Chapter Twelve Asymptotic Location of the Zeros of Exponential Polynomials

----------CHAPTER TWElVE----------

Asymptotic location of the Zeros of Exponential Polynomials

12.1. Introduction

In our study of differential-difference equations in the earlier chapters, and in particular in our proofs of the fundamental theorems on expansions in series of exponentials, we placed great reliance on information concerning the location of the zeros of the characteristic functions of the equations. It will be recalled that these functions had the form h (s)

= aos + alse-ws

+ bo + b

w8

1e-

(12.1.1)

for scalar equations of first order, and h(s)

where H(s)

=

m

L

= det H(s), (Ais

+ Bi)e-

(12.1.2)

WiS

,

(12.1.3)

i~O

for the general systems of Chapter 6. These functions are entire functions of a special type, usually called exponential polynomials or quasi polynomials. The problem of analyzing the distribution, in the complex plane, of the zeros of such functions is one that has received a great deal of attention, since it arises in many fields of both pure and applied mathematics. There are several aspects of this problem which we shall investigate in this book. First, we need to establish sufficient information concerning the geometrical distribution of the zeros of h (s), as given in (12.1.1) or (12.1.2), to enable us to prove the expansion theorems and the theorems on asymptotic behavior given in Chapters 3-6. In particular, we must show that it is possible to construct the sequences of contours described there, and we must obtain adequate information concerning the order of magnitude of H-l(S) on these contours. This will be done in the present chapter. In discussions of stability, it is frequently true that one does not need to 393

394

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

know the actual location of the zeros, but only whether all zeros have negative real parts. See, for example, §4.5, §5.4, and Chapters 10 and 11. It is of great value to be able to predict stability or instability directly from the coefficients of the given equation. In Chapter 13 we shall present methods for making correct predictions of this kind. These methods are of great value in applications of the type occurring in the theory of automatic control. 12.2. The Form of det H(s)

We shall begin our discussion by calculating the form of det H (s) , where H(s) is given in (12.1.3). For the sake of a slight later convenience, we shall first multiply by eWmS, obtaining

G(s) = exp (wms)H(s) =

m

L

(Ais

i=D

m

L

+ B i) exp

(Am_iS

i=O

(am-iS)

+ B m- i) exp

(aiS), (12.2.1)

where ai

Note that

=

Wm

o=

-

ao

i = 0, 1, ... , m.

Wm-i,

< al < ... < am = Wm.

(12.2.2) (12.2.3)

For the special equation (12.1.1), G(s) reduces to the scalar function

g(s) = e",sh(s) = aose"'S It is evident that

+ als + boews + bi.

(12.2.4)

det G(s) = eNwms det H(s),

(12.2.5)

G-I(S) = e-wmsH-I(s),

(12.2.6)

where N is the dimension of the matrix H, and that det G(s) and det H(s) have the same zeros. If we now write down the matrix G(s), and calculate its determinant, we find that it has the form

g(s) = det G(s) =

n

L

j=O

pj(s)e{3;"

(12.2.7)

where each number {3j is a combination of ao, aI, ••• , am of the form {3j

=

m

L

i-O

kiia;.

(12.2.8)

12.3

395

ZEROS OF ANALYTIC FUNCTIONS

Here each k i j is a nonnegative integer and m

L: k i=O

ij

= N,

j

= 0, 1, "., n.

(12.2.9)

Each Pj(s) is a polynomial in s of degree at most N. We shall write PieS) = Pio

+ PjlS + .. ' Pjm;Sm;.

(12.2.10)

We shall order the {3j as follows:

°=

{3o

<

{31

< .'. <

(3n

= N wm·

(12.2.11)

With this convention, it is easy to see that Po(S) = det (Ams

+ B m) ,

pn(S) = det (Aos

+ B o) .

(12.2.12)

Thus Po(s) will have degree N if and only if det Am ¢ 0, and Pn(S) will have degree N if and only if det A o ¢ 0. We shall now analyze the location of the zeros of exponential polynomials of the form in (12.2.7). The fundamental idea of the methods to be used can be illustrated by considering the function o(s) in (12.2.4). Writing o(s) = sews[ao

+ (bois) ] + seal + (blls) J,

we see that o(s) has the form (if aOal o(s)

=

(aoseW 8

0)

¢

+ als) [1 + f(S) J,

where f(S) approaches zero as I s I --7 00. This suggests that the zeros of o(s) of large modulus will be approximately equal to those of the comparison function f(s) = aoseW + alSo 8

It is sufficient, then, to find a method of locating the zeros of f( s), and to show rigorously that the zeros of o(s) are asymptotically equal to those of f(s). In the next few sections, we shall carry out this idea for the general case of a function 9 (s) given as in (12.2.7). 12.3. Zeros of Analytic Functions

In this section, we shall present several basic theorems, from the theory of functions, concerning the zeros of analytic functions. Although many readers will no doubt be familiar with these, others may perhaps find it convenient to have them stated here. Let us suppose that f(s) is a regular analytic function of the complex variable s inside and on a closed contour C, and is not zero on C. Suppose

396

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

that 8 = a is a zero of order m inside C. Then in a neighborhood of this point, f(8) = (8 - a) mc/J(s) , (12.3.1) where c/J(s) is analytic and not zero. Hence

1'(8)

m

-=---+ f(s)

8 -

a

c/J'(s) c/J(s)

.

(12.3.2)

Since c/J'/ c/J is analytic at 8 = a, I' /f has a simple pole at s = a, with residue m. Hence, the sum of the residues of f'(s)/f(8) within C is equal to the number of zeros of f(s) within C, each counted as many times as its multiplicity. In other words, if f(s) is regular and has n zeros within C,

n =

2~i ~ ;~~~)

ds.

(12.3.3)

This result can be expressed in a form more suitable for many applications. Since

d - [logf(s)] ds

I' (s)

f(s) ,

(12.3.4)

the above formula asserts that n is equal to 1/ (21Ti) times the variation in value of logf(s) as s moves once around the contour C in the positive sense. Furthermore,

logf(s) = log If(s) I

+ i arg [f(s)].

