a Laplace integral of the general form W) (1.24) F(1) = f(x, 4 exp {Sk 4) dx, s a(l)

of the maximum hood let

point, where ~(1) > 0, p( +oo) = +co, and in this neighbour-

Xx(x, 4 = %x(x0(4,4 Cl + 81(x, 41, fk 4 = mom 4 Cl +

&2(X,

41,

where lim I++m Ej(X,n) = 0 uniformly with respect to x E U(1). (1.25) then gives an asymptotic formula for the contribution of the maximum point, but there still remains the problem of estimating the ‘tail’-the integral over Z(n)\ U(A). It is quite hopeless to try to obtain a theorem of a general nature about integrals of the form (1.24), but in many concrete problems an answer is given by (1.25). Consider the Laplace transform co F(l) = exp{Ax - S(x)} dx. s0 Let S(x) be such that 1. s’(x) -+ +co, x%S”(x) + +c.o, x + +co. 2. There is a positive function p(x), ,u( +co) = +co, such that s”(y) - s”(x)(lx - yl 5 p(x)ls”(x)I-“2, If S(x) is a polynomial a > 1, or an exponential

x + +a).

of degree n 2 2, S( +oo) = +cq or a power ax’, a > 0, aebx, then conditions 1 and 2 are satisfied.

In the present case max [xi - S(x)] = $(A), X20 for 1~ 1, is attained at a unique point x = x,(l), at which S’(x) = 1. As is well known, the function g(n) is called the Young adjoint of S(x). For 1 -*+a0 2rc ,%(I) W - JS”(XO(4) (see [l 11). In these kinds of problems it is appropriate to limit ourselves to the leading term of the asymptotic formula, since the following terms of the expansion are not only cumbersome but also require conditions on the integrand which are very lengthy to formulate.

M.V.

102

Fedoryuk

II. Asymptotic

The Laplace method is based on the following facts. 1. The principle of localization. 2. Local simplification of S(x) by a change of variable. 3. The investigation of standard integrals (Watson’s lemma). In 0 3 we shall see that the method of stationary phase depends on the same facts. 2.3. The Many-Dimensional

Laplace Method.

Consider the integral (1.16).

Introduce the notations 1 s j, k 5 n. point x0 of S(x) is called nondegenerate if the Hessian

Let max,, DS(x) = S(x”) be attained at only one interior point x0 of D and let x0 be a nondegenerate maximum point. Then for il E S,, [,?I-+ co (for the sector S, see subsection 2.2 of Chapter 1) F(A) =

F

n/zldet S”(x”)l-“‘[f(xO)

+ O(n-‘)I

exp(lS(xO)}

0 This is a ‘very effective’ formula of the many-dimensional is an asymptotic expansion

(1.26)

Laplace method. There

F(A) = exp{k‘?(x”)}~-“‘2 g cki-“.

(1.27)

The proofs of these formulae are based on the fact that S(x) locally reduces to a sum of squares. Morse lemma. Let x0 be a nondegenerate stationary change of variables x = d(y) such that 40

= x0,

det#‘(O) = 1,

point of S(x). Then by a

4~ C”,

S reduces locally to the form

S(X)= s(xo)+ t J$ PjYj2.

(1.28)

Here pI,. . . , p,, are the eigenvalues of S”(x’).

The inverse vector-function y = $(x) is of class C” in a small neighbourhood of x0. If S(x) is analytic at x0, then 4(y) and Ii/(x) are analytic at y = 0 and x = x0 respectively. By the principle of localization we can assume that D is a small neighbourhood of x0. A change of variables reduces F(i) to the form

f(x,4exp{S(x,4)dx. s UC4

Suppose that the following conditions, analogous to those given in subsection 2.3 of Chapter 1, hold. 1. The function S(x, A), for 2 >> 1, has a unique, and moreover nondegenerate, maximum point x0@). As U(A) we take the neighbourhood JWIX

- x0(4 2 PC4

where ~(2) > 0, p( +co) = +cc and $ is a symmetric, positive definite matrix (if A is a symmetric, positive definite matrix, then fi is unambiguously defined). 2. For x E U(n), 2 -+ +co S&(x, 2) = s:Jx”(n), ;1)[l + &1(X, EJ],

fk4

k=O

103

The original neighbourhood of x0 can be chosen so that V is a cube lyjl < 6, 1 5 j 5 n. After this it remains to apply the one-dimensional Laplace method sequentially with respect to the variables y 1,. . . , y,. It is equally simple to analyse the case when the nondegenerate maximum point x0 lies on the boundary of the domain [ 121. Let us discuss a more complicated dependence on the parameter 1:

at x0 is

Hs(xo) E det S”(x’) # 0.

in Analysis

I dy,

F(A) =

A stationary non-zero:

Methods

where Ed, &2+ 0 uniformly Then, as 1. -+ +co,

=f(x0(4,4C1

+

&2(X,41,

with respect to x E U(A).

F(A) - (27~)“‘~ (det S:l,(x,n)(-112f(x,~)exp(S(x,~))(,=,,(,,.

Let x0 be a degenerate maximum point. Clearly we must first of all clarify to which simpler form it may be possible to reduce S(x) by a change of variables. This classical problem of analysis (the so-called resolution of singularities) was solved quite recently by Hironaka and others. We will give only a brief account of the results, since they are explained in more detail in [ 11. Let x = 0 be a stationary point of S(x) and let S(x,, . . . , x,) be an analytic function in a neighbourhood of x = 0, that is, S(x) = Can,. . U,~X;I . . . x2, where 0 5 elk < co and the series converges for complex x, 1x1 < r. The stationary point x = 0 is called isolated if the equation S’(x) = 0, for small complex (xl, has the unique solution x = 0. Make a change of variables x = d(y), that is, Xl = G4(Yl,...,Y,),...?X, = f#n,(Y,,...,Y,) and let 4(O) = 0. The mapping y + x: x = d(y) is called a diffeomorphism

of class

M.V.

104

II. Asymptotic

Fedoryuk

dS A o,=dx,

f

i C,,I-‘k(logA)’ ( I=0

At the points of r, for which g

(1.29) >

of x0. Note that V,(c) = V(c) for

A dx, A...

A dx,.

(1.30)

This form is uniquely defined on the level set &: S(x) = c if FS(x) # 0 on this set.

= P(Y)>

k=O

105

If an asymptotic expansion for V,(c), as c + + 0, is known, then using Watson’s lemma, or a modification of it, it is easy to find the asymptotic form of F(A). We will transform the integral V;(c). The Leray-Gel’fand dgferential form os is a form of degree n - 1 such that

where P(y) is a polynomial. Moreover, P can be taken to be a sufficiently long segment of the Taylor series of S. For more general conditions under which S(x) reduces locally to a polynomial see [l]. Let us consider (1.16), where S(x) is a polynomial and D is a small neighbourhood of the maximum point x0. Then as ;1+ +cc there is the asymptotic expansion F(l) = exp{lS(x’)}

in Analysis

Put f(x) = 0 outside some neighbourhood f(x) E 1.

C” if there are neighbourhoods U and V of x = 0 and y = 0 respectively, such that: 1. the vector-function x = 4(y) is a one-to-one mapping of V onto U; 2. the vector-function x = 4(y) and the inverse vector-function y = t/~(x) are infinitely differentiable in the domains U and V respectively. Let S(x) be analytic in a neighbourhood of x = 0, which is an isolated critical (stationary) point. Then there is a change of variables x = b(y) (a diffeomorphism) such that WY))

Methods

# 0, we have J

us(x) = (- l)J

.dx, A ... A dxj-l A dxj+l

A

...

dx,

A

8(x)/8x,

In the general case us(x) = l~s(~)l-~ -f (-l)‘ds(x)dxl jKiJ

A

...

A

dGj A

...

A

dx,

(1.31)

UXj

1 (the hat means that the corresponding i form has a simple geometric meaning:

factor is missing). The Leray-Gel’fand J-1..\

Here the r, are rational,

q,(x)

=

n

7 5 r, < rl < ..., lim r, = +co.

= 1r7”l

\I)

where o(x) is the area (that is, the (n - 1)-dimensional surface S(x) = c. The integral F(i) can be represented in the form

This is a typical existence theorem. In [Z] results are given which enable r, and N to be calculated, but an explicit formula for Coo has not yet been found. The problem of calculating the asymptotic form of F(I) as i -+ +co is equivalent to the following geometric problem. Let V(c), c 2 0, be the volume of the set

F(A) = exp{E(x’)}

coe-“‘q(c)dc, s0

Y-(c) =

MC = {x: S(x) - S(x”) 2 -c},

containing the point x0. For small c > 0 MC, in general position, is diffeomorphic to an n-dimensional sphere, and its boundary

F(A) = exp{lS(xO)}

v,(c) =r JMc

f(x) dx.

m e-‘cdI$(c), s0

(1.32)

f(X)%(X). s S(x)=S(d+c

Under the same conditions on x0 as above, we have

r, = {x: S(x) - S(xO) = -c) is an (n - 1)-dimensional closed surface enclosing x0. In addition VS(x) # 0 in r, for small c > 0. The many-dimensional integral F(i) reduces to a onedimensional one:

volume) on the hyper-

,

EHere the pj are rational

I

n

-2

c-+ +o

numbers, 1 spo
<“‘,

lim pk = +co. k-m

‘W e make a remark on additional parameters. In (1.16) suppose that f and S smoothly depend on parameters c( = (CQ,. . . , CX,): f = f(x, 4,

s = S(x, a).

M.V.

106

II. Asymptotic

Fedoryuk

the rapid oscillation of the exponential is the:

If S has a nondegenerate maximum point x0 for CI= 0, then for small lcll it has a nearby nondegenerate maximum point X’(U). The formulae (1.26) and (1.27) still hold for 1~1< a,, if a0 2 0 is sufficiently small. This remark also applies to all the integrals in @ 3,4 of Chapter 1. 2.4. The Logarithmic

Asymptotic

Riemann-Lehesgue

lemma. Iff(x) )yrn

Form of Laplace Integrals. In a number of

questions (for example, in probability theory when calculating large deviations) a rougher asymptotic formula for Laplace integrals is required, namely, one or two of the first terms of the asymptotic formula for log F(1). We will give some results of this kind. Let S(x) be continuous for x E R” and let f(x) have compact support, be measurable and almost everywhere bounded. We introduce the notations: J? is the set on which max,E,,,,fS(x) = it4 is attained,

b++aO

1ogw

_

b f(x)e”“dx sa

= 0.

f(x)eiLscx) dt

(1.36)

where D is a bounded domain in R”. Lemma. Let f(x) E C;(D),

A

S(x) E C”‘(R”)

and U(x)

# 0 for x E suppf. Then

F@) = O(i-m), 2 + +co.

2. If, in addition, lim loi% W) -=cc>o, c++o loge thenasil+

ei’s(X). A more general result of this form

sD

:

M

107

E L, (a, b), then

F(1) =

Denote by V(c) the volume of the set JzY~. Let .f(x) 2 6 > 0 in some neighbourhood of A. Then there are the theorems: iirn

in Analysis

It is impossible to obtain any more precise information on the speed of decrease of the latter integral under these conditions. In contrast to the Laplace integral, the asymptotic behaviour of Fourier integrals depends essentially on the differential properties of f(x) and S(x) throughout the domain of integration. The many-dimensional Fourier integrals has the form

-I”e, = {x E suppf: S(x) 2 M - c}.

1

Methods

This means that F(A) = O(AmN) for any N > 0. We give the proof: it is interesting in that it contains useful formulae. The differential operator

(1.33)

+cc logF(jl) = MA - CIlog1 + o(log2)

(1.34)

The proposition converse to 2 is true: if I/(O) = 0 and (1.34) holds, then (1.33) holds; formula (1.34) is true if S(x) is a polynomial. The set J%?may be a manifold of dimension II - 1. An example is: s(x) = -

( ’ x2mj-l~m”, jIYl j

where mj 2 1 are integers. In this case &? is the hypersurface cJ=i xjZ”‘j = 1 and V(c) - Ac1’(2mo),c + +o.

0 3. The Method of Stationary 3.1. The Principle of Localization.

Phase

Consider the Fourier integral

[ has the property that L(ei”S’“‘) = ile i’s(X). Integrating by parts we obtain h F(A) = k exp { i,@(x)} (‘Lf)(x) dx, j

sD

b where ‘L is the adjoint operator. Consequently F(A) = O(A-‘) and successive integrations by parts yields the estimate F(A) = O(K”), N > 0 arbitrary. It follows from this lemma that iff, S E Cm@), then the basic contributions to the asymptotic form of F(1) are made by: I: A. The stationary points of the phase S(x). 1, B. The boundary of the domain of integration. [ In the general case contributions are also made by the points of discontinuity 1 off and S and their derivatives. All these points are called critical for F(i). i Suppose that in D there are a finite number of critical points xl,. . . ,xk. Construct /‘ a partition of unity

b

F(I) =

s0

f(x)ei”s(x)

dx,

(1.35)

where I = [a, b] is a finite interval. The function S(x) is real-valued and is called the phase, f(x) may be complex-valued. For I >> 1 F(A) will be small because of

'

E

i:

j=l

rljCx)

+

rlrCx)

+

6(X),

XED

of class C”. The functions vi(x) have compact support each of which contains precisely one critical point xj and does not intersect r, qj(x) = 1 in some neigh-

M.V.

108

II. Asymptotic

Fedoryuk

bourhood of xj. The function qT(x) is equal to zero outside some strip close to the boundary r, qT(x) = 1 in a smaller strip containing r. The function f(x) has compact support; VS(x) # 0 on its support. An explicit form for the ‘cut-off’ functions qj, ylT, ij is not essential. Then, by the lemma, F(l) = 2

V,j(A)

(1.37)

+ VT(l) + O(Amm)

Methods

in Analysis

109

The contributions V,(n), &,(A) are calculated using integration by parts (Chapter 1, 0 1) and are given by formula (1.11) with x = a and x = b respectively. In a small neighbourhood of x0, using a change of variables, S reduces to the form S(x,) f t”, after which it remains to use Erdelyi’s lemma. We give a final formula for a nondegenerate stationary point (S”(x,) # 0). The leading term of the asymptotic formula is

j=l

Ko@)-JF n,s,,(xo),exp{.Wxo) + $6(x0) } Cf(xo)+ W-l)1

where the integrals v,,(a) =

f(X)QX) eXp{ iaS(

6(x,) = sgn S”(x,)

dX9

b(4 =sfWrA4exp~W41 dx. sD

(1.40)

There is the asymptotic expansion

(1.38)

Ko,(4 = 1-l” exp iAS

D

are called respectively the contributions due to the critical boundary r. Formula (1.37) is the principle of localization.

points xj and the

3.2. The One-Dimensional Case. The calculation of the contributions proceeds according to the same scheme as in the Laplace method: the phase S(x) is reduced to a possibly simpler form by a suitable change of variables and the standard integral obtained is then investigated. An analogue of Watson’s lemma is: Erdelyi’s lemma ([lo], [12]). Let a 2 1, /I > 0, and let the function f(x) E Cm( [0, a]) and vanish together with all its derivatives at x = a. Then, as 2 -+ +a, there is the asymptotic expansion II dx = f’ ~‘&(“+fl)/O’, xD-lf(x)ei”x” n=O s0

C”+(!!)exp{iy

“).

F(A) =

X t~-lei”‘ndt. a f(x)d s0 sm

Here the integral is taken over the ray t = x + peirr@), 0 5 p < cc in the complex t plane. We note further that for 2 = ip, p > 0, (1.39) becomes (1.17) (Watson’s lemma). Consider the integral (1.36), wheref, S E Cm(Z) and S(x) has a unique stationary point x0, a < x0 < b, of order n. Then F(1) = v,(n) + v,(n) + v,&n) + O(Xrn).

yi5(xo)

2 C&n, I n=O

where we have put

SC%x0) = JWX)

- wo))~(xoY~c4.

Formula (1.40) is without doubt very effective. For the formulae for a stationary point of order n 2 3, see [lo], [12], it has order O(n-lin). The contribution of an endpoint has order 0(2-r) and that of a stationary point has order O(1-‘i2), so the latter is generally larger. Finally, we consider a more complicated dependence of the phase on the parameter A: F(A) =

(1.39)

Moreover this expansion is valid as IA1+ co, 0 5 arg2 5 n, uniformly with respect to arg A. It is curious that Erdelyi proved this lemma using integration by parts. The first step is:

+

1

s Ik%(4l’P(4

exp(iS(x, A)} dx,

where x,(l) is a stationary point of S for 1~ 1. We will give conditions which

under

(1.42) as1+ +co. 1”. For 2 >> 1, x = x,(A) is the unique point in the interval Ix - x0(1)1 5 p(i) at which S:(x,A) = 0. 2”. AsA+ +cc p(a) -+ +a. Xx(x,(4,4

if IhI < p(i).

- CAx&)

P*@)xx(x0(4? 4 -+ +a, + h, 4 = ~[p-~(~)(s~~x,(A),

A)-“‘]

M.V.

110

II. Asymptotic

Fedoryuk

Case. Let x0 E D be a nondegenerate

3.3. The Many-Dimensional

point of the phase S(x). Then for the contribution asymptotic expansion V,,(A) = exp{ ikS(x”)}I-“‘2

f

CJk,

stationary of this point there is the A+ +Go.

(1.43)

The leading term of the asymptotic formula is n/2

ldet S”(x’)(-‘I2

x exp{~sgnY(x”))[j(xo)

in Analysis

111

where N 2 1 is an integer and the ~j are differential forms wj(x) = IVS(X)I-~ k$‘l F(‘L’-‘f)(x)dx,

A “. A d$k A ... A dx,

k

k=O

V,,(A) = exp{iAS(x’)}

Methods

+ 0(X’)]

In particular, w1 = f(x)w,, where os is the Leray-Gel’fand differential form (3 2 of Chapter 1). Consider the phase S on the boundary r, and let x0 be a stationary contraction point of S on IY If r is given by an equation g(x) = 0, Vg(x) # 0 on J’, then V’s(x) = pFg(x’). If VS(x’) # 0, then x0 is called a stationary point of type II. The idea behind such nondegenerate points is easy to formulate in local coordinates, and for their contribution there is the asymptotic formula

(1.44) V,,(L) = exp(ilS(x”)}il-(“+‘)‘2

Here sgnS”(x’) is the difference between the number of positive and negative eigenvalues of S”(x’). Formulae (1.43), (1.44) are the fundamental formulae in the many-dimensional method of stationary phase. We outline the proof. Choose a neighbourhood U of x0 small enough so that the Morse lemma (92) can be used in 0, and by a change of variables x = 4(y) reduce the phase to the form S(x) = S(x”) + $&i pjyj2, where the pj are the eigenvalues of S”(x’). We can assume that suppf c U and that, in terms of the variables y, U is a cube lyjl 5 6, 1 5 j 5 n. Then

kzo

ckn-k

(1.46)

This contribution is of lower order than the contribution of an interior stationary point, compare (1.43) and (1.46). Nevertheless the contribution of r can be very large, for example if S(x) = const. on r, see (1.45). In this case the entire boundary consists of stationary points of type II. There is an invariant formula for the coefficient Co. Let x0 be a stationary point and let r be given locally by an equation g(x) = 0. Introduce the Lagrange function L(x,p) = S(x) + pg(x), which will have a stationary point (x”,po) for some value p = ,u~ Denote Q(x, p) = det ‘:iiil’).

x det4~)

dyl . . . dy,

Then (here x = x0, ,U = po, Vg(x’) = poVS(xo))

We now apply the one-dimensional method of stationary phase to this integral successively with respect to y,, . . . , yn, which reduces to (1.43) and (1.44). These formulae have been proved by the author [12]. Let the nondegenerate stationary point x ’ lie on the boundary of D. Then formula (1.44) holds for the leading term of the contribution V,,(A) with an additional multiplier of 3 on the right. Let S(x) be analytic in a complex neighbourhood of a point x0, which is an isolated critical point. Then (1.24) holds for V,,(A) as A + +oo, the only difference being that the exponential has the form exp(iAS(xO)}. This problem is considered in more detail in [2]. Let us consider the contribution of the boundary, V,(A). If there are no stationary points of the phase on f, then evaluating the asymptotic form of V,(n) reduces to calculating the asymptotic form of integrals over the boundary. Let L be the differential operator of subsection 3.1 of Chapter 1 (LeiAS = iAeiAs). The same integration by parts as in the lemma of that subsection yields the formula V,(A) = 5 (i/l-j j=l

exp{ilS(x)}wj sr

+ O(A-N-‘).

(1.45)

co = (2n)‘“-‘Y2~oldet QI-“*exp $(sgnQ + 2) , 1 1 where the orientation of Tis such that Vg(x”) is directed along the inner normal to r (see [35]). The method of stationary phase has been applied in such a variety of problems in mathematics, mechanics, mathematical and theoretical physics, chemistry and other natural sciences, that even the compilation of a list of all these problems is only within the reach of a large collective of specialists from the different sciences. The asymptotic form of special functions, solutions of partial differential equations and pseudo-differential equations, problems in number theory, probability theory, statistical physics, diffraction and diffusion of acoustic, electromagnetic and elastic waves, waves on surfaces of fluids, waves in a plasma, neutron transitions in nuclear reactors, et cetera. We limit ourselves to giving some broad classes of integrals to which we can apply the method of stationary phase. 1. The Fourier transform of the characteristic function of a set. Let D be a bounded domain in R” and let

M.V.

112

II. Asymptotic

Fedoryuk

i(x, 0dx, CD(t) =r4x)e JD

so ii, is the Fourier transform of the characteristic function x0(x). The asymptotic form of CD(t), as 151+ co, has been fairly completely analysed in the case of convex D. An analogous problem is this: investigate the asymptotic behaviour as 2 --t +GO of the integral

The function K(x) is such that K(0) = 0, K(x) > 0 for x # 0 and the equation K(x) = 1 defines a smooth closed manifold of dimension n - 1 in R”. Integrals of these types arise, for example, in problems in number theory. 2. Principal values of integrals. If P(x) is a polynomial, then the equation P(x) = 0 defines an algebraic manifold in R” having a finite number of connected components. Let one of them, M, be a smooth manifold of dimension II - 1, and let 4(x) E Cz(R”) be concentrated on a small neighbourhood of M. Iptegrals of the form

arise, for example, in the investigation P

4; (

“\YJ

Methods

-

\%YJ

in Analysis

~

113

UthJ

where x = x(p). In symmetric form we have x = s’(p),

P = s’(x),

S(x) + s”(P) = (x,p).

(1.47) classical

The Legendre transform has been widely applied in mathematics, mechanics, thermodynamics, etc. Let us make some necessary assumptions: det S”(x’) # 0 and d(x) = 0 outside a small neighbourhood of x0. If f = $eus , then the phase function of the integral Fl,x-rpf is equal to S(x) - (x,p). Its stationary points are defined by s’(x) = p, and_ the value of the phase at a stationary point is equal to S(x(p)) - (x(p),p) = -S(p). Under our assumptions the stationary point is unique if p lies close to p” = S’(xO), an d is ’ nondegenerate. Therefore we can use (1.44) and obtain F rl,x+p(qki”S) = e- i&) 1det ytx) 1-V e W4kW

x c4c4 + w-‘u

1x=x(p).

Thus, the Fourier l-transform transforms the phase S(x) to the phase -s(p) obtained from S by the Legendre transform. In reality, the relationship between the Fourier I-transform and the Legendre transform is significantly deeper. We have from (1.47) det s”(x) = det p’(x),

of the asymptotic form of the solutions of

det S”(p) = det x’(p)

(1.48)

and we obtain the symmetric formula

u=J

F n,x-,(~(x)ldet~‘(~)11’4ex~{i~S(x)})

1

3.4. The Fourier Transform and the Legendre Transform. Let us introduce the Fourier l-transform (an idea due to V.P. Maslov), depending on a parameter

1 > 0:

=e -w4 ldet x’(~)l”~exp(

-iAs(

[4(x(p))

+ 0(X’)]

(1.49)

Here CI is the index of inertia of the symmetric matrix x’(p), that is, the number of negative eigenvalues.

exp{ -i4x,p))f(x)dx. $4. The Method of Steepest Descent

The inverse transform takes the form

CK~+.&+l(x) = (-&).,

lRn exp{iQ,p))g(p)dp.

Here ,,&? = e * irri4, (x,p) = I,?+ xjpj. Let b(x) E C;(R”), S(x) E P(R”). By calculating the asymptotic form of the Fourier I-transform of a rapidly oscillating function fjeiLs we will see a remarkable connection between the Fourier Atransform and the Legendre transform. We recall that the Legendre transform is the mapping of the pair k W))

+ (P, s”(P))

which is constructed as follows. From the equation s’(x) = p find x = x(p) and then put

4.1. Heuristic

Considerations.

Consider the integral

F(A) =

f(z)eA”(‘)dz, sY

(1.50)

where y is a finite smooth curve in the complex z plane and the functions f(z) and S(z) are analytic in a neighbourhood of y. It is required to find the asymptotic form of F(l) as i -+ co. By Cauchy’s integral theorem we can deform y, keeping the endpoints fixed; .‘le value of the integral is not changed. (To be precise we can temporarily assume lat f and S are polynomials). Assume that y can be deformed to a contour 7 ,rh

thot.

M.V.

114

II. Asymptotic

Fedoryuk

is attained at just one point z. E 7, 1”. max,.jle”S’r’I 2”. Im S(Z) K const. in some arc To containing zo. Then we can apply Laplace’s method to the integral obtained. In fact the integral over y\yo has order O(exp{I(ReS(z,) - c)}), c > 0 (see (1.17)). The integral over jjo is equal to f(z)exp{AReS(z))dz. s io This is a Laplace integral: if z = d(t), t, 5 t 5 t, is the equation of integral takes the form exp{ilImS(z,))

*’ f(&))d’(t) s fl

exp{J. Re W(t)))

70,

then the

S(Z,) = 0, and by the condition

2’, $m

S(Z,) = 0 k is the derivative (

along 9. Consequently, s’(z,) = 0.

4.2. The Local Structure of the Level Lines of Harmonic s’(z,)

dt

(1.51)

Such a point is called a saddle point of S(z). Let us explain the name. The function U(x,y) = R~s(z) is harmonic; its stationary points are those at which S(Z) = 0. The function u has no maximum or minimum points, hence all its stationary points are saddles. Take the simplest example: S(z) = .z~, then u = x2 - Y2. This equation defines a hyperbolic paraboloid in three-dimensional space, which in a neighbourhood of the origin is constructed like a mountain pass. Suppose that y can be deformed to a contour 7 SO that: 1. max,.? Re S(Z) is attained at a finite number of points zi, . . . , zk, which are either saddle points of S(z) or endpoints of the contour. 2. Im S(Z) E const. on j7 in a neighbourhood of each of the points zl,. . . , zk. Such a contour is called a saddle contour. Then we apply the localization principle (subsection 2.1 of Chapter l), and the asymptotic form of ~(2) is equal to the sum of the contributions of the points or,. . . , zk, that is, of the integrals over small arcs containing these points. There are no new analytic problems, by comparison with Laplace’s method. As before, all the asymptotic expansions can be obtained using Watson’s lemma (Chapter 1, 5 2). But first we must solve a topological problem: deforming y to a saddle path. The function S(Z) can have many (even infinitely many) saddle points, and to clarify just which ones determine the asymptotic form of the integral is a problem which, as a rule, is nontrivial. Moreover, this problem is essentially more complicated in the many-dimensional case. Therefore in connection with the method of steepest descent the word ‘example’ is not very suitable; each example is, perhaps, a small but fairly serious mathematical investigation.

in Analysis

115

The process of deformation of the contour can be pictured as follows. Let izrl be the surface u = U(X, y) in three-dimensional space, and let y* be a curve which lies on M and projects onto y. Imagine that y* is a heavy wire, with complete freedom to stretch. Then, under the action of a heavy weight is creeps downwards (the ends of Y* are fixed). Since M has no peaks or troughs, then finally from the ends the wire either goes down to a saddle point or the high point itself will be one of the ends of the wire.

u(x, Y) + io(x, y) be analytic in a neighbourhood

with a real-valued function in the exponential. Let z. be an interior point of y”. Since max,. ,-Re S(Z) = Re S(G), then :Re

Methods

= ... = s’“-“(z,)

= 0,

Functions. Let ~(2) =

of a point zo, S(Z) + const. Then S’“‘(z,)

# 0,

n 2 2,

(if n = 1, then S’(zo) # 0). If U is a sufficiently small neighbourhood by a change of variable, z = d(w), S can be reduced to the form S(z) = S(z,) + w”,

of

zo,

then

WEl/

Here V is a neighbourhood of w = 0 and d(w) is analytic and univalent in I/ Therefore locally the level lines 1,: u = c, Tc:0 = c are structured in the same way as for zn. Ifn = 1, then the level lines 1,and Tcare analytic curves, forming two orthogonal families. Let S(z) = a, U(z) the disc IzI 5 R and y a simple curve with endpoints zl, z2 lying on IzI = R. Let max ZEy Re S(z) be attained at an interior point z. of Y (and only there); for example, if y is the semicircle 1Z) = R, Re z 2 0. Let f(z) also be analytic in U. Then it is possible to change y into a segment y”= [zr , z2] and in this connection max Re S(z) < mas Re S(z) zsi ZEY Thus, if M, = max, EyReS(z) is attained at some (unique) point zo, which is neither a saddle point nor an endpoint of the contour, then y can be moved into a domain of smaller values of ReS(z). By standard arguments (for example, applying the Heine-Bore1 lemma) we can show that if among the points at which M, is attained there are neither saddle points nor endpoints, then it can be moved into a domain of smaller values of Re S(z). Let 20 be a simple saddle point, that is, S’(z,) = 0, S”(z,) # 0. Consider the simplest example: S(z) = z2, then the level lines of the real and imaginary parts of S are given by the equations u E x2 -y2=c,o-2xy=c.Thelevellineu=O consists of two straight lines dividing the plane into four sectors. In two of these u < 0, in the remaining two u > 0, since the points of the surface u = x2 - y2 lie respectively below and above the saddle point (0,O). The level lines u = C, u = c, for c # 0, form an orthogonal family of equilateral hyperboli. The level line To:u = 0 consists of two straight lines x = 0, JJ= 0. Of most interest is x = 0, which is the line of steepest descent. Along this line the function Re S(z) decreases most rapidly as we move away from the saddle point. The level

M.V.

116

Fedoryuk

II. Asymptotic

Fig. 1

lines in a neighbourhood of a simple saddle point (Fig. 1) are structured in precisely the same way. The sectors in which Re S(z) < Re S(z,), are shaded, the line of steepest descent is dotted. If y is a simple curve whose endpoints lie in different sectors in which Re S(z) < ReS(z,) (Fig. l), then y can be deformed into a saddle line. In fact, it can be replaced by an arc of the line of steepest descent ?, and the two arcs which join the ends of the contours y and &, and on which Re S(z) < Re S(z,). The contour y”constructed in this way has the minimax property (1.52) M,- = mix max Re S(z), Y’ ZEY’ where the minimum is taken over all contours such that the integrals over y and y’ are equal. The value M,- cannot be made smaller by deforming 7. A saddle contour also has the minimax property. Moreover, if there is a minimax contour $7for (1 SO), then it is equivalent to a saddle contour. If z0 is a saddle point of order n, then the level lines Re S(z) = Re S(z,) consist of 2n analytic curves; the angle between two adjacent curves at z0 is equal to z/n. They divide a neighbourhood of z0 into 2n sectors in which alternately Re S(z) < ReS(z,), ReS(z) > Re S(z,). The level lines ImS(z) = ImS(z,) are arranged in exactly the same way. If y is a simple curve, z0 E y and the endpoints of y lie in two distinct sectors Dk in which Re S(z) < Re S(z,), then y can be deformed into a saddle contour. Therefore we can introduce a definition of a saddle contour which is somewhat different to that of subsection 4.1. Namely, among the points at which max,. yRe S(z) is attained there are saddle points or endpoints of the contour, ZIP..., zk and y cannot be moved into a domain of smaller values of Re S(z) in neighbourhoods of these saddle points. Such a definition is convenient for the solution of concrete problems without the necessity to worry about whether Im S(z) is constant on the contour close to a saddle point. 4.3. Asymptotic Formulae. 1. The contribution

of an interior simple saddle point.

Let z0 be an interior point of the contour and let S’(z,) = 0, S”(z,) # 0. Then, as1+ +cc

2 v,#)=Al--

IIf

+ W-‘)I

expifW0))

(1.53)

Methods

in Analysis

117

The choice of branch of the root is as follows: arg Jm is equal to the angle between the positive direction to the tangent to the line of steepest descent 1, passing through zo, and the positive direction of the real axis. The orientation of 1 is defined by the orientation of y. For P&(1) there is an asymptotic expansion of the form (1.22). Formula (1.53) is the fundamental formula of the one-dimensional method of steepest descent. 2. The contribution of the boundary points. Let z0 be the initial point of the contour y, max,. y Re S(z) = Re S(z,)and S’(z,) # 0. Then as A--) +co I/ (A) = -f(zo) =o

+ OW’) ~s(zcJ

exp W(zo)l

(1.54)

For this contribution there is the asymptotic expansion (1.11) For formulae for the contributions of saddle points of order 3 2, see [lo], [ 121. 4.4. Examples of the Application of the Method of Steepest Descent. Example 1. Let us find the asymptotic form as x + +co of the Airy function A.(x)=;j-:cos(;+tx)dt.

The role of this function in analysis and mathematical in Chapter 2. First of all we transform the integral:

physics will be explained

di(x) = 27L L -oo O”ei(t3/3+Wdt. s

I The change of variable

t+

fi

(1.55)

leads to an integral of the form (1.50):

&i(x)=&271 -oD meAs(f) dt’

s where S(t) = i(t3/3 + t), 1 = x3/‘, y the real axis. The function S(t) has two saddle points t,,, = fi, S(t,,,) = +2/3. Since ReS(t) > 0 = maxZE,ReS(t), then the saddle point t, makes no contribution to the asymptotic form of the integral-it is necessary to deform the contour into a domain of smaller values of ReS(t) (Chapter 1, subsection 4.2). This contour of integration is infinite, therefore we must examine the behaviour of Re S(t) at infinity. We have Re(it3) < 0 in the sectors D,: 0 < arg t < 7cf3, D, : 2~13 c arg t -K rc,D, : -47~13 < arg t < -2~13, so that ReS(t) + -cc as ItI + cc along any ray, with origin t = 0, which lies in one of these sectors. In the 1 remaining sectors ReS(t) + +co along any ray. By Jordan’s lemma, y can be 1 changed, for example, to any line Im t = c > 0, parallel to the real axis. Change ( y to the line $7:Im t = 1, passing through the saddle point t = i. On jj we have

M.V.

118

II. Asymptotic

Fedoryuk

t=i+z,--co
Re S(t) = - 2/3 - ?,

thus max, Ei Re S(t) is attained only at the saddle point t = i. 7 is a saddle contour, and the asymptotic form of the integral is equal to the contribution of this point. The fact that the contour is infinite is irrelevant; it is not hard to show that the integral over y”\&, where y”is a segment [ - 6 + i, 6 + i] containing the saddle point, is exponentially small by comparison with Iexp{1S(i)}j, see (1.17). It remains to use (1.53). We have found a branch of ,/m. Close to t = i we have S(t) - S(i) - -(t - i)‘, and the line of steepest descent 1 locally has the form Im(t - i) z 0. Therefore the tangents to y”and 1coincide at the saddle point, since arg&@$ = 0. Finally we obtain 1 -1/4e-2/3x3’2[1 + o(~-3/2)], (1.56)

Methods

in Analysis

119

in the domain D, whose boundary is the ray Cl, +co) and the upper half branch of the hyperbola: 5 = ,,/m 1 5 r] < co, and in the domain D,, symmetric with D, relative to the imaginary axis. A third domain lies in the half-plane Im t < 0. By Jordan’s lemma the ray [l, +oo) can be deformed to a ray lying in D,, and the ray (-co, l] can be deformed similarly. On the contour y” thus obtained Re S(t) -+ -cc as I tl + co and y”can be replaced by a finite contour with an error of the order o(e-“), c > 0 (see (1.17)). Finally we bbtain di(x)

[sin($x1312

= &;xll,4

+ S) + 0(/x1-‘“)I.

(1.58)

The asymptotic series takes the form do2;(x) = &[exp{

x++co.

In this example we can write down the whole asymptotic series:

-:x3,)

+ iexp {gx-)

z0 (- l)nCnx3nj2

z0 CnY3n12].

(1.59)

The coefficients C, are the same as in (1.57), the choice of branches for the roots are as follows: c = T((3n + 1)/2) n 33”(2n)!

Jx

We will examine (1.55) in another way. Let us find the line of steepest descent 1 passing through the saddle point t, = i. Put t = 5 + iv, then the equation ReS(t) = ReS(i) = 0 has the form ~(35~ - q2 + 3) = 0, since 1 is the upper branch of the hyperbola: 3t2 - q2 + 3 = 0, q > 0. The integral (1.58) is equal to the integral over 1.Since S(t) is real on 1,then the asymptotic formulae (1.56) and (1.57) are also applicable for complex x: 1x1+ co, largx( 5 742 - E (see $2). As will be shown below, (1.56) and (1.57) are applicable for 1x1-+ co, (argxl 5 71- E, that is, outside an arbitrarily narrow sector containing the semi-axis (-co, 0). Example 2. We will find the asymptotic form of dol;(x) as x + --00. The change reduces (1.55) to the form of variable t + fi &i(x)

=

g

lrn

e”S(‘)dt,

S(t)

=

i(T

-

t),

= ii&l,

$5

Example 3. Let us find the asymptotic form of &i(z) for complex z, IzI + co. Let D be the complex z plane with a cut along the negative real axis (-co,O]. Choose a branch of fi in D such that Re fi > 0, z E D. This branch is positive for real z = x > 0. The saddle points for the function S(t, z) = i(t3/3 + tz) are equal to t, , 2 = f i&, and lie in the upper and lower half-planes respectively. As in example 1, we change the contour of integration to a parallel line 7, passing through the saddle point tI(z). On y”we have t=i&+z,

--co
S(t,Z)=

-iZ3,,

+ iT-T2&.

Consequently &‘i(z)

=

L!&:~z~/~

-m

O” ,it3/3-t+dt s -00

1 = (xl3’2. The saddle points tl,2 = _+ 1 lie on the contour of integration and it is possible to apply the method of stationary phase, see (1.52). It is necessary only to deal with the following complication: y is infinite, (eaS(t)I = 1 on y, hence the integral is only conditionally convergent. Let us examine the domain Re S(t) < 0 (the values of Re S less than at the saddle points). Put t = 5 + iv. The equation Re S(t) = 0 takes the form q(3g2 q2 - 3) = 0 and determines the line q = 0 and a hyperbola. Therefore Re S(t) < 0

= ei”/41*j.

c-e

1 7c

-2132312

m e-t2J; s0

t3

cos - dt.

3

Watson’s lemma applies to the latter integral and we obtain that the asymptotic formulae (1.56) and (1.57) for the Airy function are valid as IzI + co in the sector larg z I Q 71- E < 71,uniformly in arg z. The situation is more complicated in the remaining sector 0,: larg( - z)l 5 E. In example 2 it was shown that for x + -co the asymptotic form of &L’(X) is

M.V.

120

Fedoryuk

II. Asymptotic

equal to the sum of the contributions of both saddle points. We will show that this is true for z E 0, JzI -+ co, so that (1.58) and (1.59), with x replaced by z, hold. The choice of branches for & and $ is the same. In a given concrete example we can, certainly, make sense of the structure of the lines of steepest descent passing through the saddle points, and show that the Airy integral is equal to the sum of the integrals over these lines. This work is very time consuming, since the level lines are cubic curves. We will proceed somewhat differently. Denote CI= arg( - z), lcl\ 5 E, and choose a branch of & so that & = l&l ieiai2. This branch is the same as in example 2. Let a 2 0. We change the contour of integration to a polygonal arc consisting of the rays [i&, and the segment 1: C-i&, i&l. On the first i& + co), (-ifi - co, -i&] ray ReS(t,z) = -iRez3”

O~z
Ret=i&+z,

- r’Re&,

so that max Re S is attained at r = 0. The same is also true for the second ray. On 1we have t

= i&p,

-lSpSl,

S(t,z)

= (p3/3

-

form as x + co of the integral

co F(x) =

e-t2n/Zn+izx

dt,

s -00

where n 2 2 is an integer. In this case there are 2n - 1 saddle points and it is necessary to clarify which of them determine the asymptotic form of the integral. A saddle contour, if it exists, is minimax, see (1.52). In this case the contour of integration can be replaced by any straight line parallel to the imaginary axis. We will seek the minimax (1.52) over the set of these contours. Put t = 5 + iv and we find

121

71 the maximum being attained at the points <1,2(q) = +qcos We 2(2n - 1)) . ( will now find the point q = q(x) at which min v(q) - xv -cO
Both points t,,,(x) are saddle points so 7 is the required saddle contour. It remains to make the change of variable t = x iKzn-i)r, in order to reduce the integral to the form (1.50), and to calculate the contribution of the saddle point by (1.53). We then obtain

w2n-1)

_ : + 140

+ qx-

7

as x + +co, where

r

A=

2-

271 2n - 1’

a=

( > 1- &

b=

1-i (

sin&,

2n >

cos h,

40= 2(2n 7c - 1).

Thus F(x) de exponentially as x + + co, and has infinitely many positive zeros. Since F( - x) = F(x), then by the same token we have found the asymptotic form of F(x) as x + --co. Example 5. Let us find the asymptotic form as x + +cc of the Bessel function J”(x), v 2 0. We will use the Sommerfeld integral representation of the Hankel function

s m+ni

H;“(x)

=

1

7ci -m

exp{xs sinh t - vt} dt.

As contour of integration we can take a polygonal arc with three links (-oo,O], [10,ire], [in, ix + co). It is sufficient to study the asymptotic form of the Hankel function since J,(x) = Re Hil’(x). The function S(t) = sinh t has infinitely many saddle points: cash t = 0, t, =

Simple calculations show that

v(q)= -cqzn,

in Analysis

p)z3’2.

By assumption Re z312= lzj 3/2cos(3a/2) > 0 and p3/3 - p monotonely decreases and attains a maximum for p = - 1, at the saddle point t2(z). If tx 5 0, then max, Ei Re S will be attained at t,(z), so that 7 is a saddle contour for all z E D, and the asymptotic form of the integral is equal to the sum of the contributions of these points. Thus the asymptotic form of the Airy function is different in different sectors of the complex plane. This fact-different asymptotic forms for one function in different domains of the complex plane-is called the Stokes phenomenon. Example 4. Let us find the asymptotic

: Methods

7 + ikz. Choose the “nearest” one to the contour of integration c = $0~

(2(2nn-

1)y

to = “2”. The

level line of Im sinh t = 1 passing through this point is given by the equation

M.V.

122

II. Asymptotic

Fedoryuk

sinh
t, has two

7112< ij < 71,

sinh 5 cos y = 1,

the second is symmetric with it relative to t,. The contour of integration can be deformed to 1, after which it remains to use formula (1.53). The leading term of the asymptotic formula is lp(x)

z

= ll

ei(x-vn/2-n/4)

[I

+

0(~-3/2],

i

I ;/ ,I $ i ! /

f’(r)

cd

The integral F(n) has the form F(n) = &

e.,,.)$ i IZI=‘I3 S(z) = p log f (z) - log z. of&

S(Z),

+ 1- u ro” ’

O
pnz-n-l dz,

Jlzl =p

where p 2 1 is an integer, that is, the asymptotic form of the n-th coefficient of the Taylor series of [f(z)] p”. P ro bl ems of this kind arise in statistical mechanics, probability theory, etc. Let us find a saddle contour 7. The value of F(n) is not changed if we replace 1z 1= p by any simple closed curve y which lies in the disc )z( < R and goes round the point z = 0. We will look for a minimax contour (see (1.52)), taking as the contours y only the circles IzI = r. Denote M(r) = maxlZl=, 1f(z)j, then max I[f(z)]“z-‘I

f’(r0)

f-(n) = J&

with nonnegative coefficients, whose radius of convergence is R, 0 < R < co. We will find the asymptotic form as n + +cc of the integral I

(

r.

n1

)

$1 + O(n-‘)l.

Not very many classes of integrals have been studied using the method of steepest descent. We list the basic ones [12]. 1. The Laplace transform

1

m

F(A) =

exp {-S(x)

+ Ax} dx.

s 0

The function S(x) is a polymonial

or a power uxa, uxa log x (c( > 1, Re a 2 0).

2. The Fourier transform co

exp{S(x) - ilx} dx.

F(A) = s -a,

The function S(x) is a polynomial

M’(r)

or a rational function.

3. The Mellin transform

= 1.

,

121 =r

Since ck 2 0 for all k, then M(r) = f(r) and, in addition, (Pringsheim’s theorem). Consequently,

M(r) + co as r -+ R

min (M”(r)r-‘) O
exists. Since M(r) is logarithmically convex (Hadamard’s and M(r) is smooth, then for 0 < r < R o<-

point

so that as n + cc

f(z) = k$O ckzk,

,L“‘

1

s”(ro) = Pf”(rO) f(r0)

Example 6. Consider a power series

123

pfo=F’

This formula is suitable for complex x, Jarg xl s 7r/2 - E, 1x1+ co. Uniformly relative to arg x, see Chapter 1,s 2.

[f(Z)]

in Analysis

,From the arguments given above it follows that z = r,, is a saddle “> that S’(r,) = 0. Furthermore

x++co.

71X

Methods

d

dlogr

Therefore the sought after minimum root of the equation

logM(r)

three circles theorem)

= rdf’(r). dr f(r)

is attained at a unique point r,, which is a

F(A) =

O” xA-1 es(X) dx. s0

The function S(x) is a polynomial or a power ax’. A particular case is the gamma-function (S(x) = -x). In all cases we can put into the integrand a multiplier of any ‘harmless’ pre-exponential f(x), for example a polynomial. 4. Integrals of the type of the coefficients of a Laurent series

1 cn = ?z for functions f(z) which are real on the real axis.

124

M.V.

Fedoryuk

II. Asymptotic

There is also an existence theorem [ 121. Consider the integral F(z) =

e WX) dz, sY

where S(z) is a rational function, F(A) + 0. Then there is a saddle contour 7, equivalent to y. But finding 7 for a concrete function S(z); that is a problem in computational mathematics. In fact, using a computer it is possible to find saddle points and study the structure of the level lines ReS(z) = const, Im S(z) = const, passing through these points. It is then possible to find a saddle contour. There are already some papers of this kind. The Laplace method and the method of stationary phase are particular cases of the method of steepest descent if the integrand is analytic. For the Laplace method this is obvious, since the contour of integration is a saddle contour. In the case of the Fourier integral (1.35) the contour of integration can be deformed to a saddle contour. In fact, Re(iS(x)) E 0 on the contour-the segment I-and hence, among the points at which max,., Re S(iS(x)) is attained, there must be an endpoint of the contour and, possibly, a saddle point. For clarity we consider the Fourier integral 1 F(1) =

e’““*dx.

s -1 The phase function S(z) = iz2 has a unique saddle point z = 0. Further, Re(iz2) = -2xy < 0 in the first and third quadrants. The segment [ - 1, l] can be changed into a saddle contour y, consisting of the segment z = ~e’“‘~, - ,/2 5 p 5 fi, and the segments joining the endpoints f$e”14 to the points + 1 respectively. However, in many problems to which the method OS stationary phase can be applied, the phase S need not be analytic. Moreover, whereas in the one-dimensional case it is comparatively easy to deform the contour of integration to a saddle contour, in the many-dimensional case it is extremely complicated, for example, if there are degenerate stationary points, or if the contour of integration is unbounded. Therefore, as was correctly remarked by van Kampen “. . . one cannot regard the method of stationary phase as an illegitimate child of the method of steepest descent.. .“. 4.5. The Many-Dimensional

Method of Steepest Descent. Consider the integral

F@) =

s Y”

f(z) exp{Wz)}

dz,

(1.60)

where z = (zr, . ..,z,)EC”, dz = dz ,... dz,, and y” is an n-dimensional smooth compact manifold. It is assumed that the functions f(z) and S(z) are analytic in some domain D containing y”. A point z” is called a saddle point of S(z) if VS(z’) = 0, and a simple saddle point if det S”(z’) # 0. Already, for n = 2, any geometric visualization is lost, but the idea of a saddle contour as the minimax (see (1.52)) retains its validity. We will suppose, for

Methods

in Analysis

125

simplicity, that f and S are polynomials and that the boundary ay” is a connected set. For example, for n = 2, the boundary is a smooth closed curve. By PoincarC’s theorem the value of F(1) is unchanged if we replace y” by a manifold y’” with the same boundary 87”. Suppose that among such contours there is a minimax y”“,that is, such that (see (1.52)) the minimax min max Re S(z) = M,-n 7’” zep is attained on y”“.Then among the points on 7” at which Re S(z) = M,, there must either be a saddle point or a boundary point of the contour. This fact holds both in the one-dimensional and the many-dimensional cases. Later we will only consider simple saddle points. In this case Morse’s lemma (Chapter 1,9 2) is valid, and locally S(z) reduces to a sum of squares. Let S(z) = z: + ... + z,Z,then ReS(z)=xf+.*.+x,2-yy:-...-y,2 ImS(z) = 2x,y, + ... + 2x,y,. Hence it is clear that the manifold of steepest descent is an n-plane 17: x1 = 0,. . . , x, = 0 (we will not give a precise definition). We note further that the level manifold Re S(z) = c, for n 2 2, is connected for any c. Assume that z = 0 lies on the contour y” and is the only point at which max Re S(z) is attained. Then it can be shown that y” can be deformed to coincide with Ii’ in a neighbourhood of z = 0. Then, up to exponentially small quantities, the integral (1.60) will be equal to the integral over some ‘piece’ of Z7, that is, it takes the form

The asymptotic form of this integral is calculated using Laplace’s method. We formulate a final result. Let max,. ),” Re S(z) be attained only at a point z”, which is a simple saddle point and an interior point of y”. Then as 1+ +co there is the asymptotic expansion F(i) =

F

0

n’2[det S”(z”)]-‘/2

exp{U(z’)}

cJk

1

(1.61)

Formula (1.61) has been proved by the author [ 121. The choice of branch for the root depends on the orientation of the contour. The contribution of degenerate saddle points can only be calculated in certain special cases. And the question of the calculation of the contribution of boundary points of maximum Re S(z) has not been studied at all. In subsection 4.4 of Chapter 1 we saw the difficulties that already exist in the one-dimensional in the choice of a ‘necessary’ saddle point. In the manydimensional case the difficulties increase immeasurably. There are some existence

126

M.V.

II. Asymptotic

Fedoryuk

theorems [12], but no general rules for choosing saddle points exist. We will consider only certain classes of integrals. Consider the Cauchy problem for a partial differential equation with constant coefficients

2 + P(D)u = 0, uI,=o = w, la whereD = --,...,--

18

,P(~)isapolynomialin&,...,<,.Asolutionu(t,x)

i ax,1 i ax, is called a fundamental solution of the Cauchy problem (or a Green’s function) and is denoted G(t, x). Let P(c) be a homogeneous polynomial of degree 2m such that

< # 0. Rep(t) > 0, Then equation (1.62) is called parabolic. In particular,

at

The integral (1.63) is taken over the n-dimensional plane r~ = 0, lying in the n-dimensional complex space CF. From the existence theorem [12] it follows that the asymptotic form of (1.63) is equal to the sum of the contributions of the saddle points of S([,x). The required saddle contour y”” is defined by (1.52), where the minimax is taken over all contours equivalent to y” = RF. Consider a subset of this family: n-planes of the form [ = 5 + iv, 5 E R”, q = const. (for n = 1 these are lines parallel to the imaginary axis), that is, we will look for min max Re[ -P(4: + iv) + i(x, 5 + iv)]. PER”

v(q) = min ReP(5 + iv), <ER”

min C-W

s R”

ew{

- tp(5) + i(x, O> d5.

Let us study the asymptotic form of the Green’s function for fixed t > 0, 1x1+ co. We can limit ourselves to the case t = 1 in view of the homogeneity of P, thus we consider the integral F(x) = s R”

exp{ -p(5)

(1.63)

+ i(x, 0) dt.

This is easy to reduce to the form (1.60) by a change of variable 4 = Ix[~K~~-~)[, F(x) = IXp2m-1) s R”

exp (Wt,

N> di”,

where ;1 = Jx12m12m-1)and o = x/1x1. We introduce the notation i = 5 + 4,

5, rl E R”,

v’(q) = -x.

+ 0, 0.

(1.64)

of degree 2m, satisfies the estimate

and is convex upwards. This gives the following rule for choosing a saddle points: find q = q(x) from (1.64) and on the n-plane q = q(x) find a g = T(x) at which minrcR,,ReP(< + iv(x)) is attained. The points c(x) = t(x) + iv(x) are the required saddle points and the asymptotic form of (1.63) is equal to the sum of the contributions of these points. This method was suggested by S.G. Gindikin and the author [12] and completely solves the problem of choosing saddle points if v(q) is smooth. Unfortunately this is not always the case-v(q) may have corner points. In this case (1.64) does not have a solution for all x, so we fail to find the asymptotic form of (1.63) in some cone (v is homogeneous) of directions in R”. For further details see

cw. The reason why this method does ‘not work’ fully is clear-instead of the complete set of contours equivalent to RF we have taken the minimax (1.52) only with respect to a subset of them. A similar situation arises in the investigation of Green’s difference functions. Consider the equation u!+l =

SC, 4 = -P(i)

- (4 41 = 144

and is attained at the points for which

The function v(q) is homogeneous

where u(t, 5) is the Fourier transform of the Green’s function. Hence we find v and, applying the inverse Fourier transformation we obtain G(t, x) = (2n)-”

takes the form IfER”

t&q-J = 1,

g + P([)u = 0,

<sR”

We introduce the function

’

we have P(5) = tf + . .. + <.‘. Let us calculate G(t,x). We apply the Fourier transform with respect to the variables x1,. . . ,x,, then we obtain

127

VP([) = ix.

then the minimax “=Au

in Analysis

The saddle points of S are defined by the equation S; = 0, or

for the heat conduction

equation

Methods

C aluj”,l. tIllI9

(1.65)

M.V.

128

II. Asymptotic

Fedoryuk

= Herej=(j, ,..., j,),l=(I, ,..., I,) are vectors with integer coefficients, 11111 maxjlljJ,n = 0, 1,2 ,... The Green’s dgerence function y (or unit error function) is the solution of (1.65) with initial data Too = 1, 4” = 0 for j # (0,. . . ,O). For this function it is not difficult to obtain the integral representation

exp(-tPl(t))

G(t, x) = i(2$-” s R” PI

=

i:

a,25,2,

P(z) =

P”(z)z’-’

s 2-m

dz,

1 U’Z’. II’11 9

Here we are using the notation z’ = 2’1 1 . ..%I‘ml7

z = tz1,...,zftJ,

dz = dz, . ..dz.,

and T” is the ‘unit’ torus lzr I = 1,. . . , Iz,I = 1, in the m-dimensional complex space C”. We will look for the asymptotic form of the Green’s difference function for n 2 1 fixed, 1jl --t co. In this case the integrand takes the form exp{S(z,j)}z;‘...z;‘, s(z, j) = n log P(z) + 5 jk log zk. k=l

We can formulate the following rule for choosing saddle points. As family of contours, equivalent to T”, we choose the family of tori T”(p,, . . . , p,): Izl I = pr,. . . , Jz,I = pm. Denote zj = rjeidj, p = (pr,. . . , p,,,), then the problem reduces to the search for the minimax min max [nloglP(z)l P TW) The role of v(q) is played by h(p) =

max -

+ 2 jklogzk]. k=l

log(P(p,ei$J,.

in Analysis

- exp{-tP2tt)) Pl(5)

P2 = i

-

129

exp(j(x

p,(r)

<))d( >

b;5,2

k=l

k=l

ly = (27ci)-”

Methods

Let us put t = 1 and study the asymptotic form of G(l,x) as IxI+ 00. Put i = t: + iv, <, q E R”. If n = 1, then the contour of integration can be replaced by a parallel line q = q. # 0. On this line the denominator P,(t) P2(5) = (a: - b:)[’ does not vanish and (1.66) can be written as a sum of two integrals and each of them studied separately. In our case there are two phase functions S,(i, x) = -e(x) + i(x, [), each having precisely one saddle point cj(x). As a result we obtain for n = 1 that the asymptotic form of (1.66) is equal to the sum of the contributions of one or both points [i(x), i2(x). For n 2 2 it is also possible to replace the contour of integration by a ‘parallel’ n-plane of the form [ = < + iv’. But, in general, we cannot partition the integral (1.64) into a sum of two integrals. The point is that the dimension of the set D: PI([) - P2(Q = 0 is equal to 2n - 2, the dimension of the contour is n. Consequently they intersect in a set of dimension (2n - 2) + n - 2n = n - 2 > 0, and, hence, each integral individually can diverge. Let n = 2, and let ak, bk be such that the ellipses afv]: + a;tj+; = 1,

b,2v: + b;q; = 1,

intersect transversally. Then, as x: + xz + co, the asymptotic form of (1.66) is determined either by the saddle point [i(x), or the saddle point rz(x), or the saddle point cJ(x) of the whole integrand. The point c3(x), for 1x1>> 1, lies close to the discriminant set D of the polynomial P(z, [) (the symbol of the given dfferential operator). Clearly this phenomenon is essentially many-dimensional.

..,~,e’@m))

OI(j62R

0 5. Supplement. Sums and Series

and the role of (1.64) by nh’(p) = -j.

In conclusion we give one more example, emphasizing the distinction between the one-dimensional and many-dimensional cases. Consider a parabolic equation of second order in t:

5.1. Merging of Singularities. F&cc)

=

F(i, ~1)=

where ak > 0, bk > 0 are constants. The Green’s function G(t, x) is defined to be the solution of this equation with the Cauchy data GI,co = 0,

= 6(x). at r=O

E

By the same method as in (1.62), we can obtain an integral representation

Consider the Fourier and Laplace integrals b f(x,a)exp{ilS(x,a)}dx sa b f(x, a)exp(U(x, sa

(1.67) o$>dx,

where S(x, CI)is a real-valued function. Here c1is a real parameter, ICII 5 6, where 6 > 0 is small but independent of 1. For CI# 0 let S have a finite number of stationary points, all of them nondegenerate and interior to the segment I = [a, b]. Then the asymptotic form of (1.67) is easily evaluated by the methods of @2,3 of Chapter 1, and is equal to

M.V.

130

Fedoryuk

II. Asymptotic

the sum of the contributions of the stationary points and the endpoints of the segment. Let a = 0 be a critical value of the parameter, that is, either a stationary point coincides with one of the endpoints, or two or more stationary points merge. The asymptotic form of F(A, LX)can be evaluated but the formulae obtained are not uniform with respect to a. The problem is this: find asymptotic formulae, applicable for I + +co, and uniform with respect to CIE [ - 6, S], where 6 > 0 is small and does not depend on 1. Integrals of this type arise in the theory of special functions, in many-dimensional problems of the propagation and diffraction of waves, etc. The methods of investigation are as in %2-4 of Chapter 1: for small CI,S(x, a) is reduced to its simplest possible form, x here also varies in a small neighbourhood of the corresponding point xc,. Then we study the standard integrals obtained. In contrast to the results given in Q§l-4 of Chapter 1, the asymptotic formulae will include some special functions. We quote several results of this kind. 1. A stationary point close to the boundary. Consider the integral F&cc) =

‘f(x)exp{--i.(x

a>0

- a)‘}dx,

(1.68)

s0

in Analysis

= f(O)

point x0(~) = LX,which for contribution to the asympof x = 0. Therefore we can obtain

m e-‘x2 dx = f(O)@(& a).

s -cc The latter integral can be expressed in terms of a special function-the integral (or error integral)

probability

CD(x) =2J sx

f(x) = k$oak(4W - 41k+ xkfobk(4W- ~4)~ + (x(x - a))NR,(x,

a).

It is easy to show that ak, bk and RN are smooth for small 1x1,1.~1. The integral over each of the terms (for a = +co) is given in terms of @(A,U) and its derivatives, and, finally, we obtain an expansion of the form &‘(&a) = f

,I-kAk(a)@(A,

k=O

CC)+ 2 L-k-1Bk(@)e-‘a2. k=O

; The leading term of the asymptotic formula is

q.(O,O) = 0, Lyh(O,O) # 0. Xx(0, 0) z 0, i k i The Fourier integral (1.67) is analysed in exactly the same way. Similar results 1 have been obtained for the many-dimensional Laplace and Fourier integrals i W, 4 =

sD

F(l, a) =

f(x, 4 exp (Wx, a)> dx, f(x, cc)exp(iAS(x, rx)>dx,

sD

We have F(l,a) z f(0);

;(l

- @(-a$)).

\i This formula admits no further simplifications since the parameter a$ may be << 1, it is possible to expand @ in either small, large or of order 1. When l~l$ a Taylor series. When IM/$ >> 1 we can use the asymptotic formulae @(x)Tl=

-$[l

+jr$],

x-+ *co.

The coefficients c,,? are easily evaluated using integration

by parts, as in 4 1 of

+ O(l-‘@(A, a)).

Let S(x, a), for small 1~1, have a nondegenerate maximum point x = X,(U), x,(O) = 0. Then by a change of variable the Laplace integral (1.67) can be reduced to the form (1.68). Sufficient conditions on S are:

e8 dt.

71 0

131

this chapter. This leads to the asymptotic expansions obtained in $0 1,2 of Chapter 1. We outline the rigorous derivation of the asymptotic expansion. If CIis fixed, then for CI> 0 the main contribution to the asymptotic form is carried by the maximum point x = c(, and for CI< 0, by the endpoint x = 0. Therefore we expand f(x) into a series with respect to functions having zeros at x = 0 and x = Lx.

F(A, a) = f(a)@@, a) + (21x1))‘(f(a) - f(O))e-“”

The function S = -(x - IX)’ has a unique maximum CI= 0 is the boundary point x = 0. The fundamental totic form (1.68) is made by a small neighbourhood replace f(x) approximately by f(0) and a by +co to w,4

Methods

(1.69)

i when there is a nondegenerate maximum or stationary point close to the bound: ary i3D of the domain of integration. 2. Merging of saddle points. Let S, for small c( # 0, have two nearby non: degenerate stationary points which merge for CI= 0. The simplest example is given by the function S(x,

a) =

-ax

+ x3/3,

which has two nondegenerate stationary points x 1,2 = +-& for a # 0, and a degenerate stationary point x = 0 for M:= 0. Consider the Fourier integral (1.67) where f(x) E C;(I):

M.V.

132

II. Asymptotic

Fedoryuk

a F(A, a) =

s -a

exp { i;l(crx - x3/3)} f(x) dx.

(1.70)

The main contribution to the asymptotic form of the integral is carried by a neighbourhood of x = 0. Changing f(x) into f(0) and a to +co, we obtain an integral which can be expressed in terms of the Airy function (Chapter 1, 6 4) F(I, a) % 27c1-“3f(o)di(

- A2’3a).

Here we must show caution. The point is that the Airy function decreases exponentially for cI < 0, J2131a 1+ +co, but at the same time F(1,cr), in general, will not exponentially decrease. Therefore the formula is suitable as I + +co, uniformly with respect to a such that 0 5 tl 5 6. This situation can be explained in another way: for CI< 0 the saddle point of S is complex and f(x) may not be analytic. In addition, the Airy function has infinitely many real zeros, which, of course, need not coincide with the zeros of F. Therefore the leading term of the asymptotic form must be ‘modified’ near the zeros of the Airy function. Expand f in a series f(x) = k$o UkMX2 - aJk + XkZObk@W

- alk.

The function x2 - a = Sk(x, a) vanishes at the stationary points of S-this is stipulated by the form of the expansion. It is easy to show that a,(x), &(c() E C” for small a 2 0, ao(4

= tCfbl(4)

b,(a)= x

+ f(x264)1,

(a) : x2(a) Cfbl(4) 1

- f(x2@4)1,

x1,264

= i-6.

If f(x) is analytic for small 1x1, then the coefficients a,(a), b,(a) are analytic for small [al. The asymptotic expansion of (1.70) has the form (1.71) F(A, a) = 1-“36;( -A2’3a)A(A, a) + 1-2’36;‘( -,12’3a)B(;1, a) where A and B are asymptotic power series in I-‘. Let S(z, CI)be analytic in z, a in a neighbourhood s; = s;, = 0,

s:= # 0,

Methods

in Analysis

133

and A and B are analytic for small Ial. This enables us to reduce the Fourier integral to the form (1.70). Analogous results hold for the many-dimensional Fourier integral (1.69). If S has two nearby nondegenerate stationary points x,(a), x,(a), which merge for a = 0, then there is an asymptotic expansion of the form (1.71) with an extra multiplier A- (n-1)12.The phase function can be locally reduced to a sum of functions of the form (1.71) and a sum of squares CyZ2 +yf. In the general case there is a phase function S(x, a), depending on n variables x1,. . . ,x, and m parameters aI,. . ., a,, where for a = 0 the point x = 0 is a degenerate stationary point. The problem of classification of singularities arises, that is, the finding of canonical types to which the family of functions S(x, a) can locally be reduced. The basic results in this direction are given in [l], [2]. Each canonical type produces a special function. The simplest is the Airy function. The next most complicated class is the Peurcey integral W, a, P) =

m f(x) exp{iil(x4 + ax2 + bx)) dx,

s -a,

where f(x) E C;(R). For small a # 0, /I # 0 the phase function has three nearby saddle points which merge as a + 0, b --f 0. The Pearcey integral does not reduce to known special functions. It has been partially tabulated using computers. 3. Merging of other singularities. Consider the integral F&a) =

* exp{ -1x2} s --m x-ia

dx ’

(1.73)

where 0 < a 5 6. The integrand has a saddle point x = 0 and a pole x = ia, which merges with the saddle point as a -+ 0. Integrals of this kind arise in the theory of propagation of linear waves. In general we consider the Laplace or Fourier integral (1.67), where S(x, a) has a simple saddle point x,,(a), f(x, a) has a simple complex pole x,(a), and as a + 0 the saddle point and the pole merge. The asymptotic forms of these integrals are given in terms of the standard integral (1.73), which, in its turn, is given in terms of the Fresnel integral F(ll,a) = rcie”‘[l

of (0,O) and

- @(a&)].

Problems in wave diffraction lead to integrals of the form

s;;, # 0

00

F(A, a) =

at this point. Then, for c1# 0 S has two nondegenerate saddle points zi(tl), z,(a), and by an analytic change of variables z = d([, a) reduces locally to the form (1.72) s = A(a) - B(a)[ + [3/3.

ei(R+dyx)

R

s -00

&4 dx,

R = ,/A2x2 + a2f (x).

Here b(x) E C:(R), y is a real number, IyI < 1, f(x) > 0 is a smooth function and 0 < a 5 ao. The phase function S = R + iya has two nearby stationary points

In this formula 44

= Wz,(a),a)

+ m,(4,~n

B(a) = C%%(a),4 - W2W,4)12’3,

x1,2

“1

a

f(O)

J

1 -y21

which merge as a + 0. For a = 0 the phase function becomes

M.V.