(12.3.5)

Since log I f(s) I is a single-valued function of s, its variation around any closed contour is zero. On the other hand, as s varies continuously around C, the value of arg [f(s)], a multivalued function of s, may vary by a nonzero multiple of 21T. The variation in value is independent of the particular determination of arg [f(s) ] with which we start. In summary, we have proved the following theorem. Theorem 12.1. (Argument Principle). If f( s) is a regular analytic function inside and on a closed contour C, and is not zero on C, then the number of zeros of f(s) within C is equal to 1/21T times the variation of the argument of f(s) as s moves once around C in the counterclockwise sense. (It is understood that a zero of multiplicity m is counted m times.) In applying the argument principle, it is important that arg [f(s) ] varies continuously with s. There is another helpful way in which one can think of the argument principle. Consider the mapping of the complex s-plane onto tho complex w-plane by means of the relation w = f(s). This mapping carries each

12.3

397

ZEROS OF ANALYTIC FUNCTIONS

point s into a corresponding point w, and carries a closed contour C in the s-plane into a closed contour I' in the w-plane. Furthermore, the variation of arg [f(s) J, as s varies once around C, must equal 211" times the number of times I' winds around the origin in the w-plane. Therefore we have the following restatement of the argument principle. Corollary 12.1. If I( s) is analytic inside and on a closed contour C, and is not zero on C, then the number of zeros of f(s) within C is equal to the number of times the image curve of C under the mapping w = f(s) encircles the origin in the w-plane. For example, suppose that w = f(s) = (2s + 1)2, and let C be the unit circle in the s-plane. If we put s = e", we find that w

= (2 cos 8

+

1)2 - 4 sin 28

+ 4i sin 8(2

cos 8

+

1).

As 8 increases from 0 to 211", the point w moves around the curve in Fig. 12.1. Since the curve encircles the origin twice, the argument of f(s) evidently increases by 411" as s traces the circle C. Therefore, f(s) must

r

FIG. 12.1.

398

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

have two zeros within the unit circle. This, of course, checks with the fact that the only zero of f(s) is a double zero at s = -!. Let us give a simple example of the kind of problem which can be solved by means of the argument principle. We shall find the number of zeros of the function f( s) = 1 s S3 lying on or to the right of the imaginary axis. Let C represent the contour consisting of the semicircle e = re'", -71"/2 ~ fj ~ 71"/2, and the segment s = iy, -r ~ y ~ r. On the semicircle, we have

+ +

(12.3.6) Now Therefore, as fj varies from -71"/2 to 71"/2, the variation of arg (1 + r-2e-2 i6 + r- 3e:-3i6) is nearly zero, for r large enough. Hence arg [f(re i6) ] increases by nearly 371" along the semicircle. On the other hand, on the segment we have f(iy) = 1 + iCy - y3), argf(iy)

= tan-1 (y - y3) = tan" [y(l - y) (1

+

y)].

It is evident that f(s) is not zero when s is on this segment. As y varies from +r to -r, the function y - y3 varies continuously from nearly - 00 to nearly + 00, if r is large, and arg [f (iy)] increases by nearly 71". Thus the total variation of arg [f( s) ] around C is 471" (exactly 471", since it must be an integral multiple of 271" if C is a closed curve), if r is sufficiently large. Consequently f(s) has two zeros withpositive real parts. We shall conclude this section by stating two additional theorems. Theorem 12.2. (Roucht's Theorem). If f(s) and g(s) are analytic inside and on a closed contour C, and I g(s) I < I f(s) I for each point on C, then f(s) and f(s) + g(s) have the same number of zeros inside C. Theorem 12.3. The zeros of an analytic function are isolated. That is, if f( s) is analytic in a region, and is not identically zero, its set of zeros cannot have an accumulation point within the region.

EXERCISES

1. Calculate the number of zeros with positive real parts for each of these functions: (a) 1

+ s + S5.

12.4.

(e) 1

CONSTANT COEFFICIENTS: COMMENSURABLE EXPONENTS

S2

S3

2!

3!

399

+ s + - + -.

2. Use Rouche's theorem to show that f(s) = sn

+ ee'

has n zeros which approach the origin as e ~ O. 12.4. Constant Coefficients and Commensurable Exponents

We can now return to the problem of locating the zeros of a function of the form in (12.2.7) . We shall first consider the simplest special case, that in which the coefficients pj are constants, and the exponents {3j are commensurable, say, {3j = {3dj, where the d j are nonnegative integers (do = 0). Then g(s)

n

L

pj(ef3s)d;.

(12.4.1)

j~O

If we put z = ef3s, we get g(s)

=

n

L

pjZd;.

(12.4.2)

j~O

This is a polynomial of degree at most d; in z. Let Zk denote one of its zeros. Then clearly the zeros of yes) are given by the formula s

=

(3-1 log

Zk

=

~l[1og

1 Zk I

+ i(211"r + arg Zk)],

r = 0, ±1, ±2, ....

(12.4.3)

They are thus seen to lie in a finite number of chains (not necessarily distinct). Each chain consists of a countable infinity of zeros spaced 211"/ (3 units apart on a vertical line Re(s) = (3-1Iog [ Zk [. Among the notable features of this distribution of zeros are the following. First, all zeros lie within a certain vertical strip in the s-plane defined by inequalities (12.4.4) -Cl < Re(s) < Cl. Incidentally, this can be seen directly from (12.4.1), since, if Cl is large enough, the dominant term of y (s) is the one with largest exponent, ef3ns (if P» rf 0). That is, if for the moment pn denotes the nonzero coefficient with largest subscript, then

I pes) I ~ ct.!

Pn Cf3n,'

I.

Re(s)

~

Cl.