134

Fedoryuk

II. Asymptotic

Methods

in Analysis

135

non-smooth: S(x, 0) = A(1x1 + yx), and in addition the integral diverges, since the denominator is I/xl. Nevertheless, in our case the asymptotic form of the integral can be found using the formula

s

O” exp[i(Jm

-CO

+ yx)

dx = 7ciHp(aJcyq,

This series converges for small 1~1.Thus @ has a power singularity at E = 0. If a + b > 0 is not an integer, then differentiating with respect to Ethis case reduces to the following

a > 0.

JFT7)

If f(x) = const, then the asymptotic expansion of the integrals is obtained as follows. The function 4 is expanded into a Taylor series 4(x) = ~~XO d,,xn, which is integrated term-by-term. If f(x)+ const, then, by a change of variable x = g(y, a), the expression under the root reduces to the form yz + c?~(cI), p(O) > 0. There are various other types of merging of singularities, for example, a stationary point close to a corner of the boundary of the domain of integration, etc. 5.2. Integrals with Weak Singularities.

F(E) =

a > 0.

a

xp-l(x

In this case @(e;a, j) = const . .sa+B+ Ql(s) The function Ql(s) is analytic for small IsI. Let CI+ j3 = M > 0, where M is an integer. Then the expansion of @ contains log E. By (1.77) it is sufficient to consider the case a + /? = 0, so that

(1.74) We expand the integrand into a series in powers oft-‘.

Here E > 0 is a small parameter, f E C” for 0 < x 5 a, E > 0. For E = 0, f(x, E), or one of its derivatives, has a singularity at x = 0. The simplest case is a power singularity of f(x, 0). An example is the integral a (1.75) F(E) = xB-l(x + &)“qqX) dx, s0 where d(x) E P(O,a), /? > 0 and M:is not a non-negative integer. The case p = 1, c( = - 1 was considered in Chapter 1, 4 1. We investigate first of all the standard integral a (1.76) xyx + &)ddX. @(G 4 8) = s0 Ifcr+/?
m s

(1.77)

@(E; CI - N, b).

Consider the integral

n f(x, E)dx, s0

where B is the Euler beta-function. this integral and the integral

“@(c;d(J?) =N!

Further @(E;a, /?) is the difference between

l-1 t B=&T+fi(-)+ t t2 >

...)

then we obtain

(1.78) The first of these integrals is equal to

( )I

I+-@logs+ a-@loga-slog 1 +I a ’ u[ and the function in square brackets can be expanded in a power series in E. The last integrand in (1.78) has order O(tm2) as t + co, since both of the latter integrals converge. Expanding the integrand as a power series in t-’ and integrating term-by-term we obtain an asymptotic series in powers of E. Thus, if CI+ /3 = N 2 0 is an integer, then a-@log

@(E;a, /?) = const . sN log E + al (8).

+ c)adx =

into a series and integrating

(

The function @i(e) is analytic for small 1~1. Let us return to (1.75). Expand d(x) in powers of x: term-by-term,

asymptotic expansion of the latter integral in powers of E, so that

we obtain an

d(x) =5qxn +XN+‘l+bN(X), n=~ n.

M.V.

136

II. Asymptotic

Fedoryuk

Methods

in Analysis

The Poisson summation formula takes the form

$N(x) E C”‘(0, a). Then we obtain

(I +&)dt/QN(X)dX. Rr&) =sxp+N(x

(1.79)

z f(n) = 2 h4 -00 where f(t) is the Fourier transform of f(x): e-2nixcf(~) dx.

0

If m > 0 is an integer, m < c( + /? + N + 1, then there is a continuous derivative R(Nm)(c)for small E > 0 3and hence m Rck’(0) RN(c) = kTo r&t

+ o(&“+‘)v

Using the Dirac delta-function,

I3(x + A&)e(X + B&)fiqqX)dX s0

s(n)

=

kgl

=

f 6(x -m

-

n).

There is also a many-dimensional Poisson summation formula. Formula (1.79) is valid if f(x) is continuously differentiable for -co < x < co, and cYcu f’(n + x) converges uniformly for 0 5 x 5 1. For other sufficient conditions, see [S], [ll]. Consider the series c?m f(n, x), where x > 0 is a large parameter and the terms of the series oscillate strongly for x >> 1. In this case we can use the Poisson summation formula, since f(n, x) will rapidly decrease as x + +co. Example 1. Consider the series

Applying the Poisson summation $4

5.3. Sums and Series. Consider sums and series of the form

(1.80)

the Poisson formula can be written

‘33 C emzsixn -m

&-+ +o.

Because of the choice of N, the number m may be made arbitrarily large. In this example the asymptotic form has a non-local nature. In fact, the coefficient 0. RN(O) = x”+~+~$~(x) dx s0 is determined by the values of d(x) throughout the interval of integration. The same is true of R$)(O). But it is not without interest to note that the terms containing logs depend only on the values of 4 and its derivatives at x = 0. Integrals of the form a (x2 + &2)“qqX)dX, s0 may be studied in a similar way.

137

= f ih-4, -00

formula we obtain

d(n,X) =

m e-i2nnt+int(X2 + t2)-Wdt. -CC s

(1.81)

The function &n, x) is even with respect to both n and x, and for n 2 1, after the change of variable t + xt, we have

fk@).

m e-itx(t2 + 1)-1’2 dt s -00 The function e(x) can be expressed in terms of the Macdonald function: tj(x) = 2K,(x), x > 0, where the asymptotic behaviour as x + +cc is known. Namely, dh-4

where in the first case n, and in the second case 1, is a large parameter. These sums are analogues of the integrals x m f(t, A) dt. fk 4 dt, s0 s0 The arsenal of methods, which can be used to investigate the asymptotic forms of sums and series, is considerably poorer than for integrals. This is not surprising, since a series is the integral of a function having infinitely many points of discontinuity. Essentially, the only methods of a general nature are the EulerMaclaurin summation formula, considered in detail in the previous part, and the Poisson summation formula. We will not consider here the deep and delicate methods developed by I.M. Vinogradov and others in the analytic theory of numbers, since a separate part has been devoted to these questions.

= IC/(Qn - lb4

K,(x) = e?[l

t)(x) =

+ 0(x-‘)],

x-+ +co.

Thus the n-th term of the series decreases like e-n(2n-1)x for n >= 1, x + +a3, and the leading terms of (1.81) are those with indices 0,l. It is easy to show that IS(x) -4(0,x)

- #(1,x)1 < cxJnx,

so that S(x) = 2eK”“[l

+ 0(x-‘)],

An asymptotic series can also be obtained.

x++co.

M.V.

138

II. Asymptotic

Fedoryuk

139

S,(x)= f !c -k$ + ... + (- l)N-‘kN-’ XN ak (

eexn2

k=l

-00

asx+

in Analysis

then

Example 2. Let us find the asymptotic form of the theta-function S(x) = f

Methods

x

>

+O.Wehave =~l[l-(-y]$.

cc e-2nint-xi2

s -cc and from the Poisson summation

dt

=

n \i

e-n2n2/x,

Consequently

X

formula we obtain

IS(x)- S,(x)1 = F -k N /k=l

S(x)= E z e-nwx. \i 71s !f x’ \i() x

t + O(eC+),

x+

by the term with index

kN+lak

By the same token we have proved that there is an asymptotic expansion x-+

+o.

\i

Example 3. Let us find the asymptotic

form as x + +co of the series

S(n) = i

k!.

k=l

The terms increase very rapidly as the index increases, and we can expect the last term to be the leading term of the asymptotic form. We have S(n)=l+!+

1

n!

Expand the n-th term in powers of x-l :

+a.

Example 4. Consider the sum

We will give several examples in which the asymptotic form of a series is found by elementary methods.

n

+...+L

n(n - 1)

n!’

(1.83)

m (-l)knk a”kzo- Xk+l

a” n+x

and then substitute back into the original series. We obtain an expansion of S(x) in powers of x-l:

a2 = - f kak,

a3 = F k2ak,.

k=l

k=l

k=l

This series diverges for all x, but is an asymptotic series for S(x) as x + +co. In fact, put :

+

$

since the latter term converges. Consequently S(n) = n![l

+ O(n-‘)I,

It is not difficult to obtain an asymptotic in example 3. Fix N 2 1, then

S(x) = : + 2 + 3 + . . . )

=

zl

Is(x) - L‘&(x)l 5 CNx-N-1.

The asymptotic form of (1.82) as x + +0 is determined n = 0, so that

sN(x)

$1

The last series converges and, if C, is its sum, then

S(x) =

a, = F ak,

x)

-N-l

5X

In particular, we have the identity

S(x) =

(

(1.82)

+

. . . +a, XN,

n+ co.

series for S(n) by the same method as 1

.. . +n(n - l)...(n

- N + 1) Expanding each term in the sum into powers of n-l, we obtain

140

M.V.

II. Asymptotic

Fedoryuk

so that there is an asymptotic expansion

In concrete calculations it is not necessary to find all the coefficients c,; just enough of the first few. Nevertheless, we will give a formula for them, which can dk be obtained in this example: ck+r = kr, where the d, are the coefficients in the expansion eex-’ - 1 = ‘f dkXk. k=O

(2.1)

Y = (yl,...,yAT,f= (.fl,..., f,)’ (all vector functions are columns). We will only consider the homogeneous system

where

(2.2)

dX

Y(x, 4 =

(Y’k

4, f *. , Y”b,

form of integrals (see Chapter 1). Therefore the fundamental problem in linear asymptotic theory can be posed as follows. Find the asymptotic form as E -+ 0 of a fundamental system of solutions (on a given set). It is extremely rare for a system (2.2) with variable coefficients to be integrable in quadratures. At the same time, the presence of the small parameter E allows the system to be approximately integrable in quadratures. To wit, an asymptotic series for the FSS can be obtained so that each term of the series is evaluated in quadratures. This stands as the fundamental achievement of the asymptotic theory.

4 = h, where h is a constant vector, x0 E I. On the strength of known theorems from courses on ordinary differential equations, the Cauchy problem has a unique solution y(x,s) which is infinitely differentiable with respect to x, E on I x J. Consequently, for any N 2 0 we have

1.1. Statement of the Problem. Numerous problems in mathematics, mechanics, physics and the other natural sciences reduce to differential equations with a small parameter E > 0. In the linear theory we consider a system of y1 equations of the form

for the following obvious reason. Suppose that a fundamental

141

Y(x,,

0 1. Statement of the Problem. Regular Dependence on a Parameter

-dY = A(x, &)y

in Analysis

1.2. Regular Dependenceon a Parameter. Let I be the interval a 5 x 5 b, J the interval 0 < E 5 so, J = [0, so]. Throughout this section it will be assumed that A(x,E) E C”(I x J). If A(x,E) E C”(I x J), then we will say that (2.2) depends regularly on E. Consider the Cauchy problem

Chapter 2 Linear Ordinary Differential Equations

2 = 4% E)Y + f(x, 4,

Methods

matrix

Ytx9 &I = jto &jYjtX) + RN(x~ &I, I&&,4

If an FSS, or its asymptotic form as E -+ 0, is known, then the investigation of the asymptotic form of a solution reduces to the investigation of the asymptotic

E E J,

where the constant C, does not depend on x, E, and all the vector functions yj(x) are infinitely differentiable for x E I. Thus, there is the asymptotic expansion Y(X, 8) = jzo EjJ’j(X),

E-+ +o

(2.3)

uniformly with respect to x E J. Taking n linearly independent vectors h,, . . . , h, as initial data, we obtain an FSS. Let us find the coefficients in (2.3). Expand the matrix of the system in a Taylor series

4)

of (2.2) is known, where the vector functions yl, . . . , y” form a fundamental system of solutions (FSS). Then a particular solution yO(x,s) of the inhomogeneous system can be found in quadratures by the method of variation of parameters: x Y-l@, ~)f(t, E)dt. YO(X, 4 = Yb, 4 s x0

5 CNEN+1,

A(X,E)

=

2 j=O

EjAj(X),

substitute (2.3) into the system and equate the coefficients of powers of E in the equality obtained. Then we obtain a recurrent sequence of Cauchy problems dye -dx = Ao(X)Yo,

dy, _dx = AOWY,

YO(X,) = h

+ A,(X)Yo,

Yl(XO)

= 0,

142

M.V.

II. Asymptotic

Fedoryuk

If an FSS is known for the first approximation

system

dy -dx = &(X)Y,

y” - q(x,s)y

(2.4)

4,

where q,, E Cm(l x J) and the equation takes the form &“yn - qo(x94Y = 0

(2.5)

Later (Chapter 2,§ 2) we will show that it is enough to limit ourselves to the case n = 2-the order of the pole equal to the order of the equation. Making the substitution x = EZ we obtain an equation of the form y& - q(&z,&)y = 0

(2.6)

whose coefficient slowly varies for small IsI. This is the fundamental class of equation which we will discuss. In this case the dependence on the parameter is regular but, nevertheless, the methods of subsection 1.2 are inapplicable. The point is that if x varies over a finite interval [a, b], then 2 varies over a large interval [a/s, b/s] whose length tends to infinity as E + 0. Let E = 0 be an essential singular point. Since there is no classification of such singular points in analytic function theory, then even more so there is no classification of the differential equations. At present the class of equations of the form

XEI

4 2. Equations of Second Order Without Turning Points 2.1. Formal Asymptotic

Solutions. Consider the equation E2y” - WY

= 0

143

has been studied. If q(x, E) does not depend on E, q(x) = sinx, then q(x/E) = sin(x/e) and E = 0 is an essential singularity (for x # 0). Equation (2.7) is an equation with rapidly varying coefficients. If q(x, E) is periodic in x, then averaging methods, [3], can be applied to (2.7). The fundamental types of equations to be discussed here are those with slowly varying coefficients.

Consider the scalar equation

on a finite interval I = [a, b], where E lies in an interval J = (0,~~). By our assumption, q(x, E) E P(Z x J). The dependence of q on E will be singular if this function has a singularity for E = 0, for example, q(x, E) = s-l q(x). A complete classification of singularities simply does not exist if q is not analytic. Therefore we will discuss the case when q(x,E) is analytic with respect to E in the ring 0 < 1~1< E,, for fixed x E I. Then for fixed x, E = 0 will either be a pole of order II or an essential singularity. In the first case 4(x, 4 = f”qob,

in Analysis

y” - q ;,e y = 0, ( >

then all the vector functions yj(x), j 2 1, can be evaluated in quadratures. The Taylor series, in general, diverges if the elements of A(x,E) are not analytic for small 1E1,and converges if they are all analytic for 1E1< r and for some fixed x E 1. The asymptotic form of the solution of the Cauchy problem for the inhomogeneous system (2.1) is constructed in exactly the same way if f(x, E) E P(l x J). Clearly the construction of the asymptotic form of the solutions in the case of regular dependence on the parameter causes no problems. However, it is worth noting that it is necessary to know an exact solution of system (2.4) with variable coefficients, so that in the regular case, in general, it may not be possible to calculate the asymptotic form of the FSS in quadratures. 1.3. Singular Dependence on a Parameter.

Methods

= 0

(2.8)

on the interval I, q(x) E P(l), where E > 0 is a small parameter. Of course q(x) + 0. This is a case of non-regular dependence of the equation on the parameter. An equation of the form (2.8) is also called an equation with a small parameter in the highest derivative. Whereas in the case of regular dependence the limit equation (E = 0) allows us to lind a first approximation to the solution, in the present case the limit equation q(x)y = 0 contains no information about the solutions of (2.8). We will try to find the asymptotic form of some (non-trivial) solution. It is necessary first of all to guess (and no other word will do) in what form to search for the asymptotic form. Of course, this stage-guessing the form of the asymptotic form-is not subject to any formalization. Analogy, experiment, numerical simulations, physical considerations, intuition, ‘random’ guesswork; these are the arsenal of means used by any research worker. Equation (2.8) is classical, and here it is known in what form to search for the asymptotic form of the solution, and for what reason. One consideration is this. If q(x) = const, then the solutions y,,, = e *AXi’ form an FSS. By analogy, we search for solutions in the form of a series y(x,

&) =

el@@)

Jo

&jYjCx)9

(2.9)

where S(x), y,(x), yi(x). . . are unknown functions. Substitute this series into (2.8), cancel the exponential in both parts and equate to zero the coefficients of the 1 powers of s in the series obtained. Then we obtain a recurrent system of equations 6 from which we can successively determine S(x), y,(x), y,(x), . . . It is convenient i: to represent (2.9) in the form (2.10)

144

M.V.

Fedoryuk

II. Asymptotic

Putting y’Jy = w, we obtain a Riccati equation for w w’ + wz = &Pq(x), so that jzl

&jc$(X) + ( f j=

-1

Ejaj(x))2 = &Pq(x).

Equating coefficients of powers of E in this equation we get a recurrent system of equations &(x)

2ct,(x)a-,(x)

= q(x),

2al(x)olj+l(x)

=

-"l((x)

-

i

+ Mb(X) = 0

j 2 0.

ak(x)Mj-k(X),

x0(x) = -~ 4'(x) = w4 5 d2(4 32 q”“(x)’

ctj,,(X) = -___ 2&

[a;(X)

+

kio

(2.11)

ak(x)aj-k(x)l?

j

2

0.

If we put ~-r(x) = -J4(x), then in the formulae for the coefficients a,(x), we must replace & by -&. Thus equation (2.8) has two formal solutions x y,(x, A) = q-l14(x)exp i x J4odt + T l-k a,(t)dt > k=l s x0 I iS x0 (2.12)

@2(X),...

y,(x, A) = q-“4(x)exp i

in Analysis

145

has been found when an FAS has been found. In many known problems, in particular nonlinear ones, it has not yet been proved that the FAS which has been constructed is close to a genuine solution of the equation. Moreover, this proximity may not happen. Each problem on the asymptotic form of the solutions of equations (for example, (2.81)) can be divided into two parts. 1. The construction of an FAS, that is, guessing the asymptotic form. 2. The justification of the asymptotic formula, that is, the proof that the FAS is close to a genuine solution. These two problems are totally distinct, both in spirit and in the methods of their solution. We return to the FAS (2.12). The leading terms are

k=O

From the first equation we find U-~(X) = f J4(x). If we put a-i(x) = J4(x) (the choice of branch for the root is explained below), then we obtain

q(x)=----- 1 d'(x) 8 q3’2(x)

Methods

--A X .Jqodt s x0

+ kzl (-A)-”

* @,(t)dt , s x0 I

where 2 = s-i is a large parameter, I + +cc. The functions y,, y, are usually called formal asymptotic solutions (abbreviated FAS) since substituting them into the equation turns it into an identity. An FAS, up to O(eN), is a function yN(x, E) such that (y”)” - &-2q(x)y” = O(ENYN) as E + 0, x E I. We will not formulate a more precise notion of FAS; the meaning is always clear from the context. In the majority of physical work it is assumed that an asymptotic solution of an equation (ordinary differential, partial differential, integro-differential, etc.)

y,,,(x,4

- 4P4(x)exp

{ *I

pm}~

n-t +m

(2.13)

These formulae are named ‘WKB-approximations’ or WKB-asymptotic forms, after the physicists Wentzel, Kramers and Brillouin, who obtained them in 1926-28 in connection with problems in quantum mechanics. This method of constructing an FAS carries the name the WKB-method. The terms: shortwave approximation, high frequency approximation, etc, are also used. This is because many problems in the propagation of waves-electromagnetic, acoustic, elastic and probabilistic waves (quantum mechanical) etc-reduce to equation (2.8). In the mathematical literature, in particular American and English, the term ‘Liouville-Green approximation’ (or ‘LG-approximation’) is used. Liouville and Green obtained these approximations in 1837 [lS], [22]. It is clear from (2.13) that they are necessarily inapplicable at points at which q(x) vanishes. In fact, y1 and y, are smooth functions, and the right hand sides in (2.13) go to infinity for q(x) = 0. A point x0 is called a turning point (or transition point) of (2.8) if q(x,) = 0. The WKB method of constructing an FAS in the form (2.9) has been widely used in linear problems. We can apply it to equations of n-th order and systems (see subsection 2.2). Moreover, this method can be applied to many classes of linear partial differential equations; the wave equation with variable coefficients, the Schrodinger equation, etc. 2.2. Asymptotic Diagonalization of Systems. In equation (2.8) put Y = Yl>

JY; = Y2,

y =

(YI>Y2)T,

(2.14)

where 2 = s-l, then we obtain the system dY

dx = AA(x)r,

A(x) =

(2.15)

Let q(x) # 0 for x E 1. A system with n equations of the form y’ = B(x)y with variable coefficients can be integrated if B(x) is diagonal. In this case the system

M.V.

146

Fedoryuk

II. Asymptotic

decouples into n independent first order equations yj = bjj(x)yj, 1 5 j 5 n. System (2.15), in general, cannot be integrated, and hence, cannot be transformed to diagonal form. The presence of the large parameter ,I here does not complicate, but in fact simplifies, the situation. Namely, (2.15) can be diagonalized up to 0(X”) for any N > 0, that is, it reduces to a system whose matrix is diagonal up to terms of order O(I.-“). Make a substitution Y = T,(x)z, where 7’,‘,(x) is a nondegenerate matrix for x E Z of class P(Z). Then (2.15) takes the form dz dx -

=

nT,-l(x)A(x)T,(x)

-

1

Methods

Consequently, if Z’,‘,(x) is taken to be a matrix which reduces A(x) to diagonal form, then (2.15) will be diagonalizable up to O(1). The eigenvalues of A(x) are ~r,~ = fJ4(x) and are distinct for all x E Z since q(x) # 0. The corresponding eigenvectors are eI,z(x) = (1, -+J4(X))=, and therefore we can put

T,(x) =

147

Since (I + 1-l TJ’

= Z - I-’ Tl + O(P),

then the matrix of the system is equal to &l(x)

+

- T,(x)A(x)

+ A(x)T,(x)

- T,-‘(x)2

+ O(P). >

This matrix is diagonal of order O(A-‘) if the matrix in round brackets is diagonal. Hence we find

z.

T;qx)y

in Analysis

(T,(X))j,k =Pj(X) -’

dT,(x) Tom’(x)

Pk(X)

dx

>j,k’

j # k.

The diagonal elements of T,(x) remain undetermined, since the diagonal elements of - T,A + AT, are zero, and they may be put, for example, equal to zero. For the concrete system of two equations (2.15) we obtain t,, = t,, = -q’(x)/ (4q2(x)). Thus the substitution y = T,(x)(Z + A-‘T,(x))w

(2.17)

reduces (2.15) to g l/m N-4

=

= [n/i(x)

+ B(x)]z,

g -1

0 o -J&j

(

’

)

B(x)=$g

(

for x E I. The substitution

= 4)

= diagh(4,.

1

-1

A,(x) = -diag

>

. . ,P,(x))

l/i(x)

- T;‘(x)?

1 z.

We diagonalize this system up to O(A-‘) using the substitution 2-r Tl (x))w. For w we obtain a system with matrix A(Z + A-‘T,)A(Z - A-‘(I

(2.16)

y = T,(x)z reduces (2.15) to

dz - = dx

+ A-IT,)

+ A-’ T,)-’ 2.

- (I + A-IT,)-‘T;$(Z

+ Al(X) + o(n-‘)]w

1

Here we pause: the reader interested only in equation (2.8) may move on to subsection 2.3. We now turn our attention to whether this construction is suitable for systems of arbitrary order. To wit, suppose we are given a system of the form (2.15) of n equations, and let the eigenvalues pr(x), . . . ,p,(x) of A(x) be distinct for all x E I. Then they are all of class P(Z) and there is a matrix function T,(x) E C”‘(Z), nondegenerate for all x E I, such that T’,‘(xMx)T,(x)

= [kqx)

+ A-IT,)

T;‘(x)~ G(x)

(2.18)

Here we use the notation: diag A is the diagonal matrix with diagonal elements We will give another formula for A,(X). Let {e,(x),. . . ,e,(x)}, a,,,...,&. M(X)?. . . , e,*(x)} be bases of left and right eigenvectors of A(x), that is, Aejb) = Pj(X)ej(X),

ej*A(x) =

pj(X)$(X).

The vectors ej(x) are columns, the e;(x) rows, and ej*(x)eJx) = 0,

j # k. As T,(x) we can take the matrix with columns e,(x), . . . , e,,(x), then T;‘(x) will be the matrix with rows (eT(x)ej(x))-’ e;(x), j = 1,. . . , n. Consequently, A 1(x) is a diagonal matrix with diagonal elements

z = (I + p!” 3

=

ej*(x)ej(x)

’

j=l

, . . . , n.

(2.19)

The process of asymptotic diagonalization can be extended to arbitrary N 2 1 to find matrices Tl(x), . . . , TN(x), such that the substitution y = T,(x)(Z + X’T,(x)

+ ... + XNTN(x))z

M.V.

148

II. Asymptotic

Fedoryuk

g = AA(x) + A,(x) + . . . + d-N+l~ N-l(X) + n-NB,(X, 1). Here the /?i(x) are diagonal matrices, q(x), n,(x) E Cm(Z) and 5

x E I,

cN,

1112 I, > 0.

ii(;)

= (Q:x,

ph,xJ

(2.20)

which, obviously, coincides with (2.9) if Q = A2q(x). (2.20) is equivalent system

to the

= Ij-1 l~kW+

As I we choose an interval a < x < b, which may be infinite, and let

ii)(;)’

If the condition

I I Y2H

() i

1 5 2(e2P(“,X) - 1).

(2.27)

(2.28)

P(X, b) < 00 is satisfied, then (2.20) has a solution y, such that

1

1 5 2(e2p(x*b)

-

(2.29)

1).

Y2(4

(2.22)

-&@-#

(2.26)

I I Yl(4 Y,(x)

Make the transformation

mlQ() &

(2.25)

&4x) < co. Then (2.20) has a solution y, such that for x E I

Using equation (2.20) as an example we will demonstrate the method of proof of asymptotic formulae and clarify the meaning of the conditions which are imposed on the coefftcients. The first condition is obvious-the absence of turning points: (2.21) x E I. Q(x) Z 0,

Y’

Q-““(x)exp {fs:,v’i%W},

~1,2(~,~o) =

Consider the second order equation y” - Q(x)Y = 0,

Y =

149

S(xo,x) = J%[” q’!j@dt,

The number I may be complex. Neglecting neNBN(x, A)in the latter system, we obtain a system which decouples into n separate equations. Any solution of it will be an FAS of (2.15). 2.3. WKB-Estimates.

in Analysis

(A) There is a branch of the root ,,@@, smooth for x E I, such that Rem 2 0, x E I. If Q(x) is a real-valued function then this follows from (2.21). In fact if Q(x) 3 0, then we can assume that m > 0; if Q(x) < 0, then ,,@@ is purely imaginary. But if Q(x) is complex-valued, then (A) does not follow from (2.21). We introduce the notation

reduces (2.15) to

IIBN(x,n)ll

Methods

X

i

0 1:

then we obtain the system

-0 - ggo10) +c+)(-: :)I(::) Q”(x)--~ 5 (Q'W2 c((x)=18 Q3’2(~) 32 Q”2(~)

(2.23)

Analogous estimates hold for the derivatives y;, y;. Estimates of the form (2.29) are called WKB-estimates. In this relined form they were obtained by Olver. Let us prove (2.29), even though giving a rigorous proof is not quite in the spirit of this part. However, there is a reason. The fact is that the proof enables us to understand why, and in what domains of the complex x plane, the WKBasymptotic formula is valid when q(x) is analytic. Make the substitution zj = j2uj, j = 1, 2, and replace the equation obtained by the integral equations u(x) = u” + (Ku)(x),

(2.24)

u” = (LO)=,

44

=

(Ul(4,

U2@4)YY

where K is an integral operator

1

If Q = A2q, then the substitution (2.22) diagonalizes (2.15) up to O(nm2). In addition, in this case cc,(x,A) = A-la,(x), where al(x) is the function of (2.11) which gives the next term of the WKM-asymptotic formula. In addition to (2.21) we introduce the following condition.

W,(x)

=

x exp{2S(t,x)}crl(t)(ul(t) sb

W&(X)

=

-

+ U2(0)&

x a,@)(u,(t) s b

+

uz(t))dt.

(2.30)

II. Asymptotic Methods in Analysis

M.V. Fedoryuk

150

Denote IIu(x)II = supxpr max(lu,(x)l,

Iwu4l

lu,(x)l), then obviously (2.3 1)

5 P(x,w(x)ll~

We estimate ](Ku)~(x)\. Here condition (A) comes to our aid. Since ReJQcx> 2 0,

a-cx-cb,

then Re S(t, x) 5 0,

Jexp(S(t,x)}( 5 1,

xg;
and (2.31) holds for IKu,I. It thus remains to apply the method of successive approximations, to verify its convergence, and to derive (2.27) from (2.31). This procedure is quite standard. Condition (A) can be reformulated as follows. The function Re S(x,, x) does not increase along the path of integration (along the interval (x,, b)). Counterexamples are known, if this condition is not satisfied, then (2.8) may not have a solution for which the WKB-asymptotic formula is valid on the whole interval I. What form the asymptotic form of the FAS takes on the whole interval I is not known. The exception (and even then only locally, in a small interval) is the case when q(x) is analytic. If q(x) is real, then (A) is satisfied; in this case the operator -d2/dx2 + ,12q(x) is formally self-adjoint. Hence we can draw the following conclusion of a fairly general nature. Non-self-adjointness plus non-analyticity prevents the construction of the asymptotic form of the solution in the large. 2.4. The Asymptotic Form of the Solutions of (2.8). The WKB-estimate

is a deus ex machina for equation (2.8). From it we can obtain an asymptotic formula for the solutions both for large values of the parameter 2 and for large values of the argument x. Comparing (2.8) and (2.20) we see that Q = 12q, so that

a,kQ) = A-‘&;d,

P(x,, x; Q) = A-‘Ax,,

1. Let q(x) > 0 for x E I; choose the branch m the FSS Y,,2(%4

=

~1,2(x;x,,4C1

+ o(~-‘n

I+ +Go,

The estimate of the remainder is uniform in x E I. This asymptotic differentiated: =

T&hFm,2(X;Xo,;1)C1

Yi(a) = 4

y; (4 = -4&h%!

y,(b) = 4

y;(b) = N&hi@

form can be

+ an-‘)1

preserving the uniformity in x of the estimate of the remainder. Since y,, y, are solutions of (2.8), then the asymptotic form can be differentiated any number of times.

a, the solution

y, is a typical zero only in a formula for y,,

+ q’W4q(4), - q’WW)),

where A = q-‘14(a),B = q-1’4(b). Finally, the asymptotic formulae for the solutions are valid for complex I, Re I >= 0, since Re(lZJq(x)) does not change sign for x E 1. Both solutions yl, y2 are analytic in the half-plane Re 2 > 0, for each fixed x E I. Analogously, there are solutions of the form (2.32) if i lies in the left hand-plane: Rel 5 0. But this is another FSS y;, y;, different to the preceding FSS yl, y,. 2. Let Q(x) > 0, I be the semiaxis a < x < co and let the integral x (2.33) I%(X)l dx < cfz s (I be convergent. Then (2.20) has a solution y, such that

The latter integral tends to zero as x + +co, thus we have obtained an asymptotic formula Yl(4

-

x+

Yl(X,%),

+co.

Moreover, we have an estimate for the remainder YlW

I%(4 (2.32)

Y;,2(&4

Put x0 = a in y,, then yl(a) N q-“4(a), and as 1+ +co, x > decreases exponentially since S(a,x) > 0 for x > a. The solution function of boundary layer type: it is noticeably different from small, order A-‘, neighbourhood of x = a. If we put x,, = b in the then it will exponentially decrease from right to left (for x < b). We remark that the solutions y,,, satisfy the Cauchy data

x; 4.

> 0, x E I. Then (2.8) has

151

= Jl(X>-%)C1

5 c

+

&1(X)1, (2.34)

O”I%@)l~~. sx

Let us consider for which classes of functions Q(x) this formula is valid. Let Q(x) N ax”, a # 0, as x -+ +co, and let the asymptotic form be differentiable. Then q(x) = o(x-a’2-2),

X+03,

and (2.33) converges for CI> - 2. In particular, (2.33) holds if Q is a polynomial. Moreover, there are admissible functions which increase more rapidly, and ones which increase less rapidly, as x + +co, for example, Q(x) = AeBX’ (A,B,a > 0),

Q(x) = (logx)“,

-CQ
M.V.

152

Fedoryuk

II. Asymptotic

It is worth analysing (2.33) from the viewpoint

of the analytic theory of differential equations. Let Q(z) be analytic in a domain R < IzI < 00, and at z = cc let it be either analytic or have a pole. Then Q(z) - az” as z -+ co, where n is an integer. The point z = CDwill be non-singular for (2.20) for n 5 - 3, regular singular for n = - 2 and irregular singular for n 2 - 1. The condition a > - 2 means that z = cc is an irregular singular point of (2.20). Condition (2.33) is the requirement of some regularity in the behaviour of Q(x) at infinity. For simplicity, Q(x) be a polynomial, the yi(x) -+ 0 as x + +co, and the second linearly independent solution increases as x + +cc: y2(x) -+ co. The WKB-estimates allow us to construct only one of these solutions, the decreasing solution y,(x). It can be shown that x+ +cQ. y2b) - cQ-“4(x) exp(S(xo, xl>, This solution y, cannot be given by a boundary condition at infinity, since for any constant A, yz + Ay, N y,. The growth of solutions, as a rule, has no physical meaning, and usually only the fact of their existence and the other asymptotic forms for them is of importance. (2.35)

on the semiaxis x > 0. We have S(O,x) = 2/3x 3/2, hence there is the solution yl(x) = cx-114e-2’3X3’2[1 + O(x-312)],

x-+ +co.