(12.4.5)

400

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

Hence I g(s) 1 has a positive lower bound if Re(s) ~ Cl. Similarly, g(s) has a dominant, nonzero term for Re(s) ::; -Cl. Second, let R denote any region, bounded or unbounded, which is uniformly bounded away from all zeros of g (s). That is, R lies at a positive distance Ca from the set of all zeros of g(s). Then there is a positive constant C4 such that (12.4.6) s E R. I g(s) 1 ~ C4, This is obvious if R is bounded. By (12.4.5), g (s) is bounded away from zero if 1 Re(s) 1 ~ ci. Therefore, the relation in (12.4.6) must hold if R lies within the horizontal strip 1 1m (s) I ::; 7r/ (3. Since g (s) has period 27ri/(3, it follows that the relation must hold for any R. 12.5. Constant Coefficients and General Real Exponents

We now consider a function of the form g(s)

=

L

°=

n

pje~i8,

j~O

(30

<

(31

< ... < e;

(12.5.1)

°

where we no longer assume that the d, are commensurable. Without loss of (j = 0, 1, ' .. , n) . We shall allow generality, we can suppose that pj ~ the Pi to be complex. In general, we can no longer reduce the problem to an algebraic one, as we did in §12.4. We shall accordingly have to abandon the attempt to provide explicit formulas for the zeros, and shall instead seck somewhat less precise information. Theorem 12.4. Consider a function g(s) of the form in (12.5.1) with

°=

There are positive numbers

Cl

(30

<

(31

< ... <

(12.5.2)

(3n.

and Cz such that all zeros of g(s) lie in a strip

(12.5.3) and such that, if pOpn

~

0, ~

Cz 1 e~n8

I,

Re(s)

I g (s) I ~

Cz 1e~08

I,

Re(s) ::;

1

g(s)

I

~

Cl,

(12.5.4)

-Cl.

(12.5.5)

To prove this, we merely observe that if CI is sufficiently large, there is a single term in g(s) of predominant magnitude in the region Re(s) ~ CI, namely the one having the largest (3. Likewise the term with smallest (3 predominates if Re(s) ::; -Cl.

12.5

CONSTANT COEFFICIENTS: GENERAL REAL EXPONENTS

401

Suppose now that we consider a rectangle R described by the inequalities

I Re(s) I ::::;

I Imrs)

Cl,

- a

I ::::;

b.

(12.5.6)

Since g(s) is analytic, the number of its zeros in R is certainly finite, by Theorem 12.3. Not only is this true, but the number is bounded as a -. 00, as we shall now show. This fact is needed in the construction of the contours C, used in Chapters 4-6. Theorem 12.5. Consider a function g(s) of the form in (12.5.1) with pj "" 0 (j = 0, 1, "',n) and 0 = {3o < {31 < ... < (3n. Let Rbearectangle oj the form in (12.5.6) such that no zeros oj g(s) lie on the boundary oj R. Then, provided Cl is sufficiently large, the number, n (R), of zeros of g(s) in R satisfies the inequalities

-n

+ (b/7r)

({3n - (3o) ::::; nCR) ::::; n

+ (b/7r)

({3n - (3o).

(12.5.7)

The upper bound holds for all Cl > O. To prove this theorem, we use the argument principle. That is, we estimate the variation in arg [g(s) ] as e traces the boundary of R. First consider the right-hand side, R 1, of R. If Cl is large enough, g(s) = Pnelln8[1

+ f(S) ]

on R l ,

where t f(S) 1 < 1. Since the point 1 + f(S) lies close to the point 1, we see that the change in arg [1 + f(S) ] is arbitrarily small. The change in arg [ellns] is 2{3nb. Hence the variation in arg [g (8) ] on R l is arbitrarily close to 2{3nb. Similarly, the change in arg [g(s) ] on the left-hand side of R is arbitrarily close to -2{3ob. We shall now calculate the change III arg [g(s) ] on a line Im rs) = constant. Let 8 = X + iy. We find that g(s) =

n

L

i=O

p/ + ip/" then qi = p/ cos {3jY - P/' sin {3iY,

qjelliX + i

n

L

i=O

rjelliX,

(12.5.8)

where, if Pi =

ri = p/ sin {3iY

+ Pi" cos (3iY.

(12.5.9)

Without loss of generality, we can suppose that P» is real, and therefore that ro = O. We shall show that a function of the form jn(X) =

n

L

dje'"li X ,

(12.5.10)

j~O

not identically zero and with the d, and "Ii real, can have at most n real zeros. If the "Ii are commensurable, this can be deduced from Descartes'

402

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

rule of signs. In the more general case, we proceed by induction on n. We can assume that each d, is nonzero. The assertion is clearly correct if n = 0, since do ~ O. Assuming that it is true if n = k, we consider fk+l(x). The zeros of fk+l(x) are the same as those of the function k+l

L a, exp [('Yi

-

'Yo)xJ,

i~O

which contains a constant term. Since the derivative of the latter function is of the same general type as in (12.5.10), with n = k, the derivative has at most k zeros. Therefore, fk+l (x) has at most k + 1 zeros. The real and imaginary parts of g(s) cannot both be identically zero, and cannot be simultaneously zero since g(s) is not zero on the line in question. If Im [g(s) J is identically zero on this line, arg [g(s) J does not change. If Im [g(s) J is not identically zero, it has at most n - 1 zeros, since it contains at most n nonzero terms. It follows that the variation of arg [g(s) J, as e traces a segment Im(s) = constant, lies strictly between n1r and -n1r. The total variation of arg [g(s) J around the rectangle R must thus lie between the extremes 2b({3n .,- (3o) ± 2n1r. The result in (12.5.7) follows from the argument principle. The upper bound holds for all CI, since diminishing CI can only decrease the number of zeros in R. Finally, we shall extend the result in (12.4.6) to the more general function in (12.5.1). Since we no longer have periodicity, the proof is more difficult. We first prove a lemma. lemma 12.1. Let s be a complex variable restricted to a closed, bounded region R., and let the real vector x be restricted to a closed, bounded region R~ of n-dimensional space. Suppose that f( s, x) is a continuous function of z in R~ for each s in R" and an analytic function of e in R. for each x in R~. Suppose further that there is a positive integer N, not depending on x, such that for each x in R~ the function f( s, x) has at most N zeros in R•. Then in any subregion of R. in which s is uniformly bounded from the zeros and from the boundary of R., f(s, x) is uniformly bounded from zero. It is to be proved that there exist positive numbers 0 and p., independent of x, such that if (s, x) E R. X R~ and if s lies at a distance at least 0 from the boundary of R. and from the set of zeros of f(s, x), then I f(s, x) I 2: p.. For each x E R"" let E(x) be the region R. minus the points less than a distance 0 from the zeros of f(s, x) and the boundary of R•. Let 0 be so small that E(x) is nonempty for each x E R~. This is possible since the number of zeros in R. is bounded uniformly in z, Let p.(x)

= min

,eE(x)

1

f(s, x)

I.

(12.5.11)

12.5.

CONSTANT COEFFICIENTS: GENERAL REAL EXPONENTS

403

Let So = so(x) be a value for which this minimum is attained, that is,

fJ.(x) = I f(so, x)

I.

(12.5.12)

We must show that }L(x) has a positive lower bound for x E R x • The hypotheses of our lemma ensure that the zeros of f(s, x) vary continuously with x. From this it can be proved that fJ.(x) is continuous in R x • It follows that fJ.(x) attains its lower bound fJ. at some point Xo. Since fJ. = fJ.(xo) = I f(so(xo) , xo) I, and so(xo) is not a zero of f(s, xo), we find that fJ. > 0, and this completes the proof. Theorem 12.6. Consider a function y(s) of the form in (12.5.1), with pj ~ 0 (j = 0, 1, "', n) and 0 = {30 < {31 < ... < (3n. Then if s is uniformly bounded from the zeros of y (s), I y(s) I is uniformly bounded from zero. By Theorem 12.4, there are positive numbers CI and C2 such that I y(s) I ;::: C2 if 1 Re(s) I ;::: CI. We therefore can restrict attention to the strip (12.5.3). Let R; be the rectangle

I Re (s) I ::;

I Im(s)

CI,

- a

I ::;

(12.5.13)

1.

Every point of the strip is in some R a • Make the change of variable z = e - ai, which maps R; onto R o, and let z = x iy. Then

+

y (s) =

n

L: pjefJizeafJii, j-O

There exist numbers OJ, 0 ::; OJ ::; 271", such that exp (a{3Ji) We let 0 denote (00, "', On), define

fez, 0) =

n

L: pjei6iefJiz,

(12.5.14)

j~

and have y(s) = fez, 0) for sERa. Note that 0 depends on a. For some small positive number 0, let R o' be the rectangle

I Re(s) I ::;

CI

+

0,

I Im(s) I ::;

1

+

0,

which encloses R o. We now apply Lemma 12.1 to the function fez, 0) with zERo' and 0 ::; OJ ::; 271". For each 0, the function is of the kind considered in Theorem 12.5, and therefore the number of its zeros in R o' has an upper bound independent of O. It follows from Lemma 12.1 that I fez, 0) I ;::: fJ. > 0 for z bounded by 0 from the boundary of R o' and the zeros of f. Here fJ. depends on 0 but not on O. If sERa and s is uniformly bounded from the zeros of y(s), then zERo' and is bounded by 0 from the boundary of R o' and the zeros of fez, 8). Consequently, I y(s) I ;::: fJ., sERa. Since fJ. does not depend on a, I y(s) I ;::: fJ. for all e in the strip I Re(s) I ::; CI for s uniformly bounded from zeros of y (s) .

404

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

It is worth noting, in conclusion, that the distance between distinct zeros of a function of the kind in (12.5.1) can have a zero lower bound. For example, consider the difference equation The characteristic function is Its zeros are

s = 2n7r'i/ WI

and

s = 2n7fi/ W2.

If WI/W2 is irrational, there are zeros arbitrarily close to one another. 12.6. Asymptotically Constant Coefficients

For later purposes, we must consider functions with asymptotically constant (complex) coefficients, of the form g(s) =

n

L j~O

pj[l

+ E(S) Jell j ",

o=

{3o

<

{3I

< ... <

e;

(12.6.1)

where pj ;F- 0 (J' = 0, 1, ... , n). Here, we shall use the symbol E(s) as a generic symbol * for a function, analytic in a neighborhood of OCJ, such that lim

Is I~co

I E(S) I =

O.

(12.6.2)

Later on, we shall sometimes consider values of s lying only in certain subsets of the extended complex plane including the point at infinity. Then we shall require that the relation in (12.6.2) holds for points in these subsets, rather than in a full neighborhood of OCJ. It is reasonable to suppose that the zeros of g(s) are very close to those of the comparison function gI(S) =

n

L

pjell j8

(12.6.3)

j~

if I s I is large. Since the latter function is of the sort analyzed in §12.5, the nature of its zeros is described in Theorems 12.4, 12.5, and 12.6, and we can expect similar results for g (s). In the first place, Theorem 12.4 applies without alteration to g (s), since the same proof applies. Secondly, * That is, E(S) denotes a member of the class of functions, analytic in a neighborhood of

co, which satisfies the relation in (12.6.2). However, it may denote different members of

the class at different occurrences.

12.6

405

ASYMPTOTICALLY CONSTANT COEFFICIENTS

within the strip I Re (s)

I ::;

CI,

g ( s)

each exponential efJj S is bounded, and

= gl ( S)

+

(12.6.4)

E ( s) .

If s is uniformly bounded from the zeros of gl (s), bounded from zero by Theorem 12.6. Hence

I gl (s) I is

uniformly

(12.6.5) As we know, the lower bound of the distance between zeros of gl(S) may be zero. However, by Theorem 12.5, we can divide the plane into horizontal strips so small that each contains at most n zeros. If we place a circle of radius 0 about each zero, with no less than the width of a strip, then for large I s I not more than n circles can overlap. That is to say, the zeros can be enclosed in groups of at most n by a sequence of contours which are uniformly bounded from the zeros, and of diameter no greater than no. By applying Rouehe's theorem, we can see from (12.6.5) that, for I s I sufficiently large, the number of zeros of g(s) within each such contour equals the number of zeros of gl (s). Since 0 is arbitrary, we may accordingly say that the zeros of g(s) are asymptotically equal to the zeros of gl(S). It can also be seen that th~ bounds in (12.5.7) apply to g(s) as well as to gl(S). Finally, if s is uniformly bounded from the zeros of g(s), it is also uniformly bounded from the zeros of gl(S) if lsi is large. By Theorem 12.6 and Equation (12.6.5), I gl(S) I and I g(s) I are uniformly bounded from zero. We have proved the following result. Theorem 12.7. Consider a function g(s) of the form in (12.6.1). There exist positive numbers CI and C2 such that all zeros of large modulus lie in a strip I Re (s) I ::; CI, and such that

I g(s) I 2: C2 I efJns I, I g (s) I 2: C2 I efJos I,

Re(s) 2:

Re(s) ::;

(12.6.6)

CI,

(12.6.7)

-CI.