This solution, up to the constant factor c, is determined by the boundary condition at infinity: y( +co) = 0. The solution y,, normalized by the condition c = l/(2&), is called the Airy function and denoted &i(x). A second linearly independent solution grows exponentially as x + +co: y2(x) = cx-1’4e213X3’2[1 + O(X-~‘~)].

in Analysis

153

4. Let q(x) < 0, x E 1. In this case it is convenient to write (2.8), (2.20) as y” + L2q(x)y = 0,

Y” + Q(x)Y = 0,

where q > 0, Q > 0. Let I be a finite interval, then there is an FSS [l + O(Z’)]

+ilS(x,,x)}

(2.37) Both solutions oscillate strongly for 1~ 1. The function Re(ilS(x,, x)) does not change sign if 1 is complex in the upper half-plane Im 13 0. Consequently, there is an FSS y:, y:, for which (2.37) is valid as lI( -+ co, ImA 2 0. There is also another FSS y;, y;, for which these formulae are valid for 111-+ co, Im II 5 0. Let I = [a, +co) and let (2.33) hold. Since Re(iJQ(x)) = 0, there is an FSS yl,,(x,A)

=

q-1'4(x)cxp{

y,,,(x) = Q-““(x)exp

{ +i s:, Jim}

Cl + %,2(X)1,

where (2.34) holds for &j(x). These solutions can be given by boundary conditions at infinity

,lym Y’(X) Y(X)-+ i&?&i = 0. . ( >

Example 1. Consider the Airy equation y” - xy = 0

Methods

The signs -, + associated with y,, y2 respectively. Conditions of the type of (2.38) are called radiation conditions. Each of them picks out a unique, up to a constant factor, solution. For both solutions there are doubly asymptotic formulae of type (2.36). We return to Airy’s equation (2.35) and consider it on the semiaxis x < 0. Choose the branch of the root fi = ii&l, x < 0, then S(0, x) = - i2’3 1~1~‘~. Airy’s equation has an FSS consisting of oscillating solutions y,,,(x) = Ix1-114efi2’sIx13’2[1 + O(IXI-~‘~)],

x+-co.

It is obvious that there is a connection formula

Example 2. Consider the reduced Bessel equation

di(x)

=

cly3(x)

+

c2y4(x)~

where cl, c2 are constants. In Chapter 1, 5 4, it was shown that -in/4

on the semi-axis x > 0. We have S(0, x) = f ix, so the equation has two linearly independent solutions of the form Y~,~(x) = e”“[l + 0(x-2)], x+ +co. 3. Let I = [a, +co), q(x) > 0, A > 0 and let (2.33) hold. Then Yl(X,4

= ~1(x;xo,4U + ~-1&1(x,4l

(2.36)

where (2.34) holds for .sr, if i 2 & > 0. The constant c does not depend on 1. We have obtained a so-called doubly asymptotic formula. The remainder in (2.36) tends to zero if x is fixed and I + +co or if A > 0 is fixed and x -+ +OO. Moreover, this holds for complex A in the half-plane Re I 2 0.

Cl

=+c2

There the integral representation of Airy’s function was used. Finding cl, c2 by the methods of this section is impossible. 2.5. Higher

Approximations.

Additional

Parameters.

Under the conditions

q(x) # 0, x E I, and (A), (2.8) has an FSS of the form 1 + k$l

Uj,(X)~-k

+ 0(X”-‘)

1 ,

M.V.

154

II. Asymptotic

Fedoryuk

where N 2 1 is arbitrary and the plus and minus signs are taken for j = 2, j = 1 respectively. The estimate of the remainder is uniform with respect to x E I. This asymptotic form can be differentiated any number of times preserving the uniformity in x of the estimate of the remainder. The coefficients ajk can be found from the formal identities

Methods

in Analysis

155

0 3. Equations of n-th Order and Systems Without Turning Points 3.1. Systems of Equations on a Finite Interval. Consider

the system of

equations (2.39)

k=l

uk(x) as given in (2.11). The solutions ycZ are constructed as follows: equation (2.8) is reduced to a system, then diagonalized up to 0(2-“-l) and reduced to a system of integral equations of type (2.30). The method of successive approximations is then applied. The solutions Y;,~ are given by precise Cauchy data of type (2.32) and hence depend on the index N(ajk(x) do not depend on N). But in many papers the result is formulated as follows: (2.8) has solutions y,, y, for which, as E + + 0 (E = A-‘), there are the asymptotic expansions

uniform in x E I. We deliberately avoid these formulations and here is the reason why. The passage from finite sums for yj” to an asymptotic series is made by N&land’s theorem (also called Borel’s theorem). Suppose we are given an arbitrary formal series Cz==, a,,&“, and a sector D: 0 < 1~1< E,,, CI < args < /I, where p - a < 27c,in the complex E plane. Then there is a function f(s), analytic in D, for which this series is asymptotic:

on a finite interval I, A(x) E Cm(Z). Let pi(x) , . . . , p,(x) be the eigenvalues of .4(x). A point x0 is called a turning point of (2.39) if A(x,) has a multiple eigenvalue. Consider the characteristic polynomial l(x, p) = det(A(x) - pl) of A(x). A point x0 is a turning point if and only if the system Go,P) = 0, &o,P)

= 0

is consistent. We will assume that (2.39) has no turning points. Then an FAS of the system can be constructed by asymptotic diagonalization (see Chapter 2, Q2), but it is more convenient to write the FAS in the form y = exp{lS(x)}

[fe(x) + A-‘fr(x)

- ~WMX)

UE C%,a,l.

If the dependence on the parameter is smooth, there are no turning points and condition (A) is preserved, then all the above formulae remain valid. Consider the equation

(2.41)

= 0

(A(x) - S’(x)Z)fk(x) = -f&(x),

y” - A2q(x, u)y = 0,

+ . ..I.

Here S(x) is an unknown function, fe(x), j-i(x), . . . . are unknown vector functions. Substitute (2.41) into the system, cancel the exponential and equate to zero the coefficients of powers of 1-l. Then we obtain a recurrent system of equations (44

Therefore the construction of the solutions yj(x, E),expanded as the asymptotic series, is nonconstructive. Equation (2.8) may contain additional parameters, for example,

(2.40)

k 2 1.

(2.42)

It follows from the first equation that S(x) is an eigenvalue, and fe(x) is an eigenvector, of A(x). Let er(x) , . . . , e,(x) be a basis of eigenvectors. Put S(X)

=

x pj(t)dt, s x0

./i(X) =

HXbj(X)

(an eigenvector is defined up to a constant factor). The function a(x) is found from the second equation of (2.42)

y” + (k 1 + a(x))y = 0 on the semiaxis R,: 0 < x < co, where lim,,, a(x) = 0. The asymptotic form of the solution was constructed above for c1E L,(R+). There are more delicate results, which allow us to construct the asymptotic from of the solution when u E &(R+), P > 1.

The matrix of this system is degenerate and a necessary and sufficient condition for the system to be soluble is the orthogonality eT(x)fd(x) = 0. Here e;(x) is a left eigen-row-vector: ej*(x)A(x) = pj(x)ej*(x). Consequently U’(X)f?i*(X)ej(X)

+ u(x)ej*(x)ej(x) = 0,

M.V.

156

II. Asymptotic

Fedoryuk

Methods

1(x, p) = p” + q1(x)p”-l

from which we find a(x) = Cexp{jI

eT(x)ej(x) ej*(x)ej(x) ’

The functions p(/)(x) obviously coincide with the ones obtained in Chapter 2 0 2, see (2.19). Expand fi (x) relative to the basis of eigenvectors

L(x)= kil 4hMx). Multiplying

dWf&)

PjCx)- Pktx)'

+ ... + qJx)

= 0

is called the characteristic equation. Let p1 (x), . . . , p,,(x) be roots of this equation. A point x0 is called a turning point if the equation l(x,, p) = 0 has a multiple root or, what is the same thing, if the system (2.40) is consistent. Suppose there are no turning points in the interval I. We will again look for an FAS of (2.46) in the form (2.41), where this time the h(x) are scalar functions. Then we obtain a recurrent system of the equations, the first two of which take the form

the second equation of (2.42) on the left by e:(x), we obtain 4c(x) =

157

is called the A-symbol, and the equation

py)(t)dt},

kP) p(i)(t) = -

in Analysis

4% s’(x)).m

= 0

l,(x, ~(x))fd(x) + ilpp(x, ~(x))~(x)f,(x)

k # j.

The coefficient aj(x) remains undetermined, and is found from the equation of the next approximation. The vector-functions f2(x), f3(x), . . . are found similarly. Thus (2.39) has n FAS solutions of the form (2.42). Let us write out the leading terms. We introduce the notation

= 0.

Consequently s’(x) is a root of the characteristic equation: put s’(x) = pj(x). From the second equation fo(x) can be found. Put ~j(X;

X0,

~) = exp { A S:, Pj(t)dt + S:, Pj’)(l)df},

(2.47)

then we obtain an FAS

then

Yj(x9A) = jjtx; X091)[ej(X) + o(~-‘)l

The system (2.33) has a solution yj satisfying (2.44), as A-+ +co uniformly in x E I, if a condition similar to condition (A) holds: (B) The functions Re(pj(x) - pJx)), 1 5 k 5 n, do not change sign for x E I. There are counterexamples [ 131: if condition (B) is not satisfied, then there is no solution yj with asymptotic behaviour of the form (2.44). As in the case of second order equations, all the asymptotic expansions of the solutions can be differentiated any number of times with respect to x and 1. If condition (B) is satisfied for all j, then (2.39) has solutions y1 (x), . . . , y,(x) of the form (2.44). For 1 >> 1 these solutions form an FSS. A fundamental matrix Y(x, A) of the system takes the form Y(x,A) = [T(x) + O(i.l)]exp[(l

[ln(Qdr

+ 11 n,(t)dt),

Yjlx;

(2.44)

(2.45)

y@)+ Iq,(x)y’“-” on a finite interval I. The function

+ ... + A”q ” (x)y = 0

(2.46)

JjCx;

(2.48)

xO, n) Cl + o(n-‘)]

If condition (B) is satisfied, then (2.46) has a solution yj of the form (2.47), (2.48). This asymptotic form is uniform for x E I and can be differentiated any number of times with respect to x and A. The leading term of the asymptotic formula can be written in another way:

The dash means k # j. If(B) holds for all j = 1,2,. . . , n, then there are n solutions yl,. . . , y, of the form (2.48), and if I > 0 is fairly large these solutions form an FSS. Consider an even order formally self-adjoint equation 1, =(-1)”

-

(dx)

n q ( ) d

ny

( Ox (dx)

+ ... + F”q,(x)y 3.2. Equations of n-th Order on a Finite Interval. Consider the equation

xO, Al =

)

+(-l)nJ2

d

(dx)

n q (x) JC ( ’

(dxr-I’)

= 0

(2.49)

If all the coefficients are real, then I is a symmetric operator on C;(Z). The A-symbol of 1is the function kP)

= 40(X)P2”

-

41(x)P2”-2

+ .*.

+ (-l)“&(X),

M.V.

158

II. Asymptotic

Fedoryuk

and the equation l(x,p) = 0 is its characteristic equation. The conditions on the roots pi(x), . . . , pzn(x), are as before. Then (2.49) has solutions Yj (or an FSS, if(B) holds for all j) of the form (2.48), where

Yjtx;xO2 I) = C1ptx2 Pjtx))l-

S:, p,(t)dt).

llzexpji

(2.50)

This formula was obtained by the author in [13]. For the second order equation y” - /I2q(x)y = 0

we have 4x,p) = P2 - q(x),

&,P)

= 2P,

Pl,Z

=

+qGG%

and (2.48), (2.50) become the WKB-formulae (2.32). We will consider the case when the equation (system) has a turning point in 5 5. We give some examples. YC2”) + A2”q(x)y = 0 where q(x) is real-valued, q(x) # 0 for x E I. We have k

P) = P2” + q(x),

PjCx) = “/TiGi2

where the root takes all of its values. The equation has an FSS such that YjCx2

A) = (4Cx))1’2~“eXP{iCOj

S:,

fldt]

[l + O(l-‘)I,

I + +cO,

where wj are the different values of ‘m. Example 2. Consider the second order system y” = LZA(x)y. The characteristic equation has the form det(A(x) - p21) = 0, and its roots are f m, where the pj(x) are the eigenvalues of A(x). Therefore &/GE..., absence of turning points means that ,u~(x) # pJx) for all j, k, j # k, and that PjCx)f O. For simplicity, let A(x) be a real symmetric matrix. Then there is an orthonormal basis {e,(x), . . . , e,(x)> of eigenvectors of A(x), and an FSS of the form yj’ (x, A) = pJ:“4(x) exp ( + 1 S:, Ji@dt)

in Analysis

159

1”. Systems close to integrable. This means that for x >> 1 the matrix of the system is close to a diagonal or constant matrix. 2”. Systems which reduce to the type 1” by a change of the independent variable and unknown function (of the type y = T(x)z). Consider the system Y’ = cad + WIY, (2.52)

where ,4(x) = diag(p,(x), . . . , p,(x)) and the matrix functions A(x) and B(x) are continuous for x 2 0. Let fi = (0,. . . , 1, . . . , O)r, where the unity stands in the j-th place. 1. Almost diagonal systems. In this case lim IIB(x)ll = 0. x+m Suppose that the following condition holds: for x >> 1 and all k Re(pk+l(4

Example 1. Consider the two-term equation

Methods

- pL(x)) 2 c > 0.

Then (2.52) has an FSS of the form x Yjtx) = ev (Pj(t) + Pj”(t))dt Cfj + uj(x)l, iS x0 I lim pj’)(x) = lim IIuj(x)II = 0 (2.53) x+m x+00 i This result is due to Perron [25]. He also constructed a counterexample: if the 1 difference Re(p,+,(x) - p,Jx)) can change sign for x >> 1, then (2.53) is false. The asymptotic formula (2.53) is fairly rough; it is a logarithmic asymptotic. 1 \ In fact ; / yjj(x)cx~{-~~pj(l)dt)=exPlo(x)}, X-+00, and the right hand side of this formula can be fairly arbitrary as x + co. 2. L-diagonal systems. In this case a, IIB(x)ll dx < CO. s 0

Suppose for some j and all k that the following holds: the difference Re(pj(x) p,Jx)) does not change sign for x >> 1. Then (2.52) has a solution of the form

Cej(X)+ o(~-‘)l.

Yj(X)=expiS::Pj(t)dr)ll;+u,(x)l, 3.3. Large Values of the Argument. Consider a system of n equations Y’ = A(X)Y

(2.51)

on the semiaxis x 2 0. Our interest is in the asymptotic behaviour of the solutions as x + +co. However, to consider this problem in so general a form is simply hopeless. Everything that has been done here can be briefly stated as follows.

lim Iluj(x)ll = 0. x-m If the condition above is satisfied for all j, then the solutions y,(x), . . . , y,(x) form an FSS. This result is due to Levinson [S]. Perron’s counterexample is also suitable for L-diagonal systems.

M.V.

160

3. L,-diagonal

II. Asymptotic

Fedoryuk

< co.

s 0

For all j, k, j # k, suppose that

IRe(pj(x) - Prh))l 2 C > 0, Then (2.52) has a fundamental

x >> 1.

S(zo,

x [A(t) + diagB(t)] dt . 1 iS x0 and Wintner. There are certain refinements of the

This result is due to Hartman results of types l-3. All the basic results so far obtained on the asymptotic form of the solutions of (2.51), can be divided into two classes. 1) By a substitution of the form y = T(x)z, the system reduces to one of the types 1-3. Roughly speaking this can be done if all the eigenvalues of A(x) have the same order of growth and are asymptotically simple, that is, # 0, 1, 00,

j # k.

Consequently,

dt,

(2.55)

’ JQ(z)dr. sf along a canonical path y(z) we have

(2.56)

lev{S(z~t))I 5 1. If we reduce (2.54) to the system of integral equations (2.33), then all the estimates for the integral operator K are as before and the convergence of the method of successive approximations is proved in literally the same way as in the real domain. Thus, we require that the following condition hold (C) For each point z of D there is a canonical path y(z).

A,x+,

Example 1. Consider the Airy equation

w” - zw = 0.

(2.57)

Let D be the plane with a cut along the ray I, = (-co, 01. Choose a branch of & in D so that fi > 0 for x > 0. In our case S(0, z) = $z3”. Let 1,, 1-, be the rays argz = +rc/3, Do the sector largzl < 7c/3. Then S(O,z) is a one-to-one conformal mapping of Do to the right half-plane Re S > 0. Canonical paths are more conveniently represented in the S plane. Let So E Do, then the ray y”(S,): Im S = Im So, Re S 2 Re So is the image of a canonical path since Re S increases along this ray. Any ray leaving So and lying in the half-plane

$4. Equations in the Complex Domain Formulae. Consider the second order equation

- A2q(z)w = 0,

q(t)

S(z, t) =

where r 2 0 is an integer, A are constant matrices and the series converges for 1x1 > R. The asymptotic form of the FSS has been obtained in the analytic theory of differential equations [6]. In the remaining cases there are only fragmentary results. We have already mentioned that there is nothing to be expected in the construction of a general theory. All that has been said, of course, applies equally to equations of n-th order.

W”

Tr

Let y(z) be an infinite curve in D joining the points z, co. The curve y(z) is called a canonical path (or progressive path) if Re S(z, t) does not decrease along y(as a point moves from z to t). Here

k=O

4.1. WKB-Asymptotic

=

w” - Q(z)w = 0.

In addition the eigenvectors must not be ‘too rapidly’ rotating for x >> 1 (see [13]). In this case the asymptotic behaviour of the FSS is ‘regular’, that is, its leading term can be expressed in terms of the eigenvalues and eigenvectors of A(x). 2) The matrix of the system has the form A(x) = xr f

4

J% where z. E D. We will assume that D is simply-connected and unbounded, and that q(z) # 0 for z E D, so there are no turning points in D. Then the two-valued function &@ decomposes into two single-valued branches in D. Fix one of these: the second is - J4(z). Similarly we fix on one of the .four roots $“&). Finally, the integral S(z,, z) will be taken over a path lying in D, so S(z,, z) will be a single-valued analytic function in D. As in Chapter 2,s 2, we begin with WKB-estimates for equations of the form (2.54) with no parameter

Y(x) = [Z + o(l)] exp

cj,k

161

z

matrix of the form

lim Pj(x) ~ = X’a, Pk(X)

in Analysis

where q(z) is analytic in a domain D in the complex plane. We can assume, for simplicity, that q(z) is a polynomial of degree n 2 1. The asymptotic formulae (2.12), obtained in the real domain, still hold in suitable complex domains. Let us describe these domains. First of all we must be precise in the handling of many-valued functions such as J4(z), $%I and

systems. In this case co JIB(x)lj’dx

Methods

(2.54)

t

M.V.

162

Fedoryuk

II. Asymptotic

Re S > 0, is also the image of a canonical path. The domain D, satisfies condition (C). This condition also holds for the sectors D,: 7c/3 < arg z < Z, D,: --71 < argz < -7113. Moreover, condition (C) holds for the domain D - the complex plane with a cut along the ray I,. The function S(0, z) maps D onto a domain consisting of three half-planes and condition (C) is satisfied. We introduce the same notations as in Chapter 2,§ 2:

WI, Pk Y) =sIal( tCl(z;zo) = Q-“4(z)exp{-S(z,,z)),

r(z)

The infimum is taken because there are infinitely z (see example 1). Let

(2.57)

Y

many canonical paths leaving (2.58)

P(Z,D) 5 c < co, for all z E D. Then (2.55) has a solution such that I&-

11 52@2P(“‘D)-

1).

(2.59)

Q(qJ

= ... = Q’“-“(z,)

KJ~(z,~;z,) = 4-1’4(z)exp(-AS(z,,z)}, S(z,,z) =

’ ,/&idt. s 20

The notation for p remains with CI~(z; Q) replaced by a, (z; q). Then for I 2 & > 0 there is the asymptotic formula w,(z,A) = Gr(z,i;z,)[l IEl(Z,4I

+ A-l&i(z,A)],

5 CP(Z,D).

(2.60)

This is a doubly asymptotic formula since the remainder tends to zero both for z E D fixed, il+ +co, and for I > 0 fixed, z -+ co. If q(z) is a polynomial, then there is an asymptotic expansion for w1 wl(z,4 = %(z,4zdexp

{ &* (-A)- ’ i,,, r,(Wtj,

(2.61)

which can be differentiated any number of times with respect to z and il. 4.2. Stokes Lines and the Domains Bounded by them. Let z0 be a turning point

of (2.55), so that Q(zo) = 0. A Stokes line is a level line

= 0,

Q’“‘(z,)

# 0.

Then for small Iz - zel i Q(z) - 4~ - zd”,

a # 0,

S(z,, 4 - __ n2G+ 2 (z - ZOy2+l.

The equation Re(a(z - ~,#~+r) = 0 d ef mes n + 2 rays: the angle between adjacent rays being equal to 2n/(n + 2). Correspondingly, leaving a turning point of order n there are n + 2 Stokes lines, forming equal angles at -zO.In particular, three Stokes lines leave a simple turning point (n = 1); the angle between adjacent Stokes lines being 2x/3. For the Airy equation the Stokes lines are the rays argz = frc/3, argz = rc. Remark. In the literature there have been differences in the definitions. Frequently the Stokes lines defined above have been called anti-Stokes or conjugate s lines. The level lines (

Im

L-41. lead to an asymptotic formula for the solutions of (2.54).

163

leaving the turning point z0 (more precisely, a maximal connected component of it, not containing turning points). Let z0 be a turning point of order n, that is,

There is a similar estimate for the derivative w;(z). This result is due to Birkhoff The WKB-estimates Put

in Analysis

Re S(z,, z) = 0

E

p(z, D) = inf p(z, y).

Methods

‘,,&@dt s 20

=0

:alled Stokes lines. Thus, the local structure of Stokes lines is very simple. Their global structure has only been fully studied for polynomial and rational Q(z). Let Q(z) be a polynomial of degree n 2 1, z,, a turning point, and let the Stokes line 1leave zO. Then, either 1 is an infinite line, or is a finite line joining two turning points z0 and z1 . ’ Draw all the Stokes lines. They divide the complex plane into domains of two ” types. 1. Domains D of half-plane type. The function S(z,, z) maps the domain oneto-one onto a half-plane of the form Re S > c (or Re S < c). The boundary of such a domain consists of one connected component. For Airy’s equation there are three domains of half-plane type. There are n + 2 of these domains, where n is the degree of Q(Z). 2. Domains D of strip type. The function S(z,,z) maps D one-to-one onto a vertical strips a < Re S < b. The number of such domains does not exceed n - 1. ’ For examples of Stokes lines, see Fig. 2. The level lines Re S(z,,z) = const, ImS(z,,z) = const need not be closed curves. Let Q(z) be a rational function. A first order pole z0 is called a turning point of order - 1; leaving such a point there is one Stokes line. Equation (2.55) must be discussed on the Riemann sphere. In a neighbourhood of z = cc we make the

M.V.

164

II. Asymptotic

Fedoryuk

Methods

in Analysis

165

In this case Q(z) -+ 0 as Re z + -co, so that z = cc is a turning point. The family of level lines Re S = const consists of the lines y = k7-c+ 7c/2, k = 0, f 1, + 2,. . . which it is natural to call Stokes lines, and the lines ex cos y = c # 0 (z = x + iy). A clear indication that z = co is a turning point of infinite order is that:

Fig. 2

n

ez = lim n-m transformation

1+ z (

n> ’

the point z = - n is a turning point of order n. 4.3. Boundary WKB-Asymptotic

we then obtain the equation (2.62)

The point z = co is called a turning point or singular point of (2.55) if the same is true of [ = 0 for (2.62). Equation (2.55), where Q(z) is a rational function, has no turning point if and only if Q(z) is a function of the form a, a(z - zl)-2,

a(z - zJ-2(2 - z2)-2,

u(z - zl)-4<

In all of these cases (2.55) is integrable. Let (2.55) have a turning point. Then the Stokes lines divide the Riemann sphere into domains of half-plane type and strip type, and domains of annulus type and circle type. 3. Domains D of annulus type. The function w = es(‘@‘) maps D one-to-one onto an annulus 0 < a < 1WI < b < co. The boundary of such a domain has two connected components (Fig. 2). 4. Domains D of circle type. The function w = es maps D one-to-one onto a domain 0 < 1w 1 < a, or a < 1WI < co. The boundary of such a domain consists of a connected curve and a point z. which is a second order pole of Q(z) (figure 2). If z. # co, then

Conditions for Solutions and Domains of Applicability of the Formulae. The fundamental problems posed for equation

(2.55) on the semiaxis (0, +co), or the real axis (-co, 00) are the eigenvalue problem and the scattering problem. In both problems boundary conditions at infinity are given for the solutions. These problems are of two types. For simplicity, let Q(z) be a polynomial, then any solution is an entire function of z. 1. The condition of decrease of the solution. Let D be a domain of half-plane type and w(z) + 0 if z E D,

Z-CO,

ReS(z,,z) + +co.

(2.63)

Then w(z) = cw,(z), where wi(z) is the solution constructed in subsection 3.1 of Chapter 2 (see (2.46)). The second linearly independent solution grows exponentially under condition (2.63). If Q(z) is a polynomial of degree n 2 1, then there are n + 2 domains D,, . . . , Dn+2 of half-plane type. Each Dj is associated with a solution wj(z) such that wj(z) -+ 0 under condition (2.63). It is clear that these solutions are linearly independent. For (2.54) we can obtain corresponding asymptotic connection formulae for A >> 1. 2. Radiation conditions. Let D be a domain of strip type, z E D,

Z-CO,

ImS(z,,z)

-+ +co.

(2.64)

Then each of the following boundary conditions

lim (z - zo)‘Q(z) = A > 0. z-‘zo If Q(z) is a rational function, then the structure of the level lines Re S = const may be very complicated. Such a level line can be everywhere dense in some domain D. For example, if Q(z) = e2ior(z- al)-‘(z

- u2)-‘(z - uJl(z

- u4)-l,

where a, and the uj are real, the uj are distinct, and tancr is irrational, then the closure of any level line Re S = const is the whole complex z plane. If Q(z) is an entire or meromorphic function, then it is possible to show that z = co is a turning point of infinite order. An example: Q(z) = eZZ,

S(O,z) = ez - 1.

3 determines a solution which is unique up to a constant factor. These solutions are linearly independent. The same is true if Im S(z,, z) + -co. Thus in a domain of strip type there are two FSS, which are defined by their asymptotic behaviour as S(z,, z) + *cc respectively. Let us describe the maximal domains of applicability of the WKB-asymptotic formulae (2.60) for polynomials q(z). Let D be a domain of half-plane type, D, the closure of D with fixed .s-neighbourhoods of the turning points in aD removed. Then (2.60) applies for z E D,, A + +co. Let the following conditions hold: 1”. (2.54) has only simple turning points; !: 2”. there are no finite Stokes lines (that is, Stokes lines joining turning points).

M.V.

166

Fedoryuk

II. Asymptotic

In this case three Stokes lines leave each turning point zj. The maximal domain G of applicability of the asymptotic formula (2.60) is constructed as follows. Make a cut along one of the Stokes lines l(zj) leaving zj, and remove a fixed neighbourhood of this line from the complex plane. Each such line l(zj) is uniquely defined by the given initial domain D of half-plane type. The remainder in (2.60) is 0(X’) as 1+ +co, uniformly in z E G. For each z of G there is a canonical path y(z) going to infinity in D. This result is sharp: if one of the conditions l”, 2” is not satisfied, then we have to remove entire domains of half-plane type from the domain of applicability of the WKB-asymptotic formula. For example, if w(z, 1) is a solution of W”

-

12(z2

-

l)w

=

(2.65)

0,

such that w(x, 1) + 0 as x -+ +co, then (2.60), in general, is not applicable in domains of half-plane type containing the semiaxis (-co, 0) (figure 3). Consider, as an example, Airy’s equation; we retain the notation of example 1. There are precisely three domains of half-plane type D,, D,, D,, and three solutions u(z), w,(z), w,(z), such that u(z) + 0 for z E D,, z -+ co, wj(z) + 0 for z E Dj, z + co. Here Re S(0, z) + +co in these domains. We normalise the solutions by the conditions 44

1

2 i --x3/2 3

NPx-114exp 2J71

Wl (4 Nc&x)-

I

,

1’4exp ii(-~)“~ i

x+

+a,

,

x+

-00,

+

w2(4).

Consider Airy’s equation with a parameter w’! - /12zw = 0 and put Wj(Z,A) = r11’6wj(A2’3z).

For w,(x, A) there is the asymptotic formula 1 wo(z, 4 N -z-l” 2J71

167

removed. This domain (we denote it G,) is a maximal domain of applicability of the asymptotic formula (2.66). It is impossible to extend it by ‘erasing’ part of the cut 1,. This follows, for example, from the fact that u(z) is an entire function of z and the leading term of (2.66) is a many-valued function with a branch point at z = 0. Similarly, a maximal domain Gj, j = 1, 2, of applicability of the WKBasymptotic formula for wj(z,l) is the complex plane with a cut along the ray lj. Near the Stokes line Ij the asymptotic formula for Wj takes a different form. Let us find the connection formula. In this example it is not even necessary that 2 be large; let ;1 = 1. The solutions wl(z) and w2(z) are linearly independent so 44

=

Cl%(Z)

+

c,w,(z),

where cl and c2 are constants. Let S,,(z) be the branch of S(O,z) = 2/3z3j2 associated with V(Z), that is, S(0, x) > 0 for x > 0, and is defined in Go. Then Re S,(z) < 0 for z E D,, so u(z) grows exponentially in D, . The function wi(z), by construction, decreases exponentially in D,, and w2(z) increases exponentially, since Re S,(z) < 0, z E D,. Here S,(z) is the branch of S(O,z) defined in G, such that Re S,(z) > 0 in D,. Consequently c2 = lim -,44 *+a, w2(4

Re S,,(z) + +co.

z~D1,

I

L&(x)= jg(wI(x)

wo(z,2) = Pu(~2’3z),

in Analysis

Taking account of the choice of branch of z -‘I4 for u(z) and wZ(z) we find that c2 = i/2. Similarly we calculate c1 = -i/2, and obtain the identity

where x3/2 > 0 for x > 0, (-x)~/~ > 0 for x < 0; and we put Wz(Z) = WI(~). The solutions U(Z), w1(z) and w2(z) are called the Airy-Fock functions and are related to the Airy functions by

&i(x) = v(x) &’

Methods

exp

-i*z3i2

i

,

(2.66)

I

for z E D,, z + XI and any fixed 1 > 0. Let 1 >> 1, then (2.66) holds in the whole complex z plane with a fixed s-neighbourhood of the Stokes line 1, = (--co, 0)

u(z)

=

$pd4

-

w2(4).

Let S, be the sector jargz - ~1 < E, E > 0 small, containing the Stokes line 1, = (-co, 0). There are WKB-asymptotic formulae for w~,~(z)valid in this sector, and the asymptotic form of the Airy function is given by (1.59). Thus, in different sectors of the complex plane there are different asymptotic formulae for the Airy function. This fact-that a function has different asymptotic behaviour in different sectors of the complex plane-is called the Stokes phenomenon. In this formulation, to the modern reader, the Stokes phenomenon is a triviality. The Airy function has also been well studied; its properties being well known to specialists. But now we quote from a letter from Stokes to a friend-‘1 have been doing what I guess you won’t let me do when we are married, sitting up till 3 o’clock in the morning fighting hard against a mathematical difficulty. Some years ago I attacked an integral of Airy’s, and after a severe trial reduced it to a readily calculable form. But there was one difficulty about it which, though I tried till I almost made myself ill, I could not get over, and at last I had to give it up and profess myself unable to master it. I took it up again a few days ago, and after two or three days’ fight, the last of which I sat up till 3, I at last mastered it.’

M.V. Fedoryuk

II. Asymptotic Methods in Analysis

The Stokes phenomenon is insidious. The list of erroneous papers, whose authors have used an asymptotic form, valid in one domain, in domains in which it is no longer applicable, is huge (many examples are quoted in [ 151). If we adopt a broader viewpoint, then it appears that the ‘Stokes phenomenon’ is an unavoidable companion to all the natural sciences. It is enough to recall, for example, the ‘ultraviolet catastrophe’. The radiation law of Rayleigh-Jeans, obtained for long wave radiation, turned out inappropriate for short waves, and this discrepancy was resolved only by a new physical quantum-theoretical Planck hypothesis. The history of science contains many such examples.

linearly independent if ;1 2 &, >> 1. We remark that any Stokes line can be contained in some canonical domain. We have said above that a canonical domain D contains two domains of half-plane type. Let D+(D-) be the one of them which lies to the right (left) of 1 (more precisely, their images S(D+), S(D-) lie respectively to the right and left of the image S(I) of 1). Then for fixed 1 > 0, the solution u(z,A) (v(z,A)) decreases exponentially for z E D-, z -+ co (respectively, for z E D,, z -+ CO).As shown in subsection 4.3, u and v are uniquely determined by these conditions. It is no accident that we described the boundary conditions for u and u in such detail. For a mathematician it is self-evident that to select a unique solution of (2.54), it is necessary to give boundary conditions for the solution, for example, Cauchy data or the asymptotic behaviour as z + cc for fixed 1. But in many physical (and at times mathematical) papers the solutions .are given by the asymptotic behaviour relative to 1, which does not uniquely defined the solution. This often leads to errors: in [ 163 there is a list of erroneous papers. Let (uj, Vj), (uk, uk) be two FSS of (2.54). Any solution w can be written in the form

168

4.4. The Global Asymptotic Form of the Solutions of Equation (2.54). The fundamental problem of the asymptotic theory is to construct an FSS whose asymptotic behaviour as A-+ +cc is known throughout the complex z plane. When q(z) is a polynomial this problem was solved by M.A. Evgrafov and M.V. Fedoryuk [13]. Let G, be the domain obtained from the complex z plane by the removal of s-neighbourhoods of all the turning points of (2.54). We will show how to construct an asymptotic formula as 1+ +co for an FSS of (2.54) in G,. Close to turning points the asymptotic form of the FSS is constructed by known methods (see 4 5). A domain D of the complex plane is called canonical if S = S(z,, z) maps D one-to-one onto the whole complex z plane with a finite number of vertical cuts. The boundary of D consists of Stokes lines, the sides of the cuts being the images of these lines. In example 1 the domain D = D, u 1, u D, is canonical. If the branch of S(0, z) = 2/3z312is chosen to be positive on the semiaxis (0, +co), then S(D) is a plane with a cut along the ray (- ioo, 0). In this case D, u 1, u D, and D, u I, u D, are also canonical. A canonical domain consists of two domains of half-plane type and some domains of strip type. Remove from S(D) fixed .s-neighbourhoods of all the cuts, and denote by D, the inverse image of this domain in the z plane. Let 1 be a Stokes line with origin at a turning point z. in D. Choose a branch of S(z,, z) in D so that Im S(z,, z) > 0,

z E 1.

Then Re S(z,, z) > 0 ( < 0) to the right (left) of 1 and close to 1. We introduce FSS u(z, A), D(Z,A) corresponding to the triple (1,zO, D); such FSS are called elementary. These solutions have the asymptotic form u(z, A) - cq-1’4(z) exp{U(z,,

z)},

4z,4 - cq -ri4(z)exp{ -;1S(z,,z)}. Here c is a normalizing

(2.67)

W =

+

PjVj

=

ClkUk

+

BV,,

where CI,, & depend only on 2. We have (z)

=

aj!AA)($.

The matrix Qjk(A) is called the transition matrix from the FSS uj, vj to the FSS ukuk. Obviously, Qjk = sz,i’ Qjl = QklQjk9 Every transition matrix from one elementary FSS to another is a product of a finite number of the simple transition matrices which we list below. Thus, the basic problem of the asymptotic theory of (2.54) splits into three parts. 1. The topological problem. Find all turning points and Stokes lines. 2. The analytical problem. Find the asymptotic form of the elementary FSS and elementary transition matrices. 3. The algebraic problem. Multiply the transition matrices. Problem 1 is essentially ‘computational’; for a concrete equation it is possible to find the turning points and construct the Stokes lines using a computer. The asymptotic forms of the elementary FSS have been given above. Further there are four types of elementary transition matrix. 1. The transition (I, zo, Dl) -+ (I, ze, D2). Here only the canonical domain is changed, and Q(A) has the form

constant:

lim arg[cq-“4(z)] = 0. z-+zg,zE1 The asymptotic formulae (2.67) are valid for z E D, z -+ co and for each fixed A > 0: for u as Re S(z,, z) + --co; for u as Re S(z,, z) -+ +co. The solutions are

OLjUj

169

+ 0(1-i) Q(A) =(1 O(e-@)

ICI = 1,

Asymptotic w22m

O(e-‘I”) c1,c.J s-0. 1 + o(n-‘) > ’ series have been obtained for the diagonal elements ~~~(2) and

M.V.

170

II. Asymptotic

Fedoryuk

Y -1

1

w(-cc&q

Fig. 3

Thus, if it is enough for us to know the asymptotic form of the transition matrix only up to @A-‘), then it is sufficient to give the pair (I, zO) and not worry about the choice of canonical domain. 2. The transition (1,zi, D) --f (I, z2, D). In this case 1is a finite Stokes line joining the turning points zi, z2 and only the orientation changes. We have

e& = -c2

I~(Zl,Z2)I,

Cl

3. The transition (l,,z,,D) + (12,z2,D), where the rays S(I,) and S(I,) are directed to one side (Fig. 3, q(z) = a2 - z2). Let l2 lie to the left of I,, then 0 ,u

>'

n = 1,2,...,

ei40 = _c2 Cl

The formulae for the transition matrices of type 2 and 3 are exact, not asymptotic. 4. The transition (l,,z,,D2) + (/2,z,,,D2). Both Stokes lines I,, 1, leave the turning point zO. Let I, lie to the left of 1, and let ze be a turning point of order II. Then

lisin(&)!

+ ‘(I-‘)

If z,, is a simple turning point, then

Choose two different points z1 and z2, then the eigenvalue equation written

i

>

+ o(n-11,

and asymptotic series for the elements of the transition

matrix are known.

= 1

Wl(Z2?4

can be

(2.68)

.

The condition w( +cc, A) = 0 picks out a solution wl(z, A), which is unique up to a constant factor. We normalize it by the condition w,(x,4 N 4-1’4(x)exp{-~S(xl,x)},

x+

+a,

where q(x) = x2 - 1 and the branches J4(x), m are positive for x > x1. Remove from the complex plane the domain bounded by the Stokes lines l,, lz, the Stokes line I,, and small neighbourhoods of these Stokes lines. In the domain D, thus obtained the above asymptotic formula for w,(z,~) holds as II + +co, uniformly for z E D,, as follows from subsections 4.2 and 4.3. Similarly, we normalize w,(z, A) by the condition w2b, 4 N 4-“4(x) exp{lS(x,,

0 1

Q(l) = epinj6 1 (

= 0.

2, > 0,

Wl(Zl94 *w2(z2,

Re/?>O,

W)=exP{-&}\ 1

w(+m,L)

Wl(Z, A) = Aw,(z, /I).

W2(Zl?4

P = W1,zz),

= 0,

lim 1, = +co. n-+m We will investigate the asymptotic form of the eigenvalues I, as n -+ 00. Assume that in (2.65) 1 > 0 is a large parameter. Equation (2.65) has two simple turning points xl,2 = _+1, which are joined by a Stokes line 1, = (- 1,l). Another two Stokes lines, I, and 1:, leave x1 (figure 3); let 1, be the one on which Im z > 0. Similarly we denote the Stokes lines leaving x2 by 1, and lt, where Imz > 0 for z E 1,. The Stokes lines divide the complex plane into four domains of half-plane type. In this simplest example it is possible to do without the transition matrix. Let w,(z,;1), w,(z,A) be the solutions of (2.65) satisfying the boundary conditions w1(+oo, 1) = 0, w2(-co, 1) = 0. The number I is an eigenvalue if and only if these solutions are linearly dependent, that is, WA

~-J(J) = ei40‘f” (

171

It is known from the spectral theory of differential operators that there are an infinite number of eigenvalues in this problem

e

a =

in Analysis

Example 2. Consider the eigenvalue problem on -cc < x < cx)for (2.65). A number 2 is called an eigenvalue if there is a non-trivial solution w(x, 2) E L,(R); the solution is called an eigenfunction. In our example, the condition for the solution to be square integral is equivalent to the boundary conditions

11

12

Methods

x)},

x+-co,

where the branches &$) and m are positive for x < x1. As J.+ +co this asymptotic formula is valid in a domain D,, symmetric with D, relative to the imaginary axis, so that their intersection D, n D, is not empty. In (2.68) z1 can

M.V.

172

II. Asymptotic

Fedoryuk

be taken to be, for example, a purely imaginary conjugate Zr. We have w,(z,,4

4:‘4(zl)exp{-~Sl(xl,zl)}

~w,(z,,

4 =

number, and z2 its complex cl

+ o(n-l),

(2.69)

4:‘4(zl)exp{nS2(x2,zl)}

The index j means that branch associated with wj. It remains to be precise about the choices of the branches. We begin with the branches (J4(z))j. By assumption, (J4(x))i > 0 for x > x1, (&@)2 > 0 for x < x2. Consequently, for Imz 2 0, we have (JGm2 Put J4(z) = (J4(z))i

= 4&i),.

for Imz > 0, then we have Sl(Xl,Z) + SAX,,4

=

x2 &i&k J XI

Methods

in Analysis

173

where a is given by (2.70). We can also find an asymptotic formula for the eigenfunctions w,,(z) = wi(z, A,), throughout the complex plane with the exclusion of small neighbourhoods of the turning points x1 and x2. Of greatest interest is the asymptotic form of w.(x) on the real axis. Fix a small number E > 0, not depending on A. The asymptotic formula for w,(x) on the semiaxes x > x1 + E, x<x2E was found above. The asymptotic form of w,,(x) on I,= [xZ + E, x1 - E] is constructed using a very elegant procedure. Take any point x0 > x2 and analytically continue the asymptotic formula for wi(x, A) (we stress: not the function itself, but its asymptotic formula) to a point x E Z, along any path lying in the upper half-plane Im z > 0. We denote the expression obtained by w1(x + i0, A). In exactly the same way we analytically continue the asymptotic formula along a path in the lower half-plane and we obtain a value wl(x - i0, A). The semi-sum of these expressions for I = Iz, gives the asymptotic form of w,(x) for x E I,: w,(x) = $[Wl(X + io, A,) + Wl(X - io, A,)].

where the integral is taken over any curve y+ in the upper half-plane joining x1 and xz. Therefore the exponential on the right in (2.69) is equal to

The asymptotic formula (2.71) is so remarkable that it deserves much more discussion. 1” If we disregard the remainder in (2.71) we obtain the equation

If we replace z1 by z2 in (2.69), then we obtain the exponential hence we find 1, = 2n + 1.

where the contours y+ and y- are symmetric relative to the real axis. We note that the above branch of m z is single-valued in the plane with a cut along the segment lo = [xZ,xl]. Finally, on the left hand side of (2.68) we have the exponential

In the concrete example given, q(x) = x2 - 1, the eigenvalues 1, are calculated exactly and coincide with the above for n = 0, 1,2,. . . The eigenfunctions in this example are the Hermite functions. It is remarkable that the values A,,can be calculated exactly (and not just their asymptotic form) by going into the complex plane and without using exact solution [ 131. 2” Consider Schrtidinger’s equation

Here y is a closed curve enclosing I, and oriented counterclockwise, (2.70)

Similar arguments, related to the branches of Q&@, lead to the fact that the factor - 1 appears on the left hand side of (2.68), so the eigenvalue equation takes the form pa= = - 1 + ()(A-‘). Hence, we have found an asymptotic formula for the eigenvalues 2, = cI-1(n7c -t n/2) + O(n-‘),

n-+03,

(2.71)

-$y

+ (E - U(x))$ = 0.

(2.72)

Let V(x) be a ‘potential well’, that is, U(x) has precisely one minimum point x,,, for simplicity nondegenerate (V”(x,) > 0) and U’(x) > 0 for x > x0, U’(x) < 0 for x < x,,. We can assume that U(x,) = 0. Let there exist finite or infinite limits lim,, +a, U(x) = U, > 0. Consider the segment J = [Ey,Ei], where 0 < Ey < E! < min( U+ , U-) and consider the eigenvalue problem for (2.72). The spectral parameter is the energy E. We will find the asymptotic form of the eigenvalues E, E J given that h > 0 is a small parameter. Equation (2.72), for each E E J, has exactly two turning points xl(E), x2(E), x2 < xl, both simple. We will assume

M.V.

174

II. Asymptotic

Fedoryuk

xl(E)

J2m(E

- U(x))dx

= h

in Analysis

175

A number of other important spectral problems have been solved by the same methods: the asymptotic form of the scattering matrix has been found for A-+ -too, when (2.54) has an arbitrary number of real turning points, when there are no real turning points, (the so-called passing over a barrier); the asymptotic form of the eigenvalues when qtz) has poles; the asymptotic width of the gaps in the spectrum of a Sturm-Liouville with a periodic potential; etc. A survey of results is given in [13]. The possibilities of the WKB-method in the complex domain are far from exhausted.

that the potential U(x) is analytic in some neighbourhood of the real axis. Then (2.71) holds for E, = E,(h), and in this case it takes the form

s

Methods

(2.73)

.MQ

This is an equation in E which defines the eigenvalues E,,(h). Formula (2.73) is called the Bohr-Sommerfeld quantization rule; it is a famous formula in quantum mechanics. 3” Let q(z) be a polynomial, for simplicity, real for real x and q( +a) > 0, having exactly two simple zeros x2 < x1. Near the real axis the structure of the Stokes lines will be as in example 2. Take the solution w1(x, 1). It turns out that if A = 2, is an eigenvalue, then the asymptotic form of w1 is a single-valued function in the plane with a cut along the segment 1, = [xZ,x,]. As is well known, the n-th eigenfunction has exactly n zeros on the real axis, and all of them are simple. Let y be the same contour as above, then by the residue theorem

4.5. Equations of n-th Order and Systems. Consider the equation WC”)+ lql(z)w(“-‘)

+ ... + l”q,(z)w

= 0

(2.74)

where n 2 3 and the coefficients of the equation are analytic in some simplyconnected domain D in the complex z plane. We denote the roots of the characteristic equation l(p, z) = p” + 41 (z)p”-’

+ . . . + qJz) = 0

(2.75)

by plC4, . . . ,p,(z). Suppose there are no turning points of (2.73) in

Replacing w,(z) by its asymptotic form, we obtain

w’(z) --I.Jqo-~+o(/.,‘), -= WI(z) which, after integration, again leads to a formula of the form (2.71). We have restricted ourselves to the first terms of the asymptotic formulae for the eigenvalues and eigenfunctions but, of course, asymptotic series have been obtained for them [ 133. The case considered above (q(x) having two real zeros) is to some extent not very illuminating--the asymptotic formulae for the eigenvalues and eigenfunctions can be obtained by other methods. Moreover, these formulae (for example, (2.71)) hold for non-analytic functions q(x)-a finite degree of smoothness is sufficient. The method of proof is as follows. An asymptotic form, applicable for x 2 x2 + E, is constructed for the solution w,(x, 2) which decreases as x + +a3; the result is expressed in terms of the Airy function (0 5). Similarly, an asymptotic form, applicable for x 5 x2 - E, is constructed for the solution w,(x,A) which decreases as x -+ -co. If w is an eigenfunction, then w1 = Aw,, and the eigenvalue equation is the equating to zero of the Wronskian of wl, w2 at some point. Strictly speaking, this is the WKB-method. Another method is the application of the canonical operator of V.P. Maslow [23], [24]. The problem can only be solved so simply when there are exactly two turning points. If there are several and, moreover, some of them are multiple, then the above method-going straight along the real axis through the turning pointleads to depressingly cumbersome computations. Going into the complex plane enables us to solve these problems in a significantly simpler way [ 133.

1

i; 1 1 i a j

D, then all the roots pj(X) are analytic in D. We formulate an analogue of Birkhoff’s theorem for (2.73). We introduce the same notation as in 5 3: z sj(zO, z, = Pj(t) dt, sjk(zO, z, = sj(z09 z, - sk(zO, z). s Let the curve y(z) lie in z, join the points z and co, and suppose that ReS. (z t) does not decrease on moving from z to t along y(z). Then the curve is cairkd a (j, k)-canonical path. A set {yjl(z),. . ., y.,n (z)> consisting of (j, k)-canonical paths, is called a j-canonical vector-path. D is called j-admissible, if for each z E D there is a j-canonical vector path yj(z). We restrict ourselves for simplicity to the case when all the coefficients of (2.73) are polynomials. Let the following conditions hold. 1. D is j-admissible. 2. The roots pi(z), . . . , p,(z) have the same order of growth for z E D, z + co, that is, lim Pj(z) ~=Cj&#O,l,cO 2-m P!%(Z)

(j

Z Q

Then (2.73) has a solution of the form wj(z,~)=expji~~pj(t)dt+~~p:“(t)dt)CI

+A-‘Ej(z,l)]

1 The functions pj”(z) are defined by (2.47), and for I 2 I, > 0 L I

IEjCz9

n)l

5

dj(z),

lim ZED,Z+m

aj(z) = 0.

(2.76)

176

M.V.

Fedoryuk

II. Asymptotic

The asymptotic formula (2.76) is double. There is an analogous result for systems of the form w’ = AA(z)w. If qj(z) is not a polynomial, then additional conditions of type (2.33) are required. Let z0 be a turning point of (2.73), pj(zo) = pk(zO), j # k. The level line Re

’ (Pj(t) - ~k(t))dt s%

= 0,

leaving z0 is called a Stokes line. The local structure of a Stokes line, and also the global structure of individual Stokes lines, is studied equally as simply as for (2.54). Unfortunately, this concludes the analogy between equations of order n 2- 3 and order n = 2, even with polynomial coefficients. One reason is that the Stokes lines of (2.73) may intersect themselves. For n = 2 the domains of applicability of the asymptotic form are defined by one harmonic function Re !4, m dz. F or n 2 3, there are $n(n - 1) such functions. In short, at present there is no global asymptotic theory for equations of order n 2 3 and, in the author’s opinion, it is impossible to construct one in general.

Methods

in Analysis

y" - A2c(x - x,)"y = 0,

(2.79)

whose solutions are expressed in terms of special functions-the Bessel functions of order l/(n + 2). This enables us to construct an FSS yy, yi, whose asymptotic form is applicable for x E I,. A more careful analysis reveals that I, intersects Iand I,, which allows us to find an asymptotic formula for the transition matrix Q(n). Namely, we first express y;, y; in terms of yy, yi, and then these in terms of Y:, Yz’. Such a general scheme is suitable, generally speaking, for equations of n-th order and for systems. 5.2. Turning Points of SecondOrder Equations. Consider (2.77) on the interval I, where a < 0 < b and let x0 = 0 be a simple turning point. Then q(0) = 0, q’(O) # 0 and (2.79) takes the form y” - A2q’(O)xy = 0.

An arbitrary solution of this equation is y = w(A2i3(q’(0))“3x), where w(t) is a solution of the Airy equation W” - tw = 0.

0 5. Turning Points

177

(2.80)

Let q’(0) > 0, for definiteness, then q(x) < 0, x < 0;

q(x) > 0, x > 0.

5.1. The Problem of Connection Formulae. Consider the equation

(2.77)

y” - i12q(x)y = 0

with a real-valued function q(x) on a finite segment Z = [a,b]. Let there be precisely one turning point x0, a < x0 < b, of order n. Fix a small number 6 > 0. In each of the segments I- : a 5 x 5 x - 6, I+ : x0 + 6 5 x 5 b, we can apply the WKB-method and construct two FSS (y;,y;), (y:,yz) of the form (2.12). These FSS are connected by the relations Y:k4

= wl(4Y;(x?4

+ Q42(4Y,(XY4

(2.78)

Y: (x9 4 = w2 l(4Yi (x94 + 822(4Y2 (x24 which are called connection formulae. The matrix Q(1) = (~~~(1)) is called a transition matrix, or connection matrix. The problem is to find the asymptotic form of Q(n) as J + co. The WKB-method does not enable us to solve this problem, since the WKB-approximation is inapplicable near to a turning point. In a small neighbourhood I,, = (x0 - 26, x0 + 26) of x0, q(x) can be approximately replaced by a power:

4(x) = 4x - xo)n, Correspondingly

(2.77) is replaced by

c=y’“‘ozo.

We will seek an FAS of (2.77) in the form y = Aw(l-2’3<(X)) + n-2’3Bw’(12’3~(X)),

(2.8 1)

where t(x) is an unknown function and A and B are asymptotic series A =

F An(x)A n=o

z3 =

F B,(x)l-” n=O

This form of expansion was suggested by Olver. If we substitute (2.81) into (2.77), then we obtain a relation of the type A”w + Bw’ = 0, so that all the derivatives of w can be expressed in terms of w and w’. Equating A” and B to zero (w and w’ are linearly independent), we obtain n2A(y2t - q) + 2m5’5 + A.B(5’fg + A” = 0 n2.4(<‘2t - q) + n(2X5 + A(“) + B” = 0.

Substituting the series for A and B into this system and equating the coefficients of the powers of i-l to zero, we obtain a recurrent system of equations, the first of which has the form

n!

t’2w3x) Integrating,

we obtain

= 4(x).

M.V.

178

Fedoryuk

II. Asymptotic

5(x) =(2 2i,xJ4odt )” =(;s(x,,x))41,

(2.82)

where 5(x,, x) is the function of (2.25). The function t(x) is real, t(x) E P(Z), and ax) < a

x < x,;

Methods

in Analysis

179

Let us choose as a solution w(t) of the Airy equation, the Airy-Fock function u(r). Equation (2.77) has a solution y,(x, A) for which (2.81) holds, up to O(XN) (N > 0 arbitrary), as 1-+ +co, x E I. Let us write down the leading term of the asymptotic formula. We have

a4 > 0, x > x0. __ Yak 4 = &

(2.82) can be written in more detail as

u(n2’35(x)) [ 1 + o(n-‘)I

+ X”13B,(x)~‘(12’35(x))[1

+ 0(X2)],

(2.85)

where B,(x) is defined by (2.84). We will analyse this formula. If J213It(x)1 >> 1, then the Airy function can be replaced by its asymptotic form, and we obtain a WKB-approximation. Namely It is essential that function is smooth. In fact, q(x) = (x - x,)q,(x), where ql(x) is a smooth function, ql(x) > 0 for x E 1. Therefore (~~~dr)2=(X-xo)Bq2(X),

where n =O, const. (t’(x))-I/‘.

+ A,“-, = 0,

2&A,,*)

Az,+l(x) = 0,

B,,(x) = 0,

particular,

n = 0, l,...

A,(x) = (2.83)

Further

1 Bn(x) = - 2JW

x A,“-,(t)

+

o@-‘)],

decreasing as 1-+ +cc. For

iA-

Yob, 4 = __ &

114[sin(E.S(xo,x) - t) + 0(1-l)], ‘q(x)‘-

so y, is strongly oscillating. At the turning point itself we have

+ B:: = 0,

1, 2,... and A-, = 0, B-, = 0. In Hence we find that

-1/4(x)e-a3J)[1

J”4

for x 2 x0 + 6, 6 > 0, so that y, is exponentially x 5 x0 - 6,

q2(x)>Oy

independently of the choice of branch of the root, where q2(x) is a smooth function such that c(x) = (x - x0) m. For the functions A,, B,, we obtain a recurrent system of equations ~,/t%BnJ5’5)’

1-M Yo(X,4 = ~

(2.84)

s x0 Jgy$fy

where the c, are constants. Here m > 0 for x E I, and the choice of branch of m is irrelevant. For definiteness, we assume that m > 0, x > x0; Jm = &ml, x < x0. The choice of the turning point x0 as one of the limits of integration is not an accident. For any other choice A,(x) and B,,(x) would not be smooth at x0; the functions constructed by (2.84) are in P(Z). The identity (2.83) has some randomness. If instead of q(x) we take a function of the form (x - x,)q, (x, E),E = A-‘, where q1 is a smooth positive function, then the FAS (2.81) is preserved, but (2.83) will not hold.

y(xo,‘) =

A

c1+ ‘(‘-‘)I

32’3r(2/3)(qI(xo))2/3

This value is Al’6 times larger than at points x 5 x0 - 6. Equation (2.77) can be interpreted, for example, as a stationary solution of monochromatic light waves propagating in a one-dimensional medium. In this case il = w/c, where o is the frequency of the oscillation, c is the velocity of light in a vacuum and q(x) = -n2(x), n(x) being the coefficient of refraction. For x > x0 we have n2(x) < 0, so that the medium is absorbing and the waves decay (shadow zone), for x < x0 the waves propagate without decay (light zone). The boundary point x = x0 is called a caustic and, as was shown above, the flow intensity Iyo12 at this point is significantly larger than at neighbouring points in the light zone. Formula (2.85) is thus necessary only in a neighbourhood of size 12-2/3of the turning point. The second term in (2.85) contains a small multiplier X4’3 and in the first approximation it can be neglected outside Z,. Another method of investigating turning points was suggested earlier by Langer. Make a change of independent variable and function

x = d(5),

Y=&m

where 5 = 5(x) is chosen in accordance with (2.82). Then we obtain the equation

M.V.

180

II. Asymptotic

Fedoryuk

(2.86) which for ,? >> 1 is close to the standard Airy equation (see (2.79)) zic - AZ<2 = 0. Then the equation is reduced to an integral equation and the proximity of solutions of (2.77) to the standard solutions is proved. This method is called Lunger method, or the method of standard equations; it was developed in papers of Langer, A.A. Dorodnitsyn and others. The asymptotic behaviour of the solutions of (2.7) is studied similarly in neighbourhood of a turning point of order n 2 2, for equations of the form y” - [n2q(X) + f(x, P)]y

the the the the

= 0,

the function f may be expanded in an asymptotic series in powers of 1-i as 1+ +oo (see [13]). But here there is one highly nontrivial situation. For convenience we put J. = s-l, then we obtain the equation E2y” - 4(x, E)Y = 0, where q(x, E) = q(x) + O(s’). Note that the terms of order E are absent. If x0 is a simple turning point, then this need not happen and the asymptotic form of the solutions is easy to construct when 9(X,4 = 4&)

+ -q1(4 + E242(X) + ...

But if x,, is a multiple turning point, then the problem becomes much more complicated when qi(x) f- 0. We will clarify this in the example of the equation E2Y” - (x” - &)y = 0,

n 2 2,

(2.87)

but first we discuss the question of what is to be called a turning point for the equation E2yn + &a(X,&)y’ + b(X,&)Y = 0.

+ b(x,e) = 0.

in Analysis

181

E2yn - (x” - E2)y = 0

(2.90) look the same-both have a turning point of order II at x = 0. However, the asymptotic form of the solutions of (2.87) and (2.90), as E --t + 0 in a neighbourhood of x = 0, are totally different. 2” A point x = x0(&) is a turning point of (2.88) if the roots of the characteristic equation (2.89) coincide for x = x0(s). This definition is not invariant: under the change of variable and unknown function mentioned above, turning points are not mapped to turning points. However, this definition relates more to the essence of the matter. There is no essential difference between 1” and 2” only when x0 is a simple turning point (according to definition 1”). Then there is a simple turning point x0(s) (according to definition 2”) close to x0: x0(s) = x,, + O(E). If, for example,.under a change of variable x = d(t) x0 goes to to, then close to to there is a simple turning point C&(E).A simple turning point is stable with respect to small perturbations of the coefficients of the equations. We return to equations (2.87) and (2.90). For E # 0 each of them has n turning points xj(s) = 6, Xj(E) = J$?, respectively, where all the possible values of the roots are taken. The asymptotic form of the solutions of (2.90) can be given in terms of the Bessel functions [13]. Roughly speaking, the xj(s) turn out to be so close to x = 0 that they cannot be distinguished. The situation with (2.87) is different. The asymptotic form of the solutions is constructed as follows. Let D be the disc 1x15 I in the complex x plane, D, the annulus sl”’ 5 1x1 5 r. In D, (usually called the exterior) the WKB-asymptotic formula holds ($3). More precisely, D, can be divided into a number of sectors in each of which there is an FSS with a known asymptotic form. These sectors are pairwise intersecting, which enables us to find the connection formulae between the various FSS. The asymptotic expansion in the domain Di: 1x1 5 2.~“~ (usually called the interior) is constructed differently. Make a change of variable x = El/” t, then (2.87) takes the form

(2.88)

Let pl(x, E), p2(x, E) be the roots of the characteristic equation p2 + a(x,s)p

Methods

(2.89)

Here are two definitions of a turning point. 1” A point x = x,, is a turning point if the roots of the equation p2 + a(x, 0)~ + b(x, 0) = 0

coincide when x = x,,. This is an invariant definition: under a change of variable x = 4(&s), and a change of unknown function y = +(x, E)Z,where 4 and $ are smooth in all their variables, turning points are mapped to turning points. But this definition is too rough. From the point of view of this definition, (2.87) and

and Di will be the disc 1tl 5 2. Remove from Di small (independently of E) neighbourhoods U, of the turning points tk = exp{27cik/n), k = 0, 1,. . , , n - 1. These points are called secondary turning points. We introduce the notation bi = Di\ Uk U,.

Equation (2.91) is standard for (2.87), but its solutions cannot be expressed in terms of known special functions. Nevertheless, the asymptotic form of its FSS as p -+ 0 in Bi, and even in the whole complex t plane with the domains U, removed, can be found using the methods described in 4 3. The structure of the Stokes lines of (2.91) is not ditXcult to explain. In the domains U, the asymptotic form of the solutions can be expressed in terms of the Airy function. After this there is the protracted procedure of linking (or matching) the asymptotic

M.V.

182

II. Asymptotic

Fedoryuk

formulae applicable in the U, with those applicable in 4, and then the asymptotic formulae constructed in the interior domain Di with the asymptotic formulae applicable in the exterior domain 0,. Recall that the intersection D, n Di is nonempty. This is the procedure for constructing the asymptotic form of the solutions of (2.87) in a complete neighbourhood of a turning point x = 0. In general, for (2.88) a standard form (in some interior domain Di) may turn out to be any equation of the form

~‘2 +Q(t)y = 0,

(2.92)

and for small a there are two nearby simple turning points x,(a), x2(a) which, as c1+ 0, merge to form a second order turning point x0. The function q has the form

Here ,U > 0, f and g are smooth functions for small 1x1, 1E 1, f(0, 0) # 0. The standard equation takes the form &2Y"

co?

41 #O.

y” - 12(x2 - a)y = 0.

Here x 1,2(a) = &&, so the turning points are real for CI> 0 and complex for CI< 0. The asymptotic form of the solutions of (2.92) is given in terms of Weber’s function; if the turning point is complex, then q is required to be analytic with respect to (x, a). Problems such as this arise, for example, in quantum mechanics. Consider the one-dimensionalSchdinger equation

-$

+ (E - U(x))l) = 0.

Let the potential U(x) tend to zero as x + &co, and have a maximum point x0, U/(x,,) < 0. Then for values of the energy E close to U(x,), there are two nearby turning points x,,,(E), real for E < U(x,), complex for E > U(x,). The same situation holds when the potential has a minimum point x0, U”(x,) > 0. The case when there are a turning point and a first or second order pole of q close together has also been studied: -

X ‘-2f(X,

E) +

i;p2x-2

+

E2a

y,

and its solution is expressed in terms of the Bessel function of order v, where p2v2 - 1 = 4~. All the results above are given in the reference text [13], where there is a detailed bibliography.

Ey’ =

y = 0.

Ey

1

A(X,

(2.93)

E)y.

Its characteristic equation is 1(x,

p; E) =

det(A(x,

E) -

pl)

=

0.

(2.94)

Letp,(x,s),... , pn(x, E)be the roots of this equation. As for second order equations there are two definitions of a turning point. 1” A point x = x,, is a turning point of (2.93) if A(x,, 0) has a multiple eigenvalue. 2” A point x0 = x0(e) is a turning point of (2.93) if A(x,,(E), E) has a multiple eigevnvalue. The definitions of turning point for an n-th order equation ly s E”y(“)

+

f

&“-iqj(x,E)y(“-j)

= 0

(2.95)

j=l

are introduced in exactly the same way. The characteristic equation is l(X,

pi &) c p” + jil qj(x, &)p”-j = 0.

(2.96)

A turning point x0 (definition 1”) of (2.95) is called simple if the following conditions hold: 1. The equation l(x,, p; 0) has one double root p,, and all the remaining roots are simple. 2. L(-%,P,;o) z 0. Letp,(x,,,O) = p2(x,,0) = po, then thevaluesp,(x,,O) = pz,...,pn(xo,O) = p.” are distinct and p: # po. For small lx - x01 and 1~1,the symbol 1 takes the form k

Pi E) = (P2 - 24% E)P + b(x, E))(P - P&, E)). . .(P - pn(x, &)I,

where pj(X, E), j 2 3 is a function of class C”, pj(xo, 0) = p:. The roots pl, 2 have the form P1,2(X0>4

E2yrr

=

X2 >

Let x0 = 0, then q1 is a smooth function for small IxI,IEI, ql(O,O) # 0 and x~,~(M) are smooth functions of ,,& The standard example is Weber’sequations y” + 12(x2 - cr)y = 0,

183

5.3. Turning Points of Equations of n-th Order and Systems. Consider a system

y” - 12q(x,a)y = 0

x2(4)ql(x?

in Analysis

of n equations

where Q(t) is an arbitrary polynomial. Therefore, even for second order equations, the problem of constructing asymptotic formulae for FSS in a complete neighbourhood of a multiple turning point (in the sense of 1”) has not yet been investigated at all. We will give a short list of other results on the asymptotic form of a solution of (2.77). The case when (2.77) contains an additional parameter CIhas been studied:

4(x, 4 = (x - x1(4)(x -

Methods

wo,

= +>E,

0) = 0,

&/D(x,E),

D&J, 0) # 0.

D = a2 - b,

M.V.

184

II. Asymptotic

Fedoryuk

If the coefficients of (2.95) are analytic for small 1x1, 1~1,then the point x = x0 is a second order branch point for the roots ~i,~(x,O). It is a little more complicated to define a simple turning point of (2.93). With no loss of generality we can assume that A(x,,O) has been reduced to Jordan canonical form. Then a turning point x0 is called simple if conditions 1 and 2 hold, and

Methods

in Analysis

185

reduces (2.93) to the form EZ’ = B,(x)z + &B(x,E)z,

(2.97)

where B(x, E) is a smooth matrix function and B,(x) is a block diagonal matrix B,(x) = diag(W,+)

- 4x)&-,),

0 1 B(x) =(~ D(x) 0 > . We will seek a solution of (2.97) in the form Consider the simplest example-the

equation

,&”

- xy = 0

z = w(&-2’3<(x))f(x, E) + &1’3W’(E-2’3g(X))g(X,&),

and the vector functions f and g are

where c(x) is an unknown function, asymptotic series

This is equivalent to the system

f(%4 = k~or,cx,~k,

Here x = 0 is a simple turning point, since condition 1 is satisfied, 1, = - 1, and for x = 0 the matrix of the system is a second order Jordan block. If x0 is a simple turning point of (2.95) or (2.93), then for small E there is a simple turning point x0(s) in the sense of definition 2”, x0(s) = x0 + O(E). Let x0 = 0 be a simple turning point of (2.93), on a segment with no other turning points, and let the difference E) - PA E)),

dx, 4 = kzoSk(Xbk.