The zeros of g (s) are asymptotic to those of the comparison function gl (s) in (12.6.3). In any rectangle I Re(s) I ::; CI, I Im(s) - a I ::; b, in which I s I remains sufficiently large, and on the boundary of which g(s) has no zeros, the number, n(R), of zeros of g(s) satisfies the inequalities -n

+ (b/1r)

«(3n -

(30) ::; nCR) ::; n

+ (b/1r)

Finally, if s is uniformly bounded from the zeros of g(s), bounded from zero.

«(3n -

(30).(12.6.8)

I g(s) I is uniformly

406

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

12.7. Polynomial Coefficients with

mi

and

Proportional

~i

We are now in a position to analyze functions of the form in (12.2.7), g(s) =

n

L

j=>O

Pi(s)e13is,

°=

{3o

<

{31

< ... <

(3n,

(12.7.1)

having polynomial coefficients. Let mi denote the degree of PieS). Then we can write

°

g(s)

n

=

L

Pismi[1

i~O

+ f(S) ]e13i8,

(12.7.2)

where Pi ~ (j = 0, 1, ... , n). In (12.7.2), f(S) prove the following lemma.

O(ls [-1). We first

Lemma 12.2. All zeros of sufficiently large modulus of a function g(s) of the form in (12.7.2) lie within an arbitrarily narrow sector about the imaginary axis. In fact, for any 0, < < 7r/2, there is a Cl > such that

° °

and

I arg s ] ~ I arg s

In the sector I arg s [ ~ 0, we have I s !smiel3i81 = O{exp [({3i

°

I~

+ 0) Re(s)]l,

Ca Re

i

(12.7.3)

0,

- 7r

I~

0.

(12.7.4)

(s), and therefore

= 0, 1, ••. , n - 1,

for every 0 > 0. It follows that g (s) is dominated by the term in which n. Similarly, in I arg s - 7r I ~ 0, g(s) is dominated by the term with i = 0, since

i =

Ismiel3i8! = O{exp [({3j - 0) Re(s)]l,

i

=

1,2, ..• , n.

Before proceeding any further with the general case in (12.7.2), we find it expedient to examine what appears to be a rather different case. Specifically, we consider a function of the form in (12.7.2) in which there is a real number m such that

°

j = 0, 1, ... , n.

(12.7.5)

We continue to assume that = {3o < {31 < ... < (3n, but now we allow the ni, to be any integers, positive, negative, or zero. The function g(s) now takes the form g(s) =

L"

j=O

piC1

+ f(S)] exp [(3j(s + m log s)],

(12.7.6)

where log s denotes the branch which is zero when s = 1. If m = 0, this

12.7

mj, (3i PROPORTIONAL

POLYNOMIAL COEFFICIENTS:

407

function is of the form considered in §12.6, and the zeros are described in Theorem 12.7. We accordingly restrict attention to the case m ¢ O. The form of (12.7.6) suggests the change of variable Z =

s

+ m log s.

(12.7.7)

°

Furthermore, by virtue of Lemma 12.2, we can restrict attention to a region R. in which 0 < 1 arg e 1 < 1r - 0, < 0 < 1r/2, and in which lsi is large. Within R., the function of s defined in (12.7.7) is analytic and simple, and defines a one-to-one mapping of R. onto a region R, in the z-plane. Within R z , there is a unique analytic inverse function s = s(z). The image region R, is easily described. If we write e = x + iy, z = u + iv, we find that u = x

+ m log

= y

+ m arg

v

1

+ iy I, + iy).

x

(x

(12.7.8)

On a side of the sector 0 < [arg s ] < 1r - 0, we have y = ex. It is easy to see that on the image of this line, v/u ----> e as x ----> 00. Therefore, the image of the sector is asymptotically a sector. It is also clear that as I e I ----> 00 within R., I z 1 ----> 00 'within R z , and vice versa. Consequently, an analytic function of the type e (s) in R. is an analytic function of the type feZ) in R; Under the mapping in (12.7.7), the function y(s) in (12.7.6), with m ¢ 0, becomes a function fez) of the form fez) =

n

L

+ feZ) Jell;z

p;[1

=

;~o

n

L

;=0

p;[1

+ feZ) Jem;zlm.

(12.7.9)

There is a one-to-one correspondence between the zeros of large modulus of y(s) in R. and the zeros of large modulus of fez) in R z • If m ¢ 0, fez) is of the form in (12.6.1). The location of the zeros of fez) can be described with the aid of Theorem 12.7. In fact, still more precise results are obtainable, because the numbers m, are by assumption integers. The comparison function of fez) is fl(Z) =

L n

;-0

p;em;zlm

L n

pj(ez1m)m;.

(12.7.10)

;~o

If m > 0, this is a polynomial in eZlm of degree m n. If m < 0, e(-mnZ/m) fl(z) is a polynomial in ez / m , of degree 1m.; I. To each complex root w of this polynomial, there corresponds a chain of roots

z = m log [w I

+ im

(arg w

+ 2r1r),

r = 0, ± 1, ... ,

of h(z). The zeros of fez) are asymptotic to those in (12.7.11).

(12.7.11)

408

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

To obtain a description of the zeros of g(s), it is now only necessary to translate our information in the z-plane back into the s-plane. In the first place, the zeros lie asymptotically along curves Re(z) = Re(s + m log s) = constant. Let us first obtain a qualitative picture of these curves. Lemma 12.3. A curve Re(s the following characteristics.