Substituting (2.98) into (2.97) and equating the coeflicients of w and w’ to zero, we obtain the system of equations

l(x,p;&) = p2 - x.

Wpjk

(2.98)

j 2 3,

not change sign for x E I and for all k # j. Then (2.93) has solutions y,(x, E),. . . , y,(x, E) of the form (2.47). The asymptotic form of the missing solutions y,(x, E), y,(x, E) is given in terms of the solutions of Airy’s equation (2.80). Let D(x, E) be real, D:(O, 0) > 0 for definiteness. There is a matrix function T(x) E P(Z), which for x E I reduces the matrix A,(x) = A(x,O) to block diagonal form:

5’5s + Ef’ = (Bo + @f,

l’f + eg’ = (B. + cB)g.

The matrix function B(x, E) can be expanded in an asymptotic series B(x,E) = B,(x) + &B2(x) + ...

Substituting this expansion, and the expansions off and g, into the equations, and equating the coefficients of powers of E we obtain a recurrent sequence of equations. The first of them is Bofo = t’tso,

Bose = t’J,v

so that %fo

Consequently,

= cY2tfo,

&ho

= C2tso.

t’25 is an eigenvalue of B:(x), which is equal to B,2(4 =

(W4~2,

(44

-

44L2)2).

Therefore 44

The transformation

= diag(p,(x,O),...,p,(x,O)).

5’2(4t(4

= w.

Hence we tind (2.99)

M.V.

186

II. Asymptotic

Fedoryuk

and t(x) E Cm(I) (see (2.81)). The vector functions fe(x), g,,(x) are eigenvectors of B:(x) and therefore have the form

D(x) + -e @2(X) M4 = al(x)5,(x)el 5’(x) 2T so(x) = alWl

+ ~2(4e2y

e, = (1,0 ,..., O)r,

(2.100)

e2 = (O,l,O ,..., 0)r.

y$ -;oy;) = (; 1;:;) ( The homogeneous system has a non-trivial solution, so in order that the inhomogeneous system be solvable it is necessary that the right hand side be orthogonal to the solutions of the adjoint homogeneous system. A basis in the solution space of the latter is formed by the 2n-vectors

&fly(n)+ &Xu,-l(X)y(“-l)

reduces to the system of equations

Here A = b,, + b,,,

+ &2Xu,-2(X)y(“-2) + ...

(2.99)-(2.101) completely determine the leading term in the asymptotic expansion (2.98). For (2.95) and, in particular, for equations of even order 2n in the selfadjoint form (2.17), the formulae take a significantly simpler form [12]. The investigation of the asymptotic form of the solutions in a neighbourhood of a multiple turning point is a complicated problem, even for second order equations. For n-th order equations and systems only individual special cases have been examined.

+ x2ao(x)]y = 0,

WC”)- tw = 0.

If all the roots of the characteristic equation are purely imaginary, then an asymptotic form can be constructed using the canonical operator of V.P. Maslov. 2. The system (2.93) has the form: e2yr’ = iA(x)y

(2.102)

where A(x) is a Hermitian matrix for all x E I = [-a, a]. The case when A(x) has smooth eigenvalues pi(x), . . . , p,(x), and two of them coincide for x = 0, has been studied. The second order system &zy” = A(x)y

(2.103)

where ,4(x) is a Hermitian positive definite matrix for x E I, reduces to the form (2.102). Systems of the form (2.103) arise in the problems of electrodynamics, quantum mechanics, etc. The method of construction of the asymptotic form of the solutions is developed by V.V. Kucherenko [13]. Let x = 0 be a turning point of (2.93) and let A(x,E) be analytic with respect to all its variables for small 1x1,(~1.Let the eigenvalues of A(O,O) decompose into two groups, py,. . . ,pc, and J$‘+~, . . . ,p,“, with no common elements. Then using the transformation y = 7(x, E)Z (2.93) reduces to block diagonal form

B = Db,, + b,,,

and b,(x) are the elements of B,(x). The system of equations for cc,(x), IZ~(X) can be integrated:

187

where the functions Uj(X) are analytic for small 1x1. Here x = 0 is a multiple turning point. The asymptotic form of the solutions is given in terms of the solutions of the generalized Airy equation

(eT, e~l63, (6, P/S%?), condition

in Analysis

1. Equation (2.95) has the form

... + &xa,(x)y’ + c-x

The unknown functions c(~(x) and a2(x) are defined, as in 9 2, from the system of equations for the second approximation

The solvability

Methods

0 B2 (x5

4>

w + &Nc&&)w,

in a domain D: 1x1 5 r, 1.~15 E,,, for sufficiently small r, E,,. The matrix T(x, E) is holomorphic and non-degenerate in D, CN(x, E) = O(1) as E+ 0 for any N. This result was obtained by Sibuya. Therefore it is enough to study the turning point problem for systems such that A(0, 0) has a unique eigenvalue, which we can take to be equal to zero. Such a system, as was shown by Wasow, can be reduced to a normal form, but little is known about the asymptotic form of the solutions of the reduced system. 5.4. Connection Formulae for the Second Order PainlevC Equation. An equa-

8 tion of the form w” = R(z, w, w’)

(2.104)

where R is a rational function of z, w, w’ is called a Puinlev6 equation, if it does

M.V.

188

Fedoryuk

II. Asymptotic

wn = zw +

2w3

d2(k) = +og(l

(PII)

u, - 6u2u, + u,,, = 0,

References*

solution u(t, x) =

(3z)-“3

z = x(3t)-l’3,

w(z),

where w(z) is a solution of the second Painlevt equation (2.105)

wn = zw + 2w3

This is a particular case of (PII), since a = 0. We consider (2.105) on the real line and study the asymptotic form of its solutions for x --r fco. If we neglect the nonlinear term 2w3, then we obtain the Airy equation (44) which has the solution &‘@) exponentially decreasing for x + +co. Equation (2.105) also has a one-parameter family of solutions exponentially decreasing as x + + co: (2.106)

w(x) - k&i(x) Here k E R is arbitrary. If 0 < k2 < 1, then this solution asymptotic behaviour as x + -co w(x) = dlx1-1’4sin

(

+ o(lxl-7’4log

21x)-

-k’).

Similar rzsul; have been obtained for the third Painleve equation associated with the generalized Sine-Gordon equation.

In recent years interest in the Painleve equations has grown hugely, since it was discovered that they are connected with the equations of mathematical physics which can be integrated by the methods of the inverse problem of scattering theory. For example, the famous Kortweg-de Vries equation

has the similarity

189

The function c(k) is unknown; there is a conjecture that c = a / :\ argr( 1 - id2 ).

(PI) + a.

in Analysis

The additional logarithmic shift of the phase (see (2.107)) is the influence of the nonlinear term 2w3. It is clear that the constants d and c are functions of k. The problem of connection formulae for (2.105) is the finding of these functions. Using the connection between equations (2.105) and Kd V, and also the method of the inverse problem of scattering theory, it has been proved that

not have moving branch points [ 181. This means: there are a finite number of fixed points a,,..., a, in the complex plane, and only they may be branch points for a solution of (2.104). All the remaining singular points of any solution w(z) are poles or essential singular points. The classification of these equations was carried out at the beginning of this century by Painleve and Gambier. It turned out that there are exactly 50 types, which cannot be transformed to each other by changes of independent variable and unknown function. Of these exactly 6 are irreducible, that is, their solutions cannot be expressed in terms of elementary or special functions. The first two of them take the form w’! = 6w2 + z;

Methods

/3 - 4d loglxl 3’2 (’ ‘>

1x1)

has the following

+ c

’

The reader may familiarise himself with the notion of asymptotic series and their properties in any of the books [8], [lo], [ll], [12], [16], [30], [35]. [37]. Th e compact and clearly written book [lo] is, perhaps, more suitable for a first reading as an introduction to asymptotic methods. The reader will find more detail on the methods of Laplace, stationary phase and steepest descent in the books [7], [8], [lo], [11], [12], [26], [35]. In [7], [26] these methods are applied, mainly to the calculation of asymptotic formulae for special functions. The three volume work [27] is the most complete reference reference text for these methods in the one-dimensional case. The Laplace method in the many-dimensional case and the method of stationary phase may be found in [S], [12], [35], the many-dimensional steepest descent method in [ 121. For many-dimensional integrals with degenerate critical points and the structure of functions close to these points, see [l], [2]. For the asymptotic form of the Fourier I-transform, see [23], [24]. The method of the Maslov canonical operator is explained in [23], [24]. For asymptotic estimates of sums and series, see [S], [I 11, [12]. The most complete guides to the analytic theory ofdifferential equations are the books 1171, [20]; see also [6], [36]. The WKB-method for second order equations can be found in [13], [14], [16], [26], 1311, [34], [36]; there is a detailed bibliography in [13]. We note that [16], [14] are written at the physical level of rigour. The asymptotic behaviour of solutions of higher order equations and systems for large values of the argument or parameter is given in [6], [13], [33], [36], [37]. 1.

> (2.107)

The leading term of the asymptotic formula for the Airy function as x + -cc is equal to di(x) = 71-1’2lxI- 1’4sin 21x13/2/3 + $ + 0(1x1-‘I”). >

i / !

Arnol’d, V.I., Varchenko, A.N., Gusein-Zade, S.M.: Singularities of differentiable Classification of critical points, caustics and wave fronts. Moscow: Nauka, transl.: Boston: Birkhauser, 1985. Zbl. 513.58001, Zbl. 554.58001

mappings. I. 1982. English

*For the convenience of the reader, references to reviews in Zentralblatt ftir Mathematik (Zbl.), compiled using the MATH database, and Jahrbuch iiber die Fortschritte der Mathematik (Jrb.) have, as far as possible, been included in this bibliography.

190

M.V.

Arnol’d, V.I., Varchenko, A.N., Gusein-Zade, SM.: Singularities of differentiable mappings. II Monodromy and asymptotics of integrals. Moscow: Nauka, 1984. Zbl. 545.58001 3. Bakhvalov, N.S., Panasenko, G.P.: Averaging of processes in periodic media. Mathematical problems of the mechanics of composite materials. Moscow: Nauka, 1984. Zbl. 607.73009 4. Birkhoff, G.D.: Quantum mechanics and asymptotic series. Bull. Am. Math. Sot. 39,681-700 (1933). Jrb. 59, 1530 5. Clarkson, P.A., McLeod, J.B.: A connection formula for the second Painleve transcendent. Lect. Notes Math. 964, 135-142 (1982). Zbl. 502.34007 6. Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill, 1955. Zbl. 64,330 I. Copson, E.T.: Asymptotic expansions. Cambridge: University Press, 1965. Zbl. 123,260 8. de Bruijn, N.G.: Asymptotic methods in analysis. Amsterdam: North-Holland, 1958 Zbl. 82,42 (2nd ed. 1975,3rd ed., New York: Dover 1981) 9. Debye, P., Semikonvergente Entwicklungen fur die Zylinder-Funktionen und ihre Ausdehnung ins Komplexe. Mtinchen, Berlin, 40, No. 5, 1-29 (1910) 10. Erdelyi, A.: Asymptotic expansions. New York: Dover Publ., 1956. Zbl. 70,290 11. Evgrafov, M.A.: Asymptotic estimates and entire functions. 3rd. ed. Moscow: Nauka, 1979. Zbl. 447.30016 12. Fedoryuk, M.V.: Saddle-point method. Moscow: Nauka, 1977 13. Fedoryuk, M.V.: Asymptotic methods for linear ordinary differential equations. Moscow: Nauka, 1983. Zbl. 538.34001 13a. Fedoryuk, M.V.: Asymptotics: Integrals and Series. Moscow: Nauka, 1987 14. Froman, N., Froman, P.O.: JWKB approximation: contributions to the theory. Amsterdam: North-Holland, 1965. Zbl. 129,419 15. Green, G.: On the motion of waves in a variable canal of small depth and width. Trans. Camb. Phil. Sot. 6,457-462 (1837) 16. Heading, J.: An introduction to phase-integral methods. London: Methuen, New York: Wiley, 1962. Zbl. 115,71 17. Hille, E.: Ordinary differential equations in the complex domain. New York: Wiley, 1976. Zbl. 343.34007 18. Ince, E.L.: Ordinary differential equations. London: 1927. Longmans, Green & Co. Jrb. 53, 399 19. Kelvin Lord: On the waves produced by a single impulse in water of any depth, or in a dispersive medium. Phil. Mag. 5,252-255 (1887) 20. Langer, R.E.: The asymptotic solutions of certain linear ordinary differential equations of the second order. Trans Am. Math. Sot. 36,90-106 (1934). Zbl. 8,312 21. Laplace, Le Marquis de: Theorie analytique des probabilites. Paris: Mme Ve Courtier 1812. 22. Liouville, J.: Sur le developpement des fonctions ou partie de fonction en series. J. Math. Pures Appl. I, No. 2, 16-36 (1837) 23. Maslov, V.P.: Theory of perturbations and asymptotic methods. Moscow: Izdat. Most. Gos. Univ., 1965. French transl.: Paris: Gauthier-Villars 1972. Zbl. 247.47010 24. Maslov, V.P., Fedoryuk, M.V. Quasi classical approximation for the equations of quantum mechanics. Moscow: Nauka, 1976. Zbl. 449.58002 English transl.: Dordrecht Reidel 1981 25. Naimark, M.A.: Linear differential operators. (2nd ed. Moscow: Nauka, 1969. Zbl. 193, 41) New York: Ungar Vol. I (1967. Zbl. 219.34001 ), II (1968 Zbl. 227.34020) (1st ed. 1954 Zbl. 57,71) 26. Olver, F.W.J.: Introduction to asymptotics and special functions. New York-London: Academic Press, 1974. Zbl. 308.41023 27. RiekstinS, E.J.: Asymptotic expansions of integrals. Vols. 1, 2, 3. Riga: Zinatne 1974, 1977, 1981. Vol. 1 (Zbl. 292.41021), Vol. 2 (Zbl. 358.41007), Vol. 3 (Zbl. 483.41001) 28. Riemann, B.: Sullo svolgimento de1 quoziente di due serie ipergeometriche in frazione continua infinita, 1863. In: Gesammelte Werke, Leipzig: Teubner 1876. Jrb. 8,231 2.

II. Asymptotic

Fedoryuk 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

Methods

in Analysis

191

Riemann, B.: Uber die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe. Abhandl. d. Konigl. Gesellsch. der Wiss., 1867 Segur, H., Ablowitz, M.J.: Asymptotic solution of nonlinear evolution equations and a Painleve transcendent. Physica D, l-2, 105-184 (1981) Sibuya, Y.: Global theory of a second order linear differential equation with a polynomial coefficient. Amsterdam: North-Holland, 1975. Zbl. 322.34006 Stokes, G.G.: On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Camb. Phil. Sot. 10 (1857) Tamarkin, Ya.D.: On some general questions in the theory of ordinary linear differential equations and on the expansion of arbitrary functions in series. Petrograd, 1917. Jrb. 47,944 Titchmarsh, E.C.: Eigenfunction expansions associated with second-order differential equations. Oxford: Clarendon Press, Vol. 1, 1946; Vol. 2, 1958. Zbl. 61, 135. Zbl. 57,276 Vainberg, B.R.: Asymptotic methods in the equations of mathematical physics. Moscow: Izdat. Most. Gos. Univ. 1982. Zbl. 518.35002 Wasow, W.: Asymptotic expansions for ordinary differential equations. New York: WileyInterscience, 1965. Zbl. 133, 353. English transl.: Moscow: Mir 1968 Wasow, W.: Linear turning point theory. New York: Springer-Verlag, 1985. Zbl. 558.34049 Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge: University Press, 1927. Jrb. 53, 180

III.

Integral Transforms M.V. Fedoryuk Translated from the Russian by D. Newton

Contents 5 1. Introduction ......................................... .. .. . . 5 2. The Fourier Transform ................................ 2.1. The Inversion and Commutation Formulae ........... .. . . .. . . 2.2. The Spaces g(R”), Y(K) ........................... .. . . 2.3. The Fourier Transform and Generalized Functions ..... 2.4. The Cauchy Problem for Partial Differential Equations . . . . . .. . . 2.5. Fundamental Solutions of Partial Differential Equations 2.6. Integral Equations ................................ .. 2.7. The Radon Transform ............................. .. 0 3. The Laplace and Mellin Transforms ..................... 3.1. The Laplace Transform ............................ .. ...... 3.2. The Laplace Transform and Evolution Equations .. 3.3. The Mellin Transform ............................. .................................. .. 9 4. The Bessel Transform .. 4.1. The Hankel Transform ............................. .... .. 4.2. Other Transforms Connected with Bessel Functions .. 6 5. Other Integral Transforms. ............................. References ...............................................

193 196 196 200 202 204 207 213 214 216 216 220 223 225 225 228 229 231

5 1. Introduction Consider the linear integral operator rar

w-)(P) = I

J-00

mhW(4~x

(1.1)

. 194

M.V.

Fedoryuk

III.

If the integral converges, then it defines a function (T’)(p), which is called an integral transform of f(x). The function K(p, x) is called the kernel of the integral

Integral

Transforms

Applying the inverse transform we find a solution of (1.5):

[ 1

f(x) = T-1

transform.

We will consider integral transforms, not from the point of view of functional analysis, that is, not as a mapping from one function space to another, but from the point of view of the applications of integral transforms to the solution of various kinds of equations-ordinary differential, partial differential, integral, etc. The integral transforms of mathematical physics are not arbitrary linear integral operators. All of them have two properties: they have an inversion formula and a commutation formula. Namely, the inverse operator is also a linear integral operator (T-‘g)(x)

‘m &> Mp) 4 (1.2) s -m with a known kernel i?(x, p). A formula of type (1.2) is called an inversion formula. The kernels K(p, x) and I?(x, p) are connected by the relation m

(Tg)(p)

P(L(P))

.

It is by no means necessary in the commutation formula (1.4) that z be the operator of multiplication by a function of p. It is only important that z be ‘simpler’ than L, for example, so that we can explicitly find a solution of &I(P))

= k(p).

For the majority of the integral transforms of mathematical linear differential operator:

physics L is a

(1.6)

=

&x, P)K(P, y) dp = 6(x - Y),

(TLf)(P)

T-lLT

LT

= dx),

(f” = Tf, s” = Td,

g”(P) f(P) = PO)’

K(P,x)

= &)K(p,x), = QP)&,P).

Here LT(x, d/dx)y = jam (- l)j(d/dX)j(aj(X)y).

In fact, integrating by parts and taking account of the vanishing integrated out permutations, we obtain from (1.4) that

= Z(p)&

which is easily solved

)I

L X>& R&P) ( >

formula can be written in the

(1.5)

where P(L) is a given function of L. We will assume that all the mathematical operations we do are legitimate. Applying T to both sides of (1.5) we obtain = g”(P)

x>$

(1.7)

where I is the identity operator, so an integral transform may be interpreted as a similarity transform taking L to the operator of multiplication by z(p). Integral transforms are among the most powerful tools of mathematical physics, enabling us to find exact solutions of a huge number of concrete problems. Underlying all these applications of integral transforms are the commutation and inversion formulae. The general scheme of the integral transform method is as follows. Consider an equation W)f(x)

(x

(1.4)

= QP)(Tf)(P).

Here z(p) is some function of p. The commutation form

The presence of the commutation formula imposes severe limitations on the kernels K and I?. Namely, K is an eigenfunction of the adjoint operator LT, and K is an eigenfunction of L:

(1.3)

s -m where 6 is the Dirac delta-function (see subsection 2.4). For a given integral transform T there is a linear operator L, acting on functions of x, such that the commutation formula holds

W(P))f(P)

195

=sa, -a,

Thus j

J-(x)Z(P)K(P,

LT(x, d/dxMp>

of all the

x) dx.

4 = Z(P)K(P,

x).

The second formula in (1.7) is proved in a similar manner. The inversion formula (1.2) is not that different to the way in which a function f(x) is expanded as a Fourier integral relative to the eigenvalues of L:

f(x) = m @, I) s -cc

dp,

M.V.

196

III.

Fedoryuk

thus integral transforms are closely related to the spectral decomposition of linear differential operators. There is a very small collection of differential operators whose eigenvalues and eigenfunctions can be calculated exactly. Hence there is a small collection of integral transforms. It is natural to classify integral transforms according to the types of the operators L. Unfortunately, we have not encountered such a classification in the literature. There is no special terminology for the operator L; we will call it generating. Integral transforms can be divided into the following classes. 1. The generating operator L is of first order. This class includes the Fourier, Laplace, Mellin, Euler and Weber transforms. 2. The generating operator L is of second order. This class includes the Bessel, Hankel, Meijer, Kontorovich-Lebedev, Meller-Fock and Laguerre transforms. It is also natural to include the Sommerfeld transform in this class. 3. Transforms of convolution type. This class includes the Hilbert, Stieltjes, Weierstrasse and Watson transforms. This classification of integral transforms is incomplete and somewhat arbitrary, and is given for lack of anything better. All the above integral transforms are one-dimensional. The class of manydimensional integral transforms is poorer, and of those given above only the Fourier transform admits a worthwhile generalization to the many-variable case. The majority of integral transforms are connected in some way or other with the Fourier transform. The Fourier transform has a deep and far-reaching generalization: the Fourier transform on topological groups. This notion includes almost all the integral transforms considered in this part. This division of mathematics, harmonic analysis on groups, is not concerned with the subject matter of this part and will be explained in other volumes of this series. Our fundamental aim here is to acquaint the reader with the basic integral transforms of mathematical physics and to show these transforms in action. In the solution of concrete problems all the calculations will be carried out formally and only in individual cases will the question of justification be discussed. There is a vast mathematical literature devoted to integral transforms. There are many important and delicate results concerned with such questions as the analytic properties of various transforms of various classes of functions, and the conditions under which the inversion, commutation and other formulae are valid. In this part, for the reasons given above, we will restrict ourselves to rough sufficient conditions on the functions involved; but they are sufficient for the majority of concrete problems in mathematical physics.

Integral

(Ff )(5) =

Transforms

m e-‘“
2.1. The Inversion

and Commutation

Formulae.

The Fourier transform

(1.8)

(F-‘d(x) =&sy

The inversion formula is:

co

eix5g(<) d&

The kernel K(&x) = e-iX5 of the Fourier transform is an eigenfunction differential operator D = f $,

of the

tation formula (see (1.3), (1.7)) FDf = (Ff.

(1.10)

Consequently, if P(D) = ~jm,O ajDi is a polynomial

in D with constant coefficients

aj, then FP(D)f

(1.11)

= J’(S)Ff.

Formula (1.10) is one of the most important properties of the Fourier transform: differentiation transforms to multiplication by 5. There is a dual formula: D, = - f $.

Fxf = D
Consequently, if P(x) is a polynomial, FP(x)f

then = P(D{)Ff.

(1.12)

Both (1.11) and (1.12) are easily proved using integration by parts (under the assumption that f is a smooth function with compact support). The Fourier transform exists if JZm 1f(x)1 dx < co. In addition, there is the Riemann-Lebesgue

lemma. limlel -oo (Ff) (5) = 0.

The Fourier transform of a function f(x) of n variables is defined similarly (Ff )(5) =

where 5 = (5 l,. . . , &,), (x, 5) = ‘& The inversion formula is: (F-‘g)(x)

e-i(x,r)f (x) dx

(1.13)

s R" xjtj, dx = dx, . . . dx,.

= (2n)-”

ei(“*r’g(<)d& s R"

Let us introduce some standard notations: CI= (al,. negative integer components (a multi-index),

D” = 0;‘. . . 02,

xa = x=’1 ..-x, a” 2

(1.14)

. , cl,,) is a vector with non-

D = (Dl,...,Q,), of a

(1.9)

since DK = - (K, and hence there is a commu-

5 2. The Fourier Transform function f(x) is the function

197

a! = cc,!...a n..1

198

M.V.

III.

Fedoryuk

Integral

Transforms

199

The Fourier cosine transform

Let m

(F,f )(t) =

P(D) = 1 a,Da Ial=o

be a polynomial in the operators Dj with constant coefficients a,. Then the commutation formula (1.11) holds. If P(x) is a polynomial in the variables la ld xi,. . . ,x,, then (1.12) holds, where D, = - -, . . . ,r . We also mention 1 at, ) ( i at, the formulae Ff(x - a) = ei@qS)Ff(x), w?f)

W-4)(5)

Js

1

m cosx5f(x)dx, 0

and the Fourier sine transform

are also used. The inversion formulae are symmetric and take the form

= (- l)“(Ffb))(S),

(4 = f( - 4.

(1.15)

In the international literature there is a total lack of agreement both on the definition of the Fourier transform and on its notation. As a kernel, instead of exp{ - i(x, t)}, the functions exp(i(x, t)}, (27~)“‘~ exp{ + i(x, <)}, etc are used. The pair consisting of a function and its Fourier transform are denoted in various ways, for example,

(fmfb3)~ (f(x), F(P)),(f(x), g(4), (f(x), (Fx+,fM), etc. -We will denote the Fourier transform of a function f(x) either by (F’)(t)

or

by f(5).

The Fourier transform is the most frequently used of all the integral transforms. It was introduced by Fourier in 1822, and was named in his honour by his pupils. The Fourier transform has the deepest mathematical and physical significance. The pair of formulae (1.8) and (1.9) mean that ‘every’ (for example, of class L’(R)) function can be represented as a (continual) linear combination of the simplest functions, the exponentials e’@. These functions are also called plane wave, or harmonic, functions. Let us explain these terms in the case n = 1. Consider the function h(t,x) = cos(ot - xc) = Ree-i”‘+i”5, where t denotes time and x the coordinate. The level lines of this function are the straight lines ot - x4 = const; which move with velocity c = o/t and hence h is called a (travelling) plane wave. A fixed point x performs the simplest oscillations-harmonic with frequency o, and hence his called harmonic. This terminology is retained even when the dependence on time is given ‘separately’. The pair of formulae (1.8) and (1.9) mean that every oscillation is a superposition of simple harmonic oscillations. All natural scientists .have long been used to this fact, but we should not underestimate its role in the development of the natural sciences. For the contemporaries of Fourier the possibility of representing ‘any’ function as a superposition of harmonics was not only unexpected but also incomprehensible, and many important mathematicians simply did not believe it. The full admission of Fourier series and Fourier integrals into mathematics only occurred almost ten years afterwards, with the work of Dirichlet.

(F,f)(OsinbdL

These transforms are convenient when f(x) is even or odd. If f(x) is even then f, if f (x) is odd, then Ff = - i&F,f. In probability theory the Fourier-Stieltjes transform is widely used; in the one-dimensional case it takes the form Ff = &F,

f(t)

m emiXtdF(x). s -CC

=

Here F(x) is a probability distribution function, which is monotonely increasing with F(-oo) = 0, F( +co) = 1. If the probability distribution has a density, that is, dF(x) = p(x)dx, then the Fourier-Stieltjes transform is the usual Fourier transform of p(x). Let us calculate the Fourier transform in the simplest cases. Example 1. Let R(5) be a rational function having no real poles and such that the integral (F-‘R)( x ) converges. Applying Jordan’s lemma and the residue theorem to (1.9), we obtain for x > 0 (F-lb!)(x)

= i c res (R(5)e’““),

5=&l i where the sum is taken over all the poles of R(t) which lie in the upper half-plane 1 Im 4 > 0. If x < 0, then there is a similar formula in which the sum is taken over I all the poles lying in the half-plane Im 5 < 0, and the right hand side of the : formula is multiplied by - 1. In particular, if Re a > 0, x > 0, then a, e ix< -d5:

= ie?“.

’ s -mx +a Example 2. Let us calculate the inverse Fourier transform of the function e-ar2l2 , a > 0. Making the change of variable 5 - ix/a = 4, we obtain

M.V.

200

III.

Fedoryuk

e-x2/20 .

Using analytic continuation we can show that this formula is valid for Re a > 0 (here Re & > 0). Using (1.10) we can calculate F-‘(P(?j)e-““‘2), where P(5) is a polynomial.

--$(A& r))} = (271))““(det A)-1’2 exp{ -$(A-lx,

In fact, making the change of variable < = t + iA-lx,

201

for r # 0. Hence it follows that I-?(t)1 5 C(l + ItI)-‘, further integrations by parts we obtain

Example 3. Let A be a real symmetric positive definite matrix of order n. Then

F-‘{exp(

Transforms

follows (unless otherwise stated) we shall assume that f E Y. All the formulae given in this section are valid for f E 9'. The Fourier transform has the following important property: the more smooth is f(x) (for definiteness, compactly supported), the more rapidly its Fourier transform decreases as 1
where y is the line Im [ = - xfx a in the complex [ plane. It is not difficult to show that the integral over y is equal to the integral over the real axis, that is, to the well-known Poisson integral. Consequently, F-1 (e-arY2) = 1 24G

Integral

I&3 5 Cd1 + lWN>

x)}.

5 E R. Iff E C!(R), then by

5 E R.

In exactly the same way we can prove that, if f(x) E Y(R!J, then f(t) E The Fourier transformation maps Y(Rc) one-to-one onto Y(R;). We will study the properties of the Fourier transform of a function f E 9(R”). The function f(t) can be analytically continued to the whole ndimensional complex space C” as an entire function of n complex variables i 1,...,i,(i=5+ir,5~Rn,~~R”). In fact the integral

we obtain the integral

Y(R;).

which is taken over the n-dimensional plane [ = [ + iA-‘x, [ E R”, in the ndimensional complex space C”. As in example 2, we can show that this integral is equal to the integral over the real space R”, that is,

m

s R” converges for all i E C”, uniformly on compact sets, and the kernel e-i(x,c) is an entire function of i for any fixed x E R”. Let f(x) = 0 for 1x1> R. Integration by parts leads to the estimate: for any N there is a constant C, such that

Since A is real symmetric, there is an orthogonal matrix T, det T = 1, reducing A to diagonal form: T-‘AT = diag(l,, . . . , 1,). Making the change [ = TV, we obtain the integral

I.&I 5 Cdl + IW”exp{RIImU),

for all x E R”. Obviously 3(R”) c Y(R”). Both spaces are linear, topological and, furthermore, are rings. Using corresponding norms we can turn them into countably normed spaces. In all that

i E C”.

(1.17)

Therefore &‘) is an entire function of the first order of growth and finite type. The converse-the Paley-Wiener theorem-is also true [16]. For the Fourier transform there is a remarkable formula-Parseual’s equality

which is a product of n Poisson integrals. 2.2. The Spaces B(R”), Y(R”). We will discuss two function spaces. 1. C;(R”) (also denoted 9(R”) or X) is the set of all infinitely differentiable functions with compact support. 2. Y(R”) = 9’ is the Schwarz space. It elements are infinitely differentiable functions, which, together with their derivatives, decrease more rapidly than any power of 1x1as [xl+ co. Namely, for any multi-indices a, /I there is a constant CEa such that (1.16) (1 + Ixl)‘“‘I~pf(x)I 5 cap

emiCX,c)f(x) dx

=

(1.18) f(x)g(x) dx = (2x)-” &%(tl) d<. ss/ ; Here and everywhere later, the sign J means that the integral is taken over the whole space R”. It follows from Parseval’s equality that the normalized Fourier operator (2r~)“‘~F ij a unitary operator from L2(R”) to L’(R”). Consider the convolution f *g of two functions f and g: r

(f *g)(x) =

s

f(x - y)g(y)dy.

The Fourier transform takes convolution

(1.19)

to product

F(f *g) = Ff . Fg.

(1.20)

202

M.V.

Fedoryuk

III.

Integral

Transforms

203

In addition

The formal derivation is this: making a change of variables x - y = 2 and jj = y in the double integral on the left in (1.20), we obtain

6(x) =0, x#0; s

G(x)dx = 1,

F(f*g)

=

f(x - y)g(y)e-i(“*odxdy ss

f(n)g(~)e-i(r,r)e-i(~.i) ss = Ff . Fg. =

so 6(x) can be interpreted, for example, as the density of a unit mass concentrated at the point x = 0. There are, perhaps, only two basic formulae connected with the delta-function

dg dj

s

2.3. The Fourier Transform and Generalized Functions. We give a brief survey of the theory of generalized functions; for more details see [7], [ 171, [24]. Suppose we are given a function space .G?which is linear and topological. The elements 4 E !8 will be called fundamental or test functions. As !??Iwe will take either 9(R”) or Y(R”). A continuous linear functional f on 2 is called a generalized function over g. The set of all generalized functions is the linear topological space 8 conjugate to g. The value of a functionalf on an element 4 E 6? is denoted either by (f, 4) or by

dx. sf(xM(4

ei@,od[ = (27~~“6(x)

The first of these follows from the definition, and the second is a corollary, even the basic content (in modern treatments), of the inversion formula. Let n = 1; put g(5) = f(t) in (1.9), then we obtain

ss eiXr dt

(1.21)

= jf(y)(&

The latter notation is widely used in theoretical physics. Two generalized functions fi and f2 are equal, by definition, if and only if (fi, 4) = (f2, 4) for all 4 E g. If a function f(x) is continuous and bounded on R”, or even, grows no more quickly than some power of 1x1 as 1xl + co, then associated with it is the functional defined by (1.21). We identify the function f(x) and the functional f, so that Y c Y’ and 9 c g’, that is, the space of generalized functions is broader than the space of fundamental functions. The elements of S’ are called generalized functions of slow growth.

Let f, 4 E ?J. Integrating

tafiaxj,

4) = -(f,

v(h Ea?. -a)i(x)dx =d(a), s6(x The delta-function

Changing the order of integration

(1.22)

Let f E &?‘. By definition, the derivative aflax, is the functional such that (1.22) holds for all 4 E &?. This definition of derivative is designed so that it coincides with the appropriate derivative for ‘usual’ functions. Higher derivatives are defined similarly, thus all generalized functions are infinitely differentiable. One of the most important generalized functions is the Dirac delta-function 6(x - a). Its definition is

jeiCX-“o<)dy

(Ff, 4) =

[ adlaxj).

eCiytf (y) dy

= sf(y)S(x

~(~f(x)e-“‘dx)4(C)d~

s

&3&O

d5.

this integral is equal to = If(x)(Je-‘“iQ(F)dx

= 6(x),

G(ux) = lal-“qx),

(Lx # 0).