+ m log s)

=

c, m r" 0, in the s-plane has

(a) It is symmetric with respect to the real axis. (b) If s = x + iy lies on the curve, I ylx [ ~ Is[~

<1)

and

I arg s ] ~

(c) The curve is asymptotic to the curve x + m log I y I = c. (d) If m > 0, the curve lies entirely in a left half-plane, and Re (s) asls[~<1).

(e) If m +<1)

7r/2 as

<1).

<

~

-

0, the curve lies entirely in a right half-plane, and Re(s) <1).

aslsl~

Property (a) is clear. The equation x solved for y to yield

+m

log I x

+ iy I

=

<1)

~

c can be (12.7.12)

If m

> 0 ( < 0), then x is bounded above y = ±e(c- xl!m[1

(below), and as I x I ~

+ 0(1)].

<1)

(12.7.13)

From this, we see that the curve is like a simple exponential curve for I x I large. Properties (d) and (e) are now clear. Property (b) follows readily from (12.7.13). Finally, since the curves x + m log I s ] = c and x m log I y I = c approach parallelism with the y-axis, the distance between them is asymptotically the same as the horizontal distance between them. Let (X2) y) be a point on the former and (Xl) y) a point on the latter. Then

+

X2 -

Xl

= m log [ y1s I.

Since [ y1s I ~ 1, X2 - Xl = a (1), proving (c). We shall also use the following lemma:

Lemma 12.4. If Sl and S2 are points in the curvilinear strip I Re (s + m log s) I :::; Cl, and if I Sl I ~ <1), I S2 I ~ <1), in such a way that I Sl - S2 I 2:: o > 0, then the corresponding points Zl and Z2 under the mapping z = s + m log s tend to infinity in the strip I Re(z) I :::; Cl in such a way that I Zl - z21 2:: 0/2. More briefly, boundedness of distances from zero is preserved by the mapping. To show this, let s, = x, iYiJ Zj = Uj iVj (j = 1,2). We can suppose

+

+

12.7

409

POLYNOMIAL COEFFICIENTS

that these points lie in the upper half-planes. Then U2 =

Ul VI -

V2

=

(Xl -

(Yl -

+m

X2)

Y2)

log

+ m(arg SI

I SI/S21, -

arg

S2).

If I Yl - Y2 I ~ 0, then I ZI - Z2 I ~ I VI - V2 I ~ 0/2, since arg SI and arg S2 both approach 1r/2. If I u, - Y2 I < 0, the ratio Yl/Y2 must be arbitrarily close to one. Hence (sri S2) must also be arbitrarily close to one. Then I Zl - z21 ~ lSI - S2 I - I m log (SI/ S2) I ~ lSI - S2 [/2 ~ 0/2. By Equation (12.7.11), the zeros of f(z) lie asymptotically along a finite number of vertical lines Re(z) = constant, an asymptotic distance 21rm units apart. It follows that the zeros of g(s) lie asymptotically along a m log s) = c. In fact, if we put s = x iy, finite number of curves Re(s we get x + m log [ s I = Re (z) = m log I w [ + 0 (1) ,

+

Y

+ m arg s

+

= Im(z) = m (arg w

+ 2r1r) + 0(1),

where w has the meaning in (12.7.11). Since arg s ----7 ±1r/2, by Lemma 12.3, we have (12.7.14) y = m(2r1r arg w 'F1r/2) 0(1). Also,

I s I = [y I [1 + 0(1)]. x

,+

+

Therefore,

+ m log I y I =

m log [ w

I + 0 (1) .

Using the result in (12.7.14), we get x

= m(log I w I - log I 2r1rm + m arg w 'F (m1r/2) \)

+ 0(1).

(12.7.15)

From Theorem 12.7 and the above discussion, we have the following conclusion. Theorem 12.8. Consider a function g(s) of the form

g(s)

n

=

L

Pj[1

j~O

in which pj

~

+ E(s)Jsmielli',

0 (j = 0, 1, "', n) and

o=

{:lo

<

{:l1

< ... <

{:In,

(12.7.16)

and in which the m, are integers such that j = 0, 1, "', n,

for some real number m. If m

=

0, the location of the zeros is as described in

410

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

Theorem 12.7. If m rf 0, the zeros are asymptotic to those of the comparison function gl(s) =

n

L

PjSmje{jj8.

(12.7.17)

j~O

They lie asymptotically along a finite number of curves, I smesl = constant, of the type described in Lemma 12.3. The roots of large modulus in one of these chains have the form Re(s)

=

m(log I w

Im(s) = m[2r1l"

I-

log j2nrm

+ arg w =F 11"/2)

+ m arg w =F (m1l"/2) + 0(1),

\)

+ 0(1),

where w is a constant. For any m, if s is uniformly bounded from the zeros of g (s), I g (s) I is uniformly bounded from zero. Finally, let R denote a curvilinear rectangle

I Re(s + m

log s)

I :::;

CI,

I Im(s + m

log s) - a

I :::;

b.

(12.7.18)

If there are no zeros of g(s) on the boundary of R, and if n(R) denotes the number of zerosin R, then -n

+

(b/1I") ((3n - (30) :::; n(R) :::; n

+ (b/1I")

((3n - (30).

(12.7.19)

12.8. Polynomial Coefficients

Weare now ready to discuss the general case of polynomial coefficients. Consider a function of the form

g(s) =

L n

j-O

pjsm j[1

+ f(S) ]e{jjS,

o=

(30

<

(31

< ... <

(3n,

(12.8.1)

where Pi rf 0 (j = 0, 1, "', n) and where each m, is a nonnegative integer. With such a sum we now associate a polygonal graph in a Cartesian plane, by plotting the points Pi with coordinates Uh mi). The points determine a polygonal line L which (a) (b) (c) (d)

joins Po with P n , has vertices only at points of the set Ph is convex upward (or straight), and is such that no points P, lie above it.

A typical case is illustrated in Fig. 12.2. The graph obtained in this way will be called the distribution diagram of the exponential polynomial, since it provides information about the distribution of zeros. We shall find that terms of (12.8.1) corresponding to points below L

12.8

411

POLYNOMIAL COEFFICIENTS

P4

P3 L3

P2

P5

L2

L4 P6

•

Po ~o

P8 P7

131

f32

134

~3

136 f37

~5

138

FIG. 12.2.

cannot affect the asymptotic distribution of the roots, whereas terms corresponding to points on L do. Let the successive segments of L be denoted by L I , L 2, " ' , Lo, numbered from left to right, and let the slope of L; be denoted by Por. We now construct in the s-plane a number of curvilinear strips, VI, V 2, " ' , Vk , defined by the inequalities

v.. I Re(s + Por log 8) I ::;

CI,

r

= 1,2, "', k.

(12.8.2)

Each of the strips with Por 7'" 0 is bounded by curves of the type discussed in Lemma 12.3, and each strip is of retarded, neutral, or advanced type according as PoT is positive, zero, or negative. We shall show that all zeros of large modulus lie within one of these strips, and that the zeros in V r are asymptotically those of the comparison function comprised of those terms associated with points on LT' We first make a few remarks about the strips Yr. It is evident that f.lr is a decreasing function of r. From the equation in (12.7.12), it follows that the strips are disjoint, for large lsi, and that V r+1 lies to the right of V r for each r. In Fig. 12.3, we indicate pictorially the appearance of the s-plane corresponding to the diagram in Fig. 12.2. We shall denote the region between V r- I and V r by UT- I (r = 2, "', k), the region to the left of VI by Us, and the region to the right of V k by Ui. Given an arbitrarily narrow sector about the imaginary axis, all s of sufficiently large modulus in any strip V r are within the sector. The regions U; can be defined by the inequalities

Us: Re(s Uo: Re(s Us: Re(s

+ f.lT log s) > + f.ll log s) < + f.lk log s) >

Cl, -Cl.

Re(s

+ f.lr+llog s) <

-eX, (12.8.3)

Cl..

We shall now show that there are no zeros of large modulus in any U';

412

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

FIG. 12.3.

Theorem 12.9. There are positive constants Cl and C2 such that no zeros s with I s I 2: c2lie in I), (r = 0, 1, "', k), and such that within U'; one term of g(s) is of predominant order of magnitude, namely, the one corresponding to the point of the distribution diagram at the right end of the segment L; If this point is relabeled P r , then there is a positive constant C3 such that

(12.8.4) To prove this theorem, we observe that for any s E U, (r k - 1), there is a number fJ. such that

+

Re(s

+ fJ. log s)

= 0,

+

1,2, "', (12.8.5)

since Re(s fJ.r log s) 2: Cl > 0 and Re(s fJ.r+l log s) ::::; -Cl < O. The relation in (12.8.5) holds for r = 0 if the double inequality on fJ. is replaced by the single inequality fJ.r+l < fJ., and it holds for r = k with

12.8

413

POLYNOMIAL COEFFICIENTS

JL < JLr. These modifications lead to trivial alterations in the statements which follow. From the relation in (12.8.5), we obtain

j = 0, 1, •.. , n.

(12.8.6)

Thus the number m; - JL{3j is a measure of the magnitude of the jth term in g(8) at a point 8 satisfying (12.8.5). The number m, - JL{3j has a clear geometric meaning. It is the y-intercept on the distribution diagram of the segment of slope JL passing through the point P'; But if JLr+1 ~ JL ~ JLr, it is evident geometrically that the highest such intercept must correspond to one or more of the points on L; or L r+1. Points not on either of these segments have lower intercepts. In fact, let P p denote the point lying at the intersection of L; and Lr+i, and let Pi denote any point not on either L; or Lr+i' Then for each JL in the range JLr+i ~ JL ~ JLr, the intercept corresponding to P, lies above the intercept corresponding to Pi' The distance between these intercepts is a function of JL which has a positive minimum for JLr+1 ~ JL ~ JLr. From this it is clear that terms not corresponding to points on L; or L r+1 are of lower order of magnitude as I 8 I ~ 00, 8 E Us, than the term corresponding to P'; We can write

Pi

~

t;

U

L r+1 .

(12.8.7)

We shall now show that one of the terms in (12.8.1) is predominant, namely, the one corresponding to P'; Consider the ratio

for any other term "on" L; or L r+1. If the term is "on" Li; then m; JLr({3p - (3j), and J =

I exp

[({3p -

(3j) (8

ni,

+ JLr log s)J I·

Since Re(s + JLr log s) ~ C1 for s E U': and {3p > {3;, we have J exp [({3p - (3j) cr]. Similarly, if the term is "on" L r+1, then J

= I exp

[({3p -

(3j) (s

=

+ JLr+i log s) J I·

>

Since Re(s + JLr+1 log s) ~ -C1 and {3p - {3i < 0, we get J ~ e(P;-{3p)C1. Therefore, if we choose C1 large enough, depending on the constants pj and {3j, we can ensure that the magnitude of the term corresponding to the point ({3p, mp) exceeds the sum of the magnitudes of all other terms on

414

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

L, and L r+1• This establishes the inequality in (12.8.4) and completes the proof of the theorem. We now are left with the problem of describing the zeros of large modulus in each of the strips Yr. The strip V r consists of points s which satisfy a relation (12.8.8) Re(s + J.1.r log s) = c, Hence (12.8.9) f J ic. where C4 = e From the relation in (12.8.9), we at once see that the terms corresponding to points on L, are of higher order of magnitude than the other terms. Consequently,

L

g(s) =

Lr

pjsmi[l

+ f(S) ]efJ i8,

S

E

v.,

(12.8.10)

where the sum is taken over all terms corresponding to points on Ii; For convenience in writing, let us denote the points on L r , from left to right, by P rh (h = 1,2, "',
L a

+ f(S)] exp

prhsmrh[l

h=1

= smrlefJr18

L q

Prhsmrh-mrJ[l

((3rhS)

+ f(S)] exp

[({3rh - (3rl)S].

(12.8.11)

h~l

But since the points P rh all lie on a segment of slope

J.1.r,

h = 1, "',
(12.8.12)

That is to say, the exponents of e and powers of s in (12.8.11) are proportional, and the sum in (12.8.11) is of the type considered in Theorem 12.8, or Theorem 12.7 if J.1.r = o. These theorems therefore provide the description of the zeros of g(s), which we now summarize. Note that in V r , if s is uniformly bounded from zeros, I g (s) s-mr 1 exp ( - f3rlS) I is uniformly bounded from zero. 8

Theorem 12.10. Let g(s) be an exponential polynomial of the form

g(s) =

n

L j~O

Pi(s)e fJ i8 =

n

L

j=O

Pism{1

+ f(s)]e fJ i8,

(12.8.13)

where the mj are nonnegative integers, Pi ;t. 0 (j = 0, 1, ... , n), and 0 = f30 < (31 < ... < f3n. Let the s-plane be divided into regions VI, V 2 , Vk and Uo, U 1, • Uk in the manner described above. Outside a certain circle I s I = C2, the following statements apply. 000,

0

0,

(a) There are no zeros of g(s) in U; (r = 0, 1,

000,

k). If e E Us,

12.8

415

POLYNOMIAL COEFFICIENTS

I g(s) s-me-IJ, I is

uniformly bounded from zero, where smelJ' is the term of g(e) corresponding to the point of the distribution diagram at the right end of the segment L; (b) The zeros of g(s) in V r are asymptotic to those of the comparison function (12.8.14) pjsmielJi', gl(S) =

L Lr

where the notation means thal the sum is taken over all terms corresponding to points on Ii; In any region R,

I Re(s + !J.r log s) I ~

C1,

I rm(s + !J.r log s)

- a I ~ b,

(12.8.15)

with no zeros of g(s) on the boundary, the number of zeros in R satisfies

1 - n;

+

(b/7r) (3 ~ nCR) ~

(b/7r) (3

+ n; -

1,

(12.8.16)

where n; ie the number of points of the distribution diagram on L r and (3 = (3ru - (3r1 is the difference in values of (3 at the end-points of LT' (c) In any subregion of V r in which e is uniformly bounded away from all zeros, I g(s)s-me-IJ'I is uniformly bounded from zero, where smelJs corresponds to the point at the left end-point of L r • (d) In any retarded or advanced strip V r , the zeros of g(s) lie asymptotically I = constant, and are described along a finite number of curves Is~re' by asymptotic formulas of the type in (12.7.14) and (12.7.15).

We remark also that g(s) can have at most a finite number of zeros of multiplicity greater than n, since by taking b small enough we get nCR) < nr~n+l.

It is particularly noteworthy that all roots in retarded or advanced chains are asymptotically determinate. That is, such roots lie in distinct chains, and can be represented by definite asymptotic formulas. On the other hand, roots in a neutral strip need not be asymptotically determinate. For example, consider the equation 1 e' e'r8 = O. However, the roots in a neutral strip V r certainly will be asymptotically determinate if the numbers (3rh - (3r1 (h = 1, ..• , iT), corresponding to the points P rh on the associated segment L r , are commensurable. This is true, in particular, if all the (3j are commensurable to begin with, or if L; contains only two points. The latter cases are the ones most frequently arising in applications. Suppose that we let

+ +

(12.8.17)

416

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

as in Equation (12.2.10). Then as a corollary of the preceding theorem we note the fact that for all k and j

I Pikske13;8/g(S) I = 0(1)

Isl----7

as

rYJ,

(12.8.18)

provided s is uniformly bounded away from the zeros of g (s), because the import of (a) and (c) of Theorem 12.10 is that the plane can be divided into regions in each of which g(s) has the order of the one of its terms of largest order, provided s lies uniformly away from the zeros. 12.9. Examples

We shall now illustrate the preceding theory with several examples of practical interest. First consider the scalar, first-order equations discussed in Chapters 3-5. For the equation of retarded type, aou!(t)

+ bou(t) + bluet

- w)

=

we have h(s) = e-"'Sg(s) , where

+ boe"'s + b

g(s) = aose""

l

= aose"'8[1

(12.9.1)

0,

+ (bo/aos)] + b-,

(12.9.2)

The distribution diagram contains the points (0, 0) and (w, 1), showing that there is a single chain of roots of retarded type. Applying Theorem 12.8 directly, with m w-r, we find that the roots have the asymptotic form x

=

w- l

[ log

I -bllao I - log

= w- l [1og I -bllao I y

= w- l[2k7r

+ arg

-

2k7r

+ arg

log 2k7rw- l ]

(-bllao)

=r 7r/2]

+

(-brlao) w

=r 7r /2J + 0(1),

0(1),

+ 0(1),

(12.9.3) (12.9.4)

where k is any integer of large magnitude. The upper sign applies to roots for which y ----7 rYJ, the lower sign to those for which y ----7 - 00. Improved asymptotic formulas can be found by iteration (see Miscellaneous Exercises below). For the equation of neutral type,

+

aou'(t)

+ alu'(t

-

co )

+ bou(t) + bluet

- w) = 0, (12.9.5)

we have ( 12.9.6)

12.10

417

CONDITIONS THAT ALL ROOTS BE SPECIFIED

The distribution diagram contains the points (0, 1) and (w, 1), showing that all roots of large modulus lie in a neutral strip. We can also obtain an asymptotic formula for the roots in this case, since there are only two points on the distribution diagram. The roots are asymptotic to those of the comparison function Yl(S)

= (aoe""

+ al)s

(12.9.7)

and therefore have the form S

I-

= w-1/log

adao

I + i[2k1l" + arg

(- adao) J

+ 0(1)}.

(12.9.8)

As another example, we consider the equation u"(t)

+ au'(t) + bu(t) + cu(t -

w) = 0,

(12.9.9)

which arises in the study of a vibrating system with an external controlling force proportional to displacement, but subject to delay. The characteristic function is h (s) = e-Wsg (s), where y(s)

= (S2

+ as + b)e + c.

(12.9.10)

ws

The distribution diagram contains the points (0, 0) and (w, 2), showing that there is a single chain of retarded roots. Under the substitution Z = s + 2w- 1 log s, the Iunctionp (s) is transformed into fez) = ewz [ l

+ feZ) J + c.

(12.9.11)

+ 0(1), and the zeros of y(s) are w-1[log I c I - 2 log 12kw- 1 IJ + 0(1),

Therefore, »z = log (-c)

=

x

y

=

given by (12.9.12)

1l"

w- 1 [ 