= (f,F&,

so that (Ff? $4 =

(f?Fd),

(1.24)

if f; 4 E Y. This can be taken as the definition of the Fourier transform if f(x) is a polynomial,

of a generalized function f E Y’. In particular,

f = 5 aaxa, pJ=O

then

is even and homogeneous of degree - n:

d( -x)

- y)dy,

for any f E 9, from which the second formula of (1.23) follows. The formal derivation of (1.3) is the same. Let f, 4 E Y. Then f” E Y and f(t) defines a linear functional on Y’(R;) by the formula

i

by parts we obtain

(1.23)

(Ff )(O = , F. a,D”S(<). a

Ff

M.V.

204

Example 4. The (Cauchy) principal P

m fOdx s -03 X

Fedoryuk

III.

value of the integral

= lim -fodx E-+0 s 1X1>&X

[&-pp)lG=~, X

s0

u(4x) = (G*u&

X

xs_O” Wixr4(5) -e’“W5))& dx s“fwf(-4dx= so“1

4 =

transform. The right hand side of (1.24) is equal to X

s

m & -02

m -2isinxt X

0

(S

=s-a, O” d(t)(-i~sgnt)&.

,

s

-dx

0

Consequently

of the FS with the initial data; s

‘34x

- y)u,(y)dy,

(1.26)

de - = P(@, dt

Gltzo = 1,

where G = G(t, 5). Finding G and applying the inverse Fourier transformation, we obtain

We have used the known value of the Dirichlet integral: * sinxt

as initial data

which can be verified by a direct calculation. We will obtain an integral representation of the FS. By applying the Fourier transformation with respect to the variables x1,. . . ,x,, we obtain a Cauchy problem for an ordinary differential equation

a,

=

205

Gl,=o=~(x).

Thus solution of (1.25) is the convolution

This is a linear functional on 9, which we denote by 91. Let us find its Fourier

0

Transforms

function, is the solution G(t, x) with the delta-function

O” So&-is s -fX X

“fwf~-x)dx

=

Integral

G(t,x) = (27c-”

=Fsgn&

exp{tP(<)

+ i(x, <)} d&

(1.27)

s

x

This FS is a generalized function of the space s’. Example 5. Let us find the FS for the heat equation

We will now discuss some applications in mathematical physics.

of the Fourier transform to problems

2.4. The Cauchy Problem Cauchy problem

Differential

for Partial

Au.

Here P(t) = - t2, and it follows, from (1.27) and example 3, that Equations.

operator

G(t, x) = (2&$-”

Consider the

(1.25) where x E R”, t > 0, P(D) is a differential

au - = at

of order m with constant

exp{ -x2/4t}.

Substituting into (1.26) we obtain the classical Poisson formula. The Green’s function G(t, x) is the heat flow created by a point source of unit intensity situated at x = 0 and switched on at time t = 0. Example 6. The FS of the Schrddinger equation

coefficients: P(D) = t a,Da. p1=0 Equation (1.25) is called correct in the sense of I.G. Petrovski,

sup (-Rep(l))

is equal to if

< 00.

<ERn

The Cauchy problem (1.25) has a unique solution if, for example, u. E 9 (for more details, see [17]). The fundamental solution (FS) of the Cauchy problem, or Green’s

G(t, x) = (2&-“eiW4e-iX214’ Example 7. Let us find the FS of the equation

au -= -_a3u at a2

M.V.

206

III.

Fedoryuk

00 s -m

Transforms

201

Transform to spherical coordinates q r = pcosdsin8, q2 = psin4sin8, q3 = p cos 8, and integrate with respect to the angular variables. Then we obtain

It follows from (1.27) that G(t, x) = (27c-1

Integral

exp{ - it13 + ix[} dt.

The change of variables 4 = (3t)-“3q reduces this integral to the Airy integral (see part 2, Chapter 1, 9 4), so that G(t, x) = (3t)-1’3&‘i(~(3t)-“3). It is interesting to compare the behaviour of these FS for fixed t > 0 (say t = 1) and 1x1+ co. For the heat equation the Green’s function decreases exponentially, for the Schrbdinger equation it oscillates rapidly and does not decrease. For the equation of example 7, the Green’s function decreases exponentially as x + + cc and strongly oscillates as x + -co.

1 Z-------

OZ sin ctp sin rp dp

=-- 1

m

2rc2cr s 0

47f2cr s 0

= &

[cos p(r - ct) - cos p(r + ct)] dp

[b(r - ct) - 6(r + ct)].

We have used the formula

Example 8. Consider the Cauchy problem for the wave equation

co

cosxtdi”

= XC?(X),

s 0

which follows from (1.23). For r > 0, t > 0, 6(r + ct) = 0, so that where c > 0 is a constant. An FS for this equation is the solution with initial data G(t, x) = &Q

G,l,=, = 6(x).

GI,zo = 0,

The solution of the Cauchy problem is expressed in terms of the FS by the formula (1.28) u = Gt*uo + G*u,.

Using the Fourier transform we can find the fundamental solution to the Cauchy problem for equations of higher order, but with constant coefficients, which are Petrovski correct:

where the convolution is taken with respect to x. Taking the Fourier transform with respect to x, we obtain the Cauchy problem d2G -= dt2

whose solution is G = (cl (1-l) sin(ctl
s

Applying

sWfl
the inversion

= (27$”

Let us calculate the FS for IZ = 3.

sin(ctl?l)

formula

we

d5.

s

Cl?l

eiqnrdy

= 0,

am-l dt”‘G

for n 2 2, this integral is a generalized function. We will show that G is a function of the variables t and r = 1x1. The vector x can be transformed by an orthogonal transformation T, det T = 1, to the vector TX = (0,. . .,O,r). Put 4 = T-‘q. Since 151= 1~1,(x, 5) = (Tx,~) = q,, then G(t,x)

GI,=, = O,..., atmZGlt=o am-2

dG ,,I,=0 = 1,

GItEo = 0,

-C2t2G,

- 4

t=O

=

w9

Here P(z, [) is a polynomial of degree m in r and the coefficient of rm is one. If P(z, 5) is a homogeneous polynomial in z, tr,. . . ,t ,,, and the operator P is strictly hyperbolic, then the FS can be expressed in terms of an integral over the real algebraic manifold P(l, <) = O-the celebrated Herglotz-Petrouski formula (see [7]). In a similar way, the FS of Petrovski correct systems of equations can be calculated. 2.5. Fundamental

Solutions of Partial

Differential

Equations.

Consider an

equation of order m with constant coefficients (1.29) WM4 = 0) in R”. A fundamental solution E(x) of P(D) is a solution of (1.29) with right hand side in the form of a delta-function

M.V.

208

III.

Fedoryuk

(1.30)

P(D)E(x) = 6(x). It is also called an elementary solution or Green’s function. then we can find a solution to (1.29) by the formula u(x) = E *f =

If an FS is known,

s The FS is not unique; if u(x) is any solution of the homogeneous equation P(D)u(x) = 0, then the sum E(x) + u(x) is also an FS. Applying the Fourier transform to (1.30) we obtain P(&!?(t) = 1, so that E(c) = l/P(l). Applying the inversion formula, we obtain exp{~k s R"

Making

01 d5

P(l)

=--

1 m sin pr 2Pdp=L O” erpr 2n2r s o k2 + ie - p 4n2r s -m k2 + is - p2dp’

The last integral can be calculated using the theory of residues. The integrand has poles at the points p 1,2 = f ,,/m, where the branch of the root can be chosen so that Im pi > 0. Therefore, see example 1)

.

ev(i(x, O> s p(5)

f(t)@.

iJk2+ieIxI G(x,E) = -Le 4711x1 ’

(1.32)

Passing to the limit as E + +O, we obtain

R"

If P(t) has no real zeros, then u(x) is an ordinary function from Y. If P(t) has real zeros, then there is the problem of regularizing (1.32), which is considered at the end of this section. Let us discuss some examples. Example 9. We will find an FS for the Helmholtz

209

As in example 8 we can show that G is a function of t and r = 1x1 and in this integral put x = (0, 0, r). Transforming to spherical coordinates, as in example 8, and integrating with respect to the variables 0,4, we obtain

precisely the same calculations for (1.29) we obtain u(x) = (2X)-”

Transforms

G(x, E) = (27~)-~

E(x - y)f(y) dy.

E(x) = (27c-n

Integral

G(x) = -~

1 eiklxl 47clxl

.

If we put k = 0 in this formula, then we obtain an FS of Laplace’s equation in R3: AC(x) = 6(x),

G(x) = -__

1

4711x]’

operator in R3, that is, a

Formula (1.33) can also be written in the form

solution of the equation (A + k2)G(x)

= 6(x),

where k > 0 is a constant (the wave number). An FS is always non-unique, but in this case the situation is complicated by the fact that a condition of the type G(co) = 0 does not select a unique FS. In order to select a unique FS, we use the following physical considerations. The Green’s function G(x) is the field of a monopole point source situated at x = 0, in a homogeneous and isotropic non-absorbing medium. Introduce a small amount of absorption into the medium, changing k2 into k2 + is, where E > 0 is a small constant. Then the field of the point source, G(x, E), must be damped at infinity, that is, liml,l,, G(x, E) = 0. This condition selects a unique FS. We will find this solution and then let E tend to zero to obtain the required FS G(x) = lim G(x,E). &++O This method of selecting the ‘required’ FS is called the limit absorption principle and is applied in many problems of mathematical physics. For E > 0 we have

%4 = sR” k2 1’;” 52 4. P-C3

The Green’s function for the Helmholtz similar way. Example

10. Maxwell’s

operator in R” can be calculated in a

equations for a homogeneous

and isotropic

non-

absorbing medium have the form rot E = ikH,

rot H = -ikE

+ cj. C

(1.35)

Here E = (E,, E,, E3)T is the electrical field intensity, H = (H,, H,, H3)T is the magnetic field intensity and j is the current density, which we regard as given. Further, c is the velocity of light in a vacuum and k > 0 is the wave number. The system (1.35) describes steady oscillations, the dependence on time being given by a factor em’@‘.In order to find the solution of (1.35) for arbitrary j it is sufficient to solve the system rot E = ikH,

rot H = - ikE + :&~)a,

(1.36)

M.V.

210

Fedoryuk

III.

where a is a constant unit vector. A solution of the system is the field of a a point electric dipole situated at x = 0 and directed along the vector a. This field is produced by a particle with unit charge which oscillates in the direction a, near to x = 0, with frequency w. Eliminate H from (1.35) by applying the operator rot to both sides of the equation. In Cartesian coordinates rot rot E = grad div E - AE, where AE = (AE,, AE,, AE,)T.

(L, + ,oo’)G(x)

AS for Helmholtz’s

absorption principle. In Cartesian coordinates we have

where Au = (Au,, Au2, AU,)* (in curvilinear coordinates this is not the case). We take the Fourier transform with respect to the variables x1, x2, xg, then we obtain

+ graddiv E.

(PO21 - 43w AW=P~~Z+(~+PL)

grad div(h(x)a).

6(x)a - Egrad

The Green’s matrix of A + k2 is -&

$,

div(b(x)a).

and the operators A and graddiv

CR

H’“’

=

_ 4 rot

k

c

= 1

[;;:

:,i:

i;$

The matrix A(<) is real and symmetric. It has a simple eigenvalue (2 + 2p)t2, associated with which is an eigenvector equal to 4, and a double eigenvalue pc2, whose eigenvectors q fill out a plane orthogonal to 5. These facts enable us to simplify the calculations. Let e, = (l,O, O)T, e2 = (0, 1, O)T, es = (O,O, l)T be a basis of unit vectors, gj the j-th column of G(t), then

commute since A is a diagonal operator with equal diagonal elements. Therefore

. E(e) = !!e’kR+ kgraddiv

= 6(x)1.

L, = ,uA + (1 + ,u)grad div,

So that (A + k2)E = -F

211

(1.38) equation, the Green’s tensor is selected uniquely by the limit

Expressing E using the second equation of (1.35) we obtain grad div E = -g

Transforms

is the 3 x 3 matrix which is a solution of the system

Since div E = 0, then

(A + k2)E = -?6(x)a

Integral

(A - po2Z)gj

(1.39)

= ej.

Represent gj in the form gj = CX~(<)<+ qj, where (qj, 5) = 0. Taking product of both sides of (1.39) with t, we obtain

E’“’

’

A solution of the system

the scalar

((A - Pw20gj, 5) = (gj, (A - ~~~00

rot E = ikH + :6(x)a,

=

rot H = -ikE, hence we find aj = 5j[~‘((n

is called the field of a point magnetic dipole and is denoted (EC”), H(“)). It follows from (1.35) that H’“’ = E(e) E(m) = -H(e) 7 in a homogeneous and isotropic non-absorbing medium (the dependence on time being given by a factor ePiW’). Th e ch aracteristics of an elastic medium are: the density p and the Lame parameters 1, p (all positive). The system of elasticity theory has the form

=

lj,

Further, from (1.39) we find

.

Consequently, 1 gj = p(kf _ p) ej ’

1 kf - t2

1 k; - r2

(1.40)

where we have adopted the standard notations

(L, + po2)u = F(x),

k,=,jf$

(1.37)

Here x = (x1,x2,.4, u(x) = (u,C~~~W,~~(~) ’ is the displacement vector, I is the unit matrix and F(x) is a given vector of external forces. The Green’s tensor

+ 2p)t2 - po2)]-‘.

- PO’]

r].J = (pm2 - pc2)-’

Example 11. Consider the steady oscillations of frequency &

L, = (1 + 2~) grad div - p rot rot.

+ 2~)~2

aj5’[(Iz

k,=/g.

The number k, is called the longitudinal wave number, k, the transverse wave It remains to take the inverse Fourier transform, replacing po2 by

number.

M.V.

212

III.

Fedoryuk

(ljtkf)

=

&

f’(D)u(x)

k

then from (1.40) we find the components of the Green’s tensor: G,(x) = --e

1

47cpR

4(x) =fH

complex space

(@y d(x) = sexp{ik i))s(i) 4, Gw”

= f(x).

kind

Jk

of real dimension II in the n-dimensional

= cDy,/ = F-‘j

under

2.6. Integral Equations. Consider the Fredholm integral equation of the second

iktR& (1.41)

Let y be a manifold C”. The integral

213

Leray has studied the FS of a differential operator a(z, a/az), z E C”, with analytic coefficients [ll], and has constructed to this end a special integral representation.

F-'f, J

Transforms

contour integral u(x) = QY(j/p) is a solution of (1.29). In fact, differentiation the integral sign is valid (see (1.19)), and hence

po2 + i&, E > 0, as in example 5, and then pass to the limit as E-+ +O. This procedure need not be repeated, and we make immediate use of formula (1.34). Since F-1

Integral

+

m W,yM(y)dy, s -a,

where d(x) is unknown, and f(x), K(x, y) are known, functions. In a number of cases these equations can be solved using the Fourier transform. 1”. Consider the convolution equation

(1.42)

Y

(1.43) 4(x) =fW + m W - yM(y)dy. s -m This equation has the form 4 = f + K*& Applying the Fourier transform and

is called a Fourier (or Laplace) contour integral. Here c = 5 + iv d[ = d[, . . . d[,. If y = R;, then Qyg = F-‘g. A particular case of Qy is the Laplace contour integral. If g(c) is analytic in a neighbourhood of y, then y can be deformed without changing the value of the integral Qyg. For the Fourier contour integral there is the commutation formula

but no inversion formula. Using the Fourier contour integral we can find a solution of (1.29) if the right hand side has the form f(x) = QYg(x) (for example, if f(x) E 0). For n = 1 such examples have been considered in the first part of this volume. Let us return to (1.29), where f E 9(R”), Then (see subsection 2.2) the function f(l) is entire and (1.19) holds. We will call contours y1 and y2 equivalent if QYlf” = Qy2ffor anyf E 9(R”). Let us give some examples of contours equivalent to y. = R”. For n = 1 any line Im 5 = c, parallel to the real axis, is a contour equivalent to yo. For n 2 2, an n-plane y: [ = r + iv’, 5 E R”, is equivalent to y. for any fixed q” E R”. Moreover, for n 2 2, there are ‘discontinuous’ contours equivalent to yo. Example. n = 2. y consists of two half-planes, cl = t1 + ic,, 0 < t2 < GO,and Cl = t1 + ic,, --co < t2 < 0, where -cc < t1 < co, and cl, c2 are real. In fact, the line cl = t1 + ic, t1 E R, is equivalent to the real axis cl = t2 E R, so the contours y and y. are equivalent. Hbrmander has proved that for any polynomial P(c) there is a contour Y equivalent to y. such that [P(c)1 2 c > 0 on y (the so-called Hijrmander staircase, see [ 171). In addition, the quantity 1Im cl is bounded on y. Therefore the Fourier

using (1.20), we obtain 4 = f” + R .& whence we can find 4. Applying the inverse transform we obtain

:

We transform this integral to get 464 - f(x)

= & 1:

m

[ 1 $(,)

- f(t)]

eixCd5

where we have put M(5) = [l - 1?(5)]-‘Z?(<). S’mce the Fourier transform takes a product of functions to the convolution of their images, then m (1.44) cpb) = f(x) + m(x - y)f(y)dy, s -03 where m is the Fourier transform of M. In precisely the same way we solve the Fredholm equation of the first kind of convolution type:

M.V.

214

III.

Fedoryuk

f(x) =smK(x -

Transforms

215

Here x E R”, 4 is a hyperplane in Rt and dm is Euclidean measure on t. The set of hyperplanes t is denoted by P”. A hyperplane 5 can be written in the form 5 = {x E R”: (x,0) = r }, w h ere r E R and o is a unit vector: cjm,i o$ = 1. The Fourier transform and the Radon transform are connected by the relation (here s E R, )o( = 1)

YM(Y)dY.

-00

Integral

Applying the Fourier transform, we find that f = Z?. 4, so that 4 = f/K. Applying the inverse transform we find (1.45)

I

m dr exp ( - is(x, CO)}dm(x) s --a, s (x,o)=r m = -m f((o, r)e-““dz s

f(m) =

Of course, (1.44) and (1.45) are valid under certain restrictions off(x) and K(x). For sufficient conditions under which these formulae are valid, see [S]. Consider the convolution equation on the semiaxis (the Wiener-Hopf

(1.47)

Consequently the n-dimensional Fourier transform is the composition of the one-dimensional Fourier transform and the Radon transform. There is also a

equation) d(x) = f(x) +

O”w s0

- YM(Y)dY.

dual Radon transform

This equation can be solved using the Fourier transform (the Wiener-Hopf method), which is one of the most famous applications of the Fourier transform. 2”. Consider the Fox equation e4 = f(x) +

mw I -00

f"(x) = [

where dp is a measure on the compact set M, = {t E P”: x E <}, invariant relative to rotations around the point x, ,u(M,) = 1. There is an inversion formula

+ YMY)dY.

f = c-'J"-'"qf*)9

Make the change of variable y = - t, then we obtain the equation co K(x - t)qq s -CO Applying the Fourier transform we obtain 4(x) = f(x) -

t) dt

c = ( - 47c)‘“-““r(n/2)/r(

= G3

5). f(x) = (2ni)(‘-“)/2A$-1)/2

.fb (x,4) do, s p-1 and note that f,(x) = f((w,(x,o)) is a plane wave with direction w, that is, it is constant in each hyperplane orthogonal to o. Consider first the equation

+ &t,ict,,

and we find

Wh Applying the inverse transform we obtain

2.7. The Radon Transform. Closely related to the Fourier transform Radon transform A3

=

s5

f(x) dm(x).

(1.49)

tions, etc. One of the applications of the Radon transform is in the solution of equations of the form (1.29). Write the inversion formula (1.49) in the form

Replacing 5 by -t in this equality we obtain iw,

l/2)

with a suitable definition of the fractional power of the Laplace operator A. Plancherel’sformula holds, the Radon transform is defined for generalized func-

= f(x) - K(x)*(b(x).

&3 = &.E, + Rok-

(1.48)

f(5)445),

=f,

and search for its solution in the form of a plane wave with direction wave is a function of one variable (0, x); the function P(D)u, is also of one variable. Therefore equation (1.50) is an ordinary differential with constant coefficients which may be integrated. Let the solution smoothly on o, then the function

is the

(1.50) o. A plane a function equation u, depend

u(x) = (27ci)(1-“y2 u, do s p-1

(1.46) i

will be a solution of (1.29). In fact, the operators P(D) and A(n-1~2commute, so

M.V.

216

Fedoryuk

P(D)u = (27cni)“-“)‘2

III.

s S”P’

= (27ri)(‘-“)‘2 [

P(D)v,

Transforms

217

an analytic function, which allows us to use the full power of the apparatus of the theory of functions of a complex variable. The commutation formula has the form

do

f, do = f:

(1.54)

f’“‘(t) + p”F(p) - p”-If(O) - . . . - f’“-“(o),

Js-1

For further applications, properties and generalizations see [S].

Integral

where f(“)(t) E 0. The application of the Laplace transform to the solution of equations is based on this formula. As an example we consider the Cauchy

of the Radon transform,

problem for a linear ordinary differential equation with constant coeflicients:

XC”)+ a,x(“-‘) + ... + a,x = f(x),

5 3. The Laplace and Mellin Transforms -40)= x0, 3.1. The Laplace Transform. Let f(t) be a complex-valued function of a real variable t. Its Laplace transform is the function of a complex variable p:

(1.51) m e?‘f(t)dt. s0 We will use the notation (Lf)(p) = F(p). In the operational calculus it is usual to call f(t) the original and its Laplace transform F(p) the image, and to use the following notation:

IfM

I Meat,

2 0.

(1.52)

This class of functions is a linear space; denote it by 0. In all that follows we will consider functions from 0. The integral (1.5 1) converges in the half-plane Rep > a of the complex p plane, and is an analytic function therein. In addition, lim,,, F(p) = 0, if Rep --P +m. There is an inversion formula (1.53) where b > a is arbitrary, at points of continuity off(t). At points of discontinuity, the integral in (1.53) is equal to +[f(t - 0) + f(t + O)]. The analyticity of the image F(p) in the half-plane is one of the very famous properties of the Laplace transform. Applying the Laplace transform we obtain

the Laplace (1.56)

Here l(p) = p” + aIpn-l + ... + a, is the characteristic polynomial M(p) = pn-lxo + ... + x,,el + u,(~“-~x,

+ ... + x,-J

Thus, the differential equation for x(t) is transformed equation, (1.56), for the image X(p) so that

Let f(t) satisfy the conditions:

t

(1.55)

l(~)X(p) = F(P) + M(P).

F(P) + f(t).

1”. f(t) is continuous on the real axis R except, possibly, for discontinuities of the first kind, and there are only finitely many points of discontinuity in each finite interval. 2”. f(t) E 0 for t < 0. 3”. f(t) grows no faster than exponential as t -+ +co, that is, there are constants M and a such that

=x,-1.

Denote by X(p) and F(p) the images of x(t) and f(t). Applying transform and using linearity and (1.54), we obtain

@f)(p) =

f(t) + f-(P);

$0) = x1,...,$(O)

X(P) =

/ 1 [ / 1 / 1 1 Ii 1 : :

F(P) + M(P) l(P)

of (1.55),

+ ... + a,-,~,.

to a linear algebraic

(1.57)

Now it remains to recover x(t) from its image X(p), which can be done, for example, by using the inversion formula. This method of solution is called the operational calculus or operational method. In practice, the operational method is applied in the following way. First of all it is necessary to calculate the image F(p) of f(t). There are vast tables of Laplace transforms and the image F(p) is found from these tables. The image X(p) is given by the final formula (1.57), and the original x(t) is recovered from the image again using tables. Here, of course, one must use the linearity of the Laplace transformation and certain other properties. Thus, the search for the solution of the Cauchy problem (1.55) for a differential equation reduces to algebraic operations. The algorithm for using the operational method is unusually simple, and it is not by chance that this method is highly popular among electrical engineers, radio technicians, and others. For its origin the operational calculus is significantly indebted to electrical engineers who, in the calculation of electrical circuits, had to solve many systems of linear differential equations with constant coefficients (Kirchoff’s law). This method, without any justification, was developed by the eminent American electrical engineer and physicist Heaviside, at the beginning of this century. The rigorous justification of the method was done only in the twenties.

218

M.V.

Fedoryuk

III.

Integral

Example 1. Let n 2 0 be an integer, a any complex number. Then n!

t”f(t)

x(0) = 0,

i(0) = 0

-4 ‘@)

In addition, there is the shift formula f(t - h) + emphF(p),

where h > 0, f(t) = 0 for t < h and the formula (the multiplication

COSOA

f(at)+ $),

Let w # oO, then it is impossible + ~ P p2 + co2) . directly from the table. But, expanding X(p) in partial fractions we find

x(t) =

F P al; - co2 ( p2 + w2

If f, g E 0, then their convolution

P p2 + 0; 1

(f*d(t)

x(t) = 2.

Example 5. Consider convolution type:

t sin o0 t.

a > 0.

is =

f*s

If o = oe, then

theorem)

h - 4gWd~. s0 takes the convolution of originals to the product of

The Laplace transformation their images:

F (cos ot - cos coot). 0); - co2

F

m F(q)dq. s P

t’

X(P)= F(p2+ 02;p2+ co;)’

X(P) =

nF(p). )

and the dual formula is

where we > 0, w > 0, F # 0 are constants. This equation describes, for example, the small oscillations of a charge suspended on a string under the action of external periodic forces. Passing to images, we obtain

x(t)

-;

The Laplace transform of the integral of the original is equal to

Example 2. We will solve the Cauchy problem

to find

+ (

(the integral is calculated using integration by parts). Using Euler’s formula, we can find the Laplace transforms oft” cos cot, t” sin ot and form a small table. This is quite sufficient to solve the Cauchy problem (1.55) with a right hand side which is a quasi-polynomial (f(t) = I;=1 pj(t) e’j’, where pj(t) is a polynomial, ,uj are constants).

using the table formula

219

In particular, we can calculate the Laplace transform of any step function and any spline. We will give some more properties of the Laplace transform. There is the formula dual to (1.54):

Let us calculate the Laplace transform in the simplest cases.

I + co&x = Fcosot,

Transforms

(1.58)

+ F(P)G(P).

the Volterra

integral

equation of the second kind of

f

0

x(t) = f(t) +

In this case the solution x(t) is unbounded on the semiaxis t 2 0 (the phenomenon

s0

k(t - 7)x(z) dz.

Passing to images, we obtain

of resonance). Example 3. Let a, a be complex numbers, a arbitrary, Re c(> - 1. Then

taP

qa + 1) + (p + a)a+l.

Example 4. Let 0(t) be the Heauiside function: d(t) the Dirac delta-function, h > 0. Then f3(t - h) + p-‘eeph,

(J-6448/0399/Kll/WSL/pp.214-218)

X(P) = F(P) + K(P)X(P),

: e(t) = 0, t < 0, Q(t) = 1, t 2 0,

d(t - h) + emph.

where X, F and K are the images of x, f and k, so that X(P) =

F(P) 1 - K(P)’

The function x(t) is found by the inversion formula. The Volterra equation of the first kind of convolution type is solved similarly.

220

M.V.

Fedoryuk

III.

The analyticity of F(p) enables us to calculate the original using residue theory. Let F(p) be a mermorphic function not having poles in the half-plane Rep > a, and let the integral lZm IF@ + iy)l dy converge for any b > a. Let there exist a sequence of circles C,,; (pj = R,, such that R, < ... < R, -+ co, maxcn IF(p)1 + 0 as n + co. Then there is the formula (called the seconddecom-

Example 6. We will solve the mixed problem for the heat equation

au a2u %=a 2jp’

(1.59)

u(t,O) = 0,

u(t, 1)= 0,

4094 = f(x),

in the domain 0 < t < co, 0 < x < 1,where a, I > 0 are constants, f(x) is a smooth function, f(O) = f(l) = 0. Applying the Laplace transform we obtain a boundary value problem for an ordinary differential equation

P’Pn

where the sum is taken over all the poles of F(p). The Laplace transform can be considered as a special case of the Fourier transform. In fact, comparing (1.8) and (1.51) it follows that ccf(t)) (PI = ~(fww)

221

Transforms

or the equations of mathematical physics, we will restrict ourselves to two examples and some remarks of a general nature concerning (1.62).

position theorem)

f(t) = C res eP’F(p)

Integral

u2v” = pv - f(x),

VI,=0 = U,=l = 0,

where v(p, x) is the image of u(t, x). Hence we find

( - iP)

where O(t) is the Heaviside function. The inversion formula for the Laplace transform is a corollary of the inversion formula for the Fourier transform.

eP,x)

=

G(x, if; PV(~) d5, J0

where G is the Green’s function: 3.2. The Laplace Transform and Evolution Equations. The fundamental applications of the Laplace transform are related to evolution equations, that is, to

those in which there is a distinguished variable which varies between 0 and +co. We denote it by t and interpret it as time. It is convenient to present the application of the Laplace transform in a fairly general form. Consider the Cauchy problem p = Ax + f(t),

vl(x) = sinhdx a

~lWV2(5)

x

I

l,

A

v2(4v1(5)

x

2

5,

v2(x) = sinha(lJp

’

- x),

A = -u&sinhal.

Jp

We note that G is a single-valued function of p. Applying the inversion formula, we obtain

(1.60)

x(0) = xg

1

G(x, 5;~) = -

b+im

.kx)=&

for 0 < t < co. Here x0, x(t) and f(t) are elements of some Banach space W (for 0 5 t < co), A is a linear operator acting in some domain g(A) c %!?‘,and not depending on time. Apply the Laplace transform to (1.60), then we obtain an equation

R,(P)

eP’G(x, 5; P) dp.

s0 s b-iw The poles of G are found at the points pn = - (n~u//)~, n = 1,2,. . , (p = 0 is not a pole). For t > 0 the integral over (b - ice, b + ioo) is equal to the sum of the residues at the points p = pn (see (1.59)), and, finally, we obtain

PX(P) - x0 = AX(P) + F(P)?

where X and F are the images of x and f. Hence we find X(p) = (A - PI)-’ (x0 + F(p)), where I is the identity operator. We introduce the notation

Id5

u(t, x) = f

e-(rmni’)2tfn sin y x, 1

n=l

>

where the f, are the coefficients in the Fourier sine series of f(x):

(1.61)

= (A - PII-‘;

this is called the resolvent of A. Applying the inversion formula (1.53) we obtain b+icc x(t)=&

__ rm

b+im

eP’R,(p)dp.xo + i eP’R,WW 27ri s b-im

dp

(1.62)

s This is the general scheme for the application of the Laplace method to evolution equations of any order with respect to t. Since concrete applications of the Laplace method to partial differential equations are usually contained in standard courses on partial differential equations, b

Example 7. In the theory of a collisionless plasma, there is the nonlinear system of equations of A.A. Vlasov, which have recently been given more and more

attention in the mathematical literature. In the simplest, one-dimensional linearized, case the system takes the form [12] f, + VL = m4

E,=

-41

m fdv. s -m

and (1.63)

222

M.V.

Fedoryuk

III.

Here t is the time, x is the coordinate, u is the velocity, f(t, x, v) is the function of concentration of electrons and E(t, x) is the electromagnetic field intensity in the plasma. More precisely, f and E are perturbations of these functions and f = fe(v), E = E, are the unperturbed states. The function f0 E Cm(R) and decreases rapidly as [VI + co. For f there is given rapidly decreasing initial data fit+ = g(x, u) and for E there is the condition E IX=-oD= 0. Applying to (1.63) the Fourier transform F with respect to x and the Laplace transform L with respect to t (here p = - iw), we obtain i<E= -47~

iof” + i&f” = f;(u)B + s”((, ?I),

O” f”du, s -00

in which u is a parameter. Here f” = FLf, E” = FLE. Hence we can find f and El; of particular importance are the functions

E(o, 5) = Hb, ~ 0 4~5)

’

A = 1-471i

O” ~fd(4 du s -m 0 + (u

m __ a594 du H = hi s -m w + 50

translation of the contour of integration). Further, let the resolvent R,(p) have a finite number of poles in the strip a < Rep < b: Rep, 5 Rep, 5 ... 5 Rep,. Then x(t) = f res (eP’R,(p))x, + O(eCbt), (1.66) j=l p=pj I and the asymptotic form of x(t) as t -+ +cc is determined by the pole (or poles) with the least real part. One of the most famous applications of this method is connected with Vlasov’s system (1.63). Let &(u), g(<, u) be entire functions of u, and let A(o, 5) (5 E R fixed, 5: # 0) have no zeros in the half-plane Im o 2 0 (see (1.64) for d). Then A can be analytically continued to some half-plane of the form Im o > -c, c > 0, and the asymptotic form of E(t, l), as t + +co, is determined by the zeros of A(o, 5) with the largest imaginary part (see (1.65)). In particular, E(t, 5) decreases exponentially as t + +cc (this, the so-called ‘Landau decay’ is a fact which, at one time, appeared unexpected to physicists).

(1.64)

M(z) =

&co, <)eiw’dco.

=

IIUP)ll

5 cu

+ IPI)-*,

f(x) = &

(1.65)

s -w+ia The application of the Laplace transform enables us to comment on the behaviour of the solutions of (1.63) for large values oft and, in particular, on the stability with respect to linear approximation of a variety of problems in hydrodynamics, plasma theory, etc. The general scheme for applying the Laplace transform in this case is as follows. Let f(t) = 0 in (1.60). In many cases there is an estimate for the resolvent in the form O
in some half-plane Rep > a. Even when A is a self-adjoint operator on a Hilbert space and CY= 1, the integral in (1.62), in general, is not absolutely convergent. Integrating by parts, we obtain

dx.

(1.67)

s”_‘i” M(z)x-“dz.

(1.68)

O”f(x)x=-l s0

The inversion formula takes the form

m+in 5)

223

Transforms

3.3. The Mellin Transform. This transform is given by the formula

We remark that in these formulae 5 E R, Im w > 0. If we use the inversion formula we find f and E. In particular, W,

Integral

b

The simplest sufficient condition continuous for 0 < x < co and

100

for the validity

coIf(x)Ixb-‘dx

of these formulae is: f(x) is (1.69)

< co.

s 0

Since the Mellin transform and its applications have been considered in detail in the first part of this volume, we will only pause on it briefly. The commutation formula is (1.70) under the condition

that (1.69) is satisfied for f(x) and xf ‘(x), and that

b+im x(t)

=

-f

ep’Ri(p) dp.

s b-im If cI > 3, then Ilx(t)ll 5 ebr and for b < 0 x(t) is exponentially damped as t + +a. If 0 < a 5 3, then we can integrate by parts several times and we get to the same result. Another approach to the investigation of the asymptotic form of the solution x(t) is as follows. We suppose that the integral (1.62) converges on any line Rep = c, a 5 c 5 b, and that these integrals are equal (we can do a parallel

lim x’f(x)

= 0,

x-0

lim x’f(x)

x-00

= 0.

The Mellin transform is closely related to the Fourier and Laplace transforms. Making the change of variables x = ey, f(x) = g(y) in (1.67) we obtain co M(z) =

-oog(ykzY dy = W(iz),

s where F is the Fourier transformation.

224

M.V.

We note one more important forms of f(x) and g(x), then

Fedoryuk

III.

formula: if F(z) and G(z) are the Mellin

trans-

(1.71) Let us give some applications

[

transform, we obtain

Of(x) dx,

1-s

(H,f )(O =

Example 9. Consider Laplace’s equation in a sector 0 < 4 < d,, < 271, 0 < r < GOwith Dirichlet conditions on the sides of the sector:

UI#ybo = UlW.

we require the boundedness of the solution

at infinity

z=o.

a=s-2,

(1.74)

xl”1 f(x)1 dx < co.

0

If, in addition, for each a > 0 the integral fz 1f ‘(x)1 dx converges then there is an inversion formula W;‘g)(x)

=

m s(WM)5&> s0

(1.75)

where g = H,f. The commutation formula takes the form H,L,f

= -12H,f,

(1.76)

where L, is the Bessel operator

fib, 400)= h(s).

Solving this boundary value problem and applying the inversion formula, we find the solution u(r, 4). A very elegant application of the Mellin transform is the construction of a fundamental system of solutions for the equation y(“) - x”y = 0, for any n and any complex CI.

dx,

03 s

au

tio(.s) and z&(s) are defined similarly. In polar coordinates we have Au = u, + r-l u, + r-‘udd, and from (1.70) we obtain an ordinary differential equation for ii

m f(x)J,(xOx s0

trans-

where J,,(x) is the Bessel function of order v. Integrals of the form (1.72), (1.73) are called Fourier-Bessel integrals. The Hankel transform is usually considered for real values v > -3, although it is defined for all all complex v. It follows from the asymptotic formulae for the Bessel function as x -+ +co, that the Hankel transform off(x) exists if

and the

so-called ‘Meixner condition on the edge’, lim & = 0. These conditions az r+o guarantee the uniqueness of the solution. Apply the Mellin transform with respect to the variable z: m rselu(r, 4) dr. w, 4) = s0

qgs, (b) + Lx2qs, 4) = 0,

(1.72)

( )I

g+;g+

G(z) 1 + F(z)’

Applying the inversion formula (1.68), we find the required solution y(x).

u”(s,O)= Go(s),

m Z(xtMx, s0

In (1.72) h is an elementary function. The Hankel transform of order v (also called a Bessel or Fourier-Bessel form) is the integral transform

where Y, F and G are the images of y, f and g, hence we find

In addition,

The Bessel transform is an integral transform of

4.1. The Hankel Transform.

where Z,(x) is the Bessel function of order v, that is, is the solution of Bessel’s equation

Y(z) + F(z)Y(z) = G(z).

4&O = %I(~),

225

the form (Kf )(5) =

where f and g are known functions. Applying the Mellin

Y(z) =

Transforms

$4. The Bessel Transform

of the Mellin transform.

Example 8. Consider the integral equation

Integral

(1.77) (1.76) holds under the following conditions: f E C2 for 0 < x < co, the integral m x”21LYfldx

s0

III. Integral Transforms

M.V. Fedoryuk

226

converges, and

,

xv+lf’(x) + 0, X”2f’(X) + 0,

x’f(x) + 0, x”2f(X) -+ 0, We mention one more important

formula

9 (4 dx =

52)

= ss

Transforming

r2 ap

m f(,,&~e6Xl~lmix~~Zdx1 m -co -02

x2 = rsin&

Example 1. Let G(r) be the Green’s function of the Helmholtz that is, the solution of the equation

dx,.

=

s

2n e-irp

om.fWr

cos(f$

-e)

(1.83)

This function was calculated in 5 2 (example 5). We will obtain an expression for G(r) in the form of a Fourier-Bessel integral. Apply the Fourier transform with respect to x, y to both sides of (1.83) and use (1.80) and (1.76). Then for the image G we have an ordinary differential equation

we obtain (here g(p, 0) = f(tcr,, t2)) db 0) =

operator in R3,

(A + k2)G(r) = 6(r).

t2 = p sin 8,

t1 = pcose,

az2'

We also introduce the notation r = (x, y,z), R = Jm

to polar coordinates

x1 = rcosfj,

(1.82)

d5.

d~~+e!+Ls!Y+~

(1.79)

coSw,f)(Smd(~)~~. s0

Let us give the relationship between the Hankel and Fourier transforms. Consider the Fourier transform f(< r, &) of a function f depending only on the radius: m,

5H6%5)f(5) Y

The Hankel transform is used in the solution of equations in R3 which have axial symmetry, that is, are invariant relative to rotation about some axis. All the basic operations of vector analysis-gradient, divergence and curl-are invariant relative to rotation about any axis passing through the origin. Therefore the Laplacian d = div grad, the operator of elasticity theory (see (1.73)) and Maxwell’s operator are invariant relative to rotation around any such axis. Choose the z axis as the axis of symmetry and introduce cylindrical coordinates r, 4, z. In these coordinates

x+CO.

which follows from the indentity J;(x) = --Jr(x). For the Hankel transform there is a Parseual identity s0

= ; s

H1;= -5Ho.L

00 a4

Wof)(x)

x- +o,

221

d$

$

-t k2 - c2 G = 6(z).

s0

27wof)(P)

(1.80)

We again use the principle of limit absorption (5 2, example 5), that is, replace k2 by k2 + ie and require that lim,,l,,i G(<,z,.s) = 0. Having calculated G we pass to the limit as E --t +O. Put K(E) = Jm and choose a branch of IC(E) such that Re K(E) > 0, Im K(E) < 0, which is possible since Im ICY < 0. Then we obtain

We have used the known integral representation for J,(x). Similarly, the Fourier transform of f( xf + ... + xf) can be expressed in terms of the Hankel transform of order n/2 - 1. In applications, in addition to Fourier-Bessel integrals, Hankel contour integrals are used

(K, J-1 (4=stH:“(x, M-03 dt.

G=-e

(1.81)

Y

Here Hil’ is a Hankel function of the first kind, y is a contour in the complex 5 plane. The commutation formula is preserved. We will express a Fourier-Bessel integral in terms of a Hankel contour integral in the most important special case. Let v = 0 and let f(t) be an even function. We can take y to be the real axis, where Hi”(l) = - H&‘)( 151)for 5 < 0. Sincef( - c) = f(5) and J,(5) = Re H&“(5) for 5 > 0, then

-1

-K(E)2

47uc(&)

Applying the inversion formula (1.70) and passing to the limit, we obtain G(r) = -&

I F

m5 e-KIZIJO(
lC=pT,

(1.84)

Os<
(1.85)

and the branch of the root is chosen so that K> f0,

In particular,

it follows that

t>k;

ImIc
M.V.

228

III.

Fedoryuk

Integral

Transforms

229

The inversion formulae take the form

eikR -=

R

s

m ~e-ktzb,,(~~) 0 K

(1.86)

d&

Ki,(X)z sinh rtr F(T) dz,

In the theory of Bessel functions, this formula is called the addition theorem. Example 2. We will find the Green’s function G(r, r”) in the half-space z > 0, under the boundary condition G = 0 on the plane z = 0. G can be interpreted as the sound field of a point monopole emitter, situated in a half-space filled with a fluid (the z axis is directed downwards), which borders on air (at the boundary the flow is zero). Since this problem is invariant relative to shifts parallel to the plane z = 0, then

W,r”)

= G(x - xo,y - Y~,~;O,O,~~),

zo > 0.

Therefore it is suflicient to calculate G for z” = (O,O, h), h > 0; we denote this function by G(r,h). Put G(r,h) = Go(r,ro) + u(r), where G,(r, r”) is the Green’s function of free space. Acting in the same way as in example 1, we obtain a boundary value problem on the semiaxis (0, co):

(g-+=0,

W&hi?(s) ds,

where I,(x) is the Besselfunction with an imaginary argument. Both of these transforms are related to the Laplacian A in R3. Namely, if we look for solution of Au = 0 in the form u = eT(e-ibZ w(r), then we obtain w(r) = Ki,(lalr), which reduces to the Lebedev-Kontorovich transform (1.87). The Meijer transform is close to the Hankel transform. If we search for a solution of Laplace’s equation in the form u = e’“‘~e”flw(a), in toroidal coordinates c(,z, 4, then we obtain W(U) = cl PBm_1,2(cosh~1)+ c,Q8”_,,,(sinh a), where P and Q are Legendrefunctions. This reduces to the Meller-Fock transform m

P~z=o=&e-“”

Change k2 into k2 + is, E > 0, and impose the condition z + co. Then we obtain

at infinity:

J’(P) =

fi + 0 as

s1

P-1/2+iF(x)f(x)dx,

(1.90)

P 2 0,

with inversion formula f(x) = ptanhnp

u”= &,-N+M

(1.91)

mP-1/2+ip(x)+(p)dp-L. s1

5 5. Other Integral Transforms

In this formula E = 0 and the branch of rcis as in (1.85). Finally, we obtain “5 o i J,([r) [e-(K+z)h - e-K(Z-h)] d&

G(r, h) = &

(1.89)

An important

class of integral transforms are transforms of convolution type

s

The problem of a point source in an elastic half-space (Lamb’s problem) is solved in exactly the same way.

m G(x - t)f(t)dt = G*f(x) (1.92) s -a, The function G(x) is called the kernel of the transform. Transforms of the type c?(x)=

00

4.2. Other Transforms Connected with Bessel Functions. These are the Kontorovich-Lebedev transform F(z) =

m Kiz(x)f(x) dx, s0

(1.87)

CD(x)= s0 (in particular, the changes

and the Meijer transform

t

K,(st)&fWdt,

the Laplace transform) reduce to convolution = e-‘,

x = et 2

%
type. In fact, after

US = f(emT)e-’

we obtain &3

where K,(x) is the Macdonald function.

k(xt)f (t) dt

=

O” G(t - +Wd~, s -Co

G(t) = k(e5).

M.V.

230

III.

Fedoryuk

Integral

Transforms

231

4”. The Somm&feld transform

The inversion formula formally follows from the fact that the Fourier transform F takes a convolution of functions to the product of their images, so that

e-ikrsin8cos(~-p)S(p)d~.

(1.93) Here e(t) is the Fourier transform of the kernel G. In general here F must be understood to be not simply the Fourier transform but rather a Fourier contour integral, for example,

(F,+&))(5) = over some line 1 in the complex

g(W”“dt s1

t plane (for example, over the line t =

iz, z E R,

s Y

Here y is a contour in the complex plane. It consists of two simple infinite curves y+, y-, Imp > 0 (< 0) on y+ (y-). The curve y+ has two asymptotes Rep = d-n-S, Rep=d+6, w h ere 6 > 0 is small, the curve y- has asymptotes Rep = C$- 6, Re p = C$+ rc + 6. Moreover, r, 4, z are cylindrical coordinates in R3. The function S is a solution of Helmholtz’s equation (A + k2)S = 0. Using the Sommerfeld transform the problem of plane wave diffraction in a wedge has been solved exactly-by Sommerfeld for Dirichlet and Neumann conditions, and by G.D. Malyuzhints for the third boundary value problem [ 133.

F is the two-sided Laplace transform).

If P(t) = l/c(<) is a polynomial,

then the formula can be written in the form (1.94)

f(t) = w%(t)>

where P is a differential operator. In general, P is a pseudo-differential operator. We will list several more frequently encountered transforms of convolution type. 1”. The Hilbert transform

OD s(5)d(. (H-lg)(t) =-1 71 s-,t-5

where the integral is in the sense of a principal value. The inversion formula is

In particular, Hz = -I. For more detailed properties and applications

of H see

PI. 2”. The Weierstrass

transform

(Wf)(x) =1

O”ev{-(x

24% s -cc

-

Y)')~(Y)~Y.

The kernel G(x) is the value for t = 1 of the Green’s function of the heat conduction equation. The Hilbert and Weierstrass transforms have been generalized to the case of several variables.

References* There is a vast literature devoted to integral transforms and their applications. Somewhat conventionally it can be divided into three types. 1. Reference works. 2. References in which the properties of integral transforms are studied. 3. References devoted to the applications of integral transforms to the problems of mathematical physics, etc. It is simply impossible to survey all the monographs, textbooks and articles in which integral transforms are applied to concrete problems. In the first type there are the monographs [l], [4], [S], [14], [15], [25], in the second [3], [6], [S], [9], [lo], [ll], [13], [16], [19], [24], [26], 1271, in the third [2], [18], [20], [21], [23]. This division is highly conventional since the three types of literature are closely interleaved. The most extensive are the references devoted to the Fourier transform: [3], [S], [7], [16], [17], [18], [19], [24], 1261, [27], and the Laplace transform: [4], [S], [lo], [25], then the Mellin transform: [S], [19], 1211. The monographs [S], [18], 1201, [21], [27] are devoted to the Bessel transform, and 1131 to the Sommerfeld transform. Transforms of convolution type are considered in [9]. The Hilbert transform, both one-dimensional and many-dimensional, and the many-dimensional Laplace transform may be found in [S], [24], the Radon transform in [9] and the Laguerre transform in [S]. For onedimensional integral transforms in the complex domain see [6]. There is a detailed bibliography in

I% 1. 2. 3.

3”. The Euler transform 4.

(Ff)(x)= (t - x)"fW& sY where y is a contour in the complex t plane. This is used in the study of the solutions of ordinary differential equations with rational coefficients.

Bateman, H., Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of integral transforms, Vols. 1, 2. New York: McGraw-Hill, 1954. Zbl. 55, 364; Zbl. 58, 341 Brekhovskij, L.M: Waves in layered media. Moscow: Nauka, 1973. English transl. (of an earlier edition) New York: Academic Press, 1960 Bochner, S.: Vorlesungen tiber Fouriersche Integrale. Leipzig: Akad. Verlagsges. 1932 Zbl. 6, 110. New York: Chelsea Publ., 1948. English transl.: Lectures on Fourier integrals. Princeton: University Press, 1959 Ditkin, V.A., Prudnikov, A.P.: Operational calculus in two variables and its applications.

*For the convenience of the reader, references to reviews in Zentralblatt fur Mathematik (Zbl.), compiled using the MATH database, and Jahrbuch fiber die Fortschritte der Mathematik (Jrb.) have, as far as possible, been included in this bibliography.

232

5.

6. 7.

8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

M.V.

Fedoryuk

Moscow: Fizmatgiz, 1958. Zbl. 126, 314 English transl.: Oxford: Pergamon Press, 1962. Zbl. 116,309 Ditkin V.A., Prudnikov, A.P.: Integral transforms and operational calculus. Moscow: Nauka, 1974. Zbl. 298.44007 English transl. (of an earlier edition) New York: Pergamon Press, 1965. Zbl. 133,62 Dzhrbashyan, M.M.: Integral transforms and representations of functions in the complex domain. Moscow: Nauka, 1966. Zbl. 154,377 Gel’fand, I.M., Shilov, G.E.: Generalized functions and operations over them. Moscow: Fizmatgiz, 1959. Zbl. 91, 111 English transl.: New York London: Academic Press, 1964. Zbl. 115,331 Helgason, S.: The Radon transform. Boston: Birkhauser, 1980. Zbl. 453.43011 Hirschman, II., Widder, D.V.: The convolution transform. Princeton: University Press, 1955. Zbl. 65,93 Lavren’tev, M.A., Shabat, B.V.: Methods of the theory of functions of a complex variable. (4th ed.) Moscow: Nauka, 1973. Zbl. 274.30001 Leray, J.: On prolongement de la transformation de Laplace qui transforme la solution unitaire dun operateur hyperbolique en sa solution elementaire, Probltme de Cauchy IV. Bull. Sot. Math. Fr. 90,39-156 (1962). Zbl. 185,343 Lifshits, E.M., Pitaevskij, L.P.: Physical kinetics. Moscow: Nauka, 1979. English transl.: New York: Pergamon Press, 1981 Malyuzhinets, G.D.: Sommerfeld integrals and their applications. Moscow: Rumb., 1981 Marichev, 0.1.: A method of calculation of the integrals of special functions (theory and tables of formulas). Minsk: Nauka i Tekhnika, 1978. Zbl. 473.33001 Oberhettinger, F.: Fourier transforms of distributions and their inverses. New York-London: Academic Press, 1973. Zbl. 306.65002 Paley, R.E.A.C., Wiener, N.: Fourier transforms in the complex domain. New York: American Math. Sot., 1934. Zbl. 11, 16 Shilov, G.E.: Mathematical analysis. A (second) special course. Moscow: Nauka 1965. Zbl. 97, 36. English transl.: Oxford: Pergamon Press. Zbl. 137,262 Sneddon, I.: Fourier transforms. New York: McGraw-Hill, 1950. Zbl. 38,268 Titchmarsh, E.C.: Introduction to the theory of the Fourier integrals. Oxford: Clarendon Press, 1937. Zbl. 17,404 Tranter, C.J.: Integral transforms in mathematical physics. London: Methuen, 1956. Zbl. 74,319 Uflyand, Ya.S.: Integral transforms in the problems of elasticity theory. Moscow: Nauka, 1963. Zbl. 126, 199 Vilenkin, N.Ya.: Special functions and group representation theory. Moscow: Nauka, 1965. Zbl. 144, 380 English transl.: Transl. Math. Monographs 22 (Am. Math. Sot., Providence 1968) Vladimirov, V.S.: The equations ofmathematical physics. Moscow: Nauka 1967; (2nd ed. 1971) (3rd ed. 1976) English transl.: New York Marcel Dekker Zbl. 207,91 Vladimirov, V.S.: Generalized functions in mathematical physics. Moscow: Nauka, 1976, 2nd ed. Moscow: Nauka 1979; Trans: Moscow: Mir 1979 Widder, D.V.: The Laplace transform. Princeton: University Press, 1941. Wiener, N.: The Fourier integral and certain of its applications. Cambridge University Press, 1933. Zbl. 6, 54 Zemanian, A.H.: Generalized integral transformations. (Pure and Applied Mathematics. Vol. 18) New York Wiley 1968. Zbl. 181, 127

Author Index Abel, N.H. 15 Ablowitz, M.J. 191 Abramowitz, M. 79 Adams, J.C. 42 Arnol’d, V.I. 189, 190

Fischer, Fourier, Froman, Froman,

\ Gambier, B. 188 Gauss, C.F. 69 Gel’fand, LM. 79,232 Gel’fand, S.I. 79 Gel’fond, A.O. 80 Gindikin, S.G. 127 Green, G. 145, 190 Gusein-Zade, SM. 189, 190

Bakhvalov, N.S. 190 Bateman, H. 231 Bemshtein, I.N. 79 Bieberbach, L. 79 Birkhoff, G.D. 190 Bochner, S. 231 Borel, E. 79 Bourbaki, N. 27,79 Brekhovskij, L.M. 231 Bremmer, H. 80 Brillouin, L. 145 Bromwich, T.J. 79 Bureau, F. 79 Biirmann, H. 37

Hadamard, J. 80 Hardy, G.H. 7,80 Hartman, P. 160 Hausdorff, F. 12,80 Heaviside, 0. 217 Heading, J. 190 Helgason, S. 232 Hilbert, D. 20 Hille, E. 190 Hironaka, H. 103 Hirschmann, 1.1. 232 Hopf, E. 214 Hormander, L. 212 Hurwitz, A. 12

Cauchy, A.L. 9,36,37,39 Cesaro, E. 79 Clarkson, P.M. 190 Coddington, E.A. 190 Copson, E.T. 190 De Brujin, N.G. 190 Debye, P. 94,190 Dienes, P. 79 Dirichlet, P.L. 198 Ditkin, V.A. 231,232 Dorodnitsyn, A.A. 180 Dzhrbashyan, M.M. 232 Edwards, R.E. 79 Erdelyi, A. 108, 190,231 Euler, L. 2, 10, 12, 15, 16,22,23,34, 42,43, 44,45,46, 69, 81 Evgrafov, M.A. 79, 168,190 Faber, G. 76 Fedoryuk, M.V. Fejer, L. 13,?9

E. 79 J. 17 N. 190 P.O. 190

Ince, E.L. Jeans, J.

35, 37

190 168

Kelvin, Lord 94, 190 Kirchoff, G.R. 217 Klein, F.C. 14 Knopp, K. 80 Kolmogorov, A.N. 18 Kramers, H. 145 Krull, W. 26,80 Kucherenko. V.V. 187 Kushnirenko, A.G. -80

168, 190 Lagrange,

J.L.

37

234 Landau, E. 21 Langer, R.E. 180,190 Laplace, P.S. 93, 190 Lavrent’ev, M.A. 232 Lebesgue, H. 8,9 Leont’ev, A.F. 80 Leray, J. 213,232 Levinson, N. 80, 159, 190 Lifshits, E.M. 232 Liouville, J. 145, 190 Luke, Y. 80 Maclaurin, C. 85 Magnus, W. 231 Malyuzhinets, G.D. 231,232 Marichev, 0.1. 232 Markushevich, A.I. 80 Maslov, V.P. 112, 174, 190 McLeod, J.B. 190 Mercator, N. 34 Mittag-Lelller, G. 13 Naimark, M.A. 190 Newton, I. 2, 29, 30, 31, 33 Ndrlund, N.E. 80 Oberhettinger, F. 232,232 Olwer, F.W.J. 80, 149, 177, 190 Painlevt, P. 188 Paley, R.E.A.C. 80,232 Panasenko, G.I. 190 Perron, 0. 159 Pitaevskij, L.P. 232 Planck, M. 168 Poincart, H. 29, 80,85 Pol, B. van der 80 Pblya, G. 6, 80 Prudnikov, A.P. 231,232 Ramanujan, S. 22,23 Rayleigh, J.W. 168 RiekstinS, E.J. 190 Riemann, B. 12, 16, 47, 69, 94, 190, 191 Riesz, F. 80

Author Index Riesz, M.

80

Schwartz, L. 80 Segur, H. 191 Seidel, L. 14 Shabat, B.V. 232 Shapiro, Z.Ya. 79 Shilov, G.E. 79,232 Sibuya, Y. 191 Sneddon, I. 232 Sommerfeld, A. 23 1 Stegun, I. 79 Steiltjes, T.I. 29 Steinhaus, H. 12 Stirling, J. 85 Stokes, G.G. 14,167,191 Struik, D. 80 Szeg6,G. 6,80 Tamarkin, Ya.D. 191 Titchmarsh, E.C. 80, 191,232 Toepler, A. 80 Toeplitz, 0. 11 Tranter, C.J. 232 Tricomi, F.G. 231 Uflyand, Ya.S. 232 Vainberg, B.R. 191 Varchenko, A.N. 189,190 Vilenkin, N.Ya. 80,232 Vinogradov, LM. 136 Vladimirov, V.S. 232 Walker, R. 81 Wasow, W. 191 Watson, G.N. 81,191 Weierstrass, C. 12, 14, 16 Wentzel, G. 145 Widder, D.V. 81,232 Wiener, N. 80,81,214,232 Wintner, A. 160 Whittaker, E.T. 81, 191 Zemanian, A.H. 232 Zygmund, A. 8 1

Subject Index I Abscissa of absolute convergence of Dirichlet series 48 - of convergence of Dirichlet series 48 Approximation, high frequency 145 - Liouville Green (LG) 145 - shortwave 145 -WKB 145 Asymptotic formula, doubly 152 - logarithmic of Laplace integrals 106 Bernoulli numbers

44

Composition of series, Hadamard 77 - Hurwitz 77 Condition, Meixsner on the edge 224 - of decrease of the solution 165 Conditions, Dirichlet 224 -radiation 153, 165 Continuation, analytic 16 Contour, Hijrmander staircase 212 - saddle 114 Contours, equivalent 212 Contribution of a boundary point 117 - due to a critical point 108 - due to the boundary 108,117 - of an interior simple saddle point 116 -of a point 97 Convergence in the algebraic sense 27 - in the arithmetic sense 27 - mean square 20 uniform 14 Convex hull 31 Convolution 201 Criterion, Cauchy 4 Critical strip 47 Curves, homotopic 72 Decay, Landau 223 Density of a sequence, lower 75 - maximal 75 - minimal 75 -upper 75 Dependence, non-regular, of an equation on a parameter 143

- regular, on a parameter 96, 141 - singular, on a parameter 96, 142 Diagonalization, asymptotic of systems 145 Disc of convergence of a series 15 Domain, canonical 168 -exterior 181 -interior 181 -j-admissible 175 - of annulus type 164 - of circle type 164 - of half-plane type 163 -of strip type 163 Equation, Airy 152 - - generalized 187 -characteristic 157, 158 -convolution 213 - evolution 220 -Fox 214 - Fredholm integral, of the first kind of convolution type 213 - - -, of the second kind 213 - functional 47 -heat conduction 126,205,221 - hypergeometric 69 - - generalized, 66 - Korteweg-de Vries 188 - Laplace 224 - of convolution type 213 - Painlevt 187 - - second 188 - parabolic 126 - - of second order 128 - Petrovski correct 204 - reduced Bessel 152 - Riccati 144 - Schrodinger 173,205 - -, one-dimensional 182 - Volterra integral, of the second kind of convolution type 219 - wave 206 - Wiener-Hopf 214 - with a small parameter in the highest derivative 143

Subject Index

236 - with rapidly varying Equations, Maxwell’s - Weber’s 182 Euler constant 42 Expansion, asymptotic

coefficients 209

143

87

Finite part of a function 24 Form, Leray-Gel’fand differential 105 Formula, Abel-Plana 57 -Cauchy 36 -commutation 194,19S,197,217,223,22S - Euler-Maclaurin 22,44 - for the coefficients of a power series 36 - Hankel’s 45 ~ Herglotz-Petrovskii 207 -inversion 194,195,215,216,223,225,229,230 - - for Fourier transform 17, 196 - Lagrange 38 - non-local asymptotic 95 - Parseval’s 17 - Plancherel 21 S - Poisson 58,205 - - summation 137 -shift 219 -Stirling’s 43, 100 Formulae, asymptotic 28 - connection 176 - Fourier 17 FSS elementary 168 Function, Airy 117, 152 - analytic 16 - Bessel 225 -- with imaginary argument 229 -beta 42 -delta 18, 202 -elementary 208 -error 92 - fundamental 202 -gamma 42 - - incomplete 92 - generalized 202 - - of slow growth 202 - Green’s 128,205,208 - - of a second order parabolic equation 128 - - difference 128 -harmonic 198 - Heaviside 218 - hypergeometric 70 -image 216 - Macdonald 228 -Mobius 47 - of bounded variation 18 - original 216

-phase 106 - plane wave 198 - Riemann zeta 46 -test 202 -theta 138 - transcendental 34 - unit error 128 - Weber’s 182 -Young adjoint 101 Functions, Airy-Fock - Legendre 229

Subject - oscillating 96 - with a weak singularity Kernel

166

General term of a series 4 Group, fundamental 72 - homotopy 72 - monodromy 72,73 Hausdorff means 12 Hessian 102 Homotopy class 72 Hypothesis, Lindeliif 48 - Riemann 47 Index of inertia 113 Integral, Dirichlet 204 - divergent 23 -error 130 -Fourier 90 - - contour 212 - Fresnel 92 -improper 7 -- absolutely convergent 7 - - convergent 7 - Laplace 90 - - contour 212 - - many-dimensional 97 - - one-dimensional 97 - Pearcey 133 - Poisson 93,200 - principal value 9, 112 -probability 130 - regularized 23 Integral representation, Euler 70 - Mellin-Barnes 71 - of the beta function 43 - of the gamma-function 43, 100 - Sommerfeld’s, for the Hankel function 121 Integrals, Fourier 17 - Fourier-Bessel 225 - Hankel contour 226 - Laplace 97 - of the type of the coefftcients of a Laurent series 123

of an integral

194,229

I-transform, Fourier ‘12 \ I-symbol 157 Law, Kirchoff’s 217 Lemma, Erdelyi’s 108 -Morse 102 - Riemann-Lebesgue 107 -Watson’s 98 Line, anti-Stokes 163 -Stokes 162,176 - - conjugate 163 Matrix, transition 169 Merging of saddle points 131 -of singularities 129, 133 Method, Abel-Poisson 12 - arithmetic means 10, 11 - Euler’s 10, 11 -Fourier 17 - Langer 180 - Laplace 59,93 - - many-dimensional 102 - of regularisation of integrals 23 - of standard equations 180 - of stationary phase 94, 106 - of undetermined coefficients 27,30 - operational 217 - regular summation 11 - steepest descent 123 - - - many-dimensional 124 - Wiener-Hopf 214 -WKB 145 Number wave - - longitudinal - - transverse

Quantization,

Parceval identity 226 Parceval’s equality 201 Partial sum of a series 4 Path, canonical 161

Bohr

Radius of convergence Resolvent 220

208 211 211

Operational calculus 59, 217 Operator, Bessel 225 - differential with constant coefficients - generating 196 - - first order 196 -- second order 196 - Helmholtz 208

237

- ( j, k)-canonical 175 - progressive 161 Phase 106 Phenomenon, Stokes 120 - of resonance 218 Point, critical 107 - saddle 114 - singular 164 - stationary, close to the boundary 130 -- nondegenerate 102 --oftypeI1, 111 -turning 145,155,157,164,177,180,181 - - of a system 183 - - simple of a system 183, 184 --secondary 181 Points, moving branch 188 Polygon, Newton 31 Principle, limit absorption 208 - of analytic continuation 16 - of localisation 97,106, 108 Problem, Cauchy, for linear ODE with constant coefficients 217 - -, for PDE 204 - -, for PDE with constant coefficients 126, 204 -fundamental, in linear asymptotic theory 141 -Lamb’s 228 - of classification of singularities 133

96, 134

transform

Index

204

Sommerfeld of a series

Sequence, asymptotic 87 - measurable 75 - summable by a matrix method Series, absolutely convergent 4 - asymptotic 29,87 - binomial 33 - Biirmann-Lagrange 37 - conditionally convergent 6 -convergent 4 - Dirichlet 46 - - generalized 49 - Euler-Maclaurin 44 -formal 25 - formal power 25 -Fourier 17 -function 14 - - convergent 14 -Gauss 69 - hypergeometric 69

rule 15

11

174

238 - Lagrange 37 - numerical 4 - Stirling’s 43 Singularity, power 134 Solution, elementary 208 -formal asymptotic (FAS) 143,144 -fundamental, of the Cauchy problem 204 - fundamental 207 Space of formal power series 26 - Schwartz 200 -2qR”) 200 -Y(R”) 200 Star, Mittag-Leftler 13 Sum of a series 4 Symbol of a differential operator 129 System of equations, Vlasov’s 221 Systems almost diagonal 159 -close to integrable 159 - L-diagonal 159 - I,,-Diagonal 160 Tensor, Green’s 210 Test, Abel’s 6 - Cauchy’s 5 --integral 5 - comparison 4 - -, for improper integrals 7 - D’Alembert’s 5 - Dini’s 18 - Dirichlet’s 18 - Leibnitz’s 18 - Weierstrass 15 Tests for convergence of Fourier series Theorem, addition 228 - an analogue of Birkhoff’s 175 Borel’s 154 - Fabry’s 76 - Fejtr’s 13 - Hadamard’s three circles 122 monodromy 72 - multiplication 219 - Norlund’s 154

Subject

126,

18

Index - Paley-Wiener 201 - PolyB’s 76,77 - Pringsheim’s 76,122 - Riemann’s 6 - Riesz-Fischer 19 - second decomposition 220 - Toeplitz’s 11 Theorems, Abel’s 15 - Weierstrass’ 15 Topology of formal series 26 Transform, Bessel 225 -Euler 230 - Fourier 49,123,196,226 - -, of a characteristic function 111 - -, of a generalized function 203 - - Stieltjes 199 - Hankel 225 - Hilbert 230 - integral 194 - Kontorovich-Lebedev 228 - Laplace 49,123 - -, two-sided 230 - Legendre 112 - Meijer 228 - Meller-Fock 229 - Mellin 49, 123, 223 - of convolution type 196,229 -Radon 214 --dual 215 - Sommerfeld 23 1 - Weierstrass 230 Transition 169,170 Trivial zeroes of the zeta function 47 Types of dependence on parameters 96 Value, regularized 23 Vector-path, j-canonical WKB-approximation WKB-asymptotic WKB-estimates Zeta function,

forms 149 Riemann

175 145 145

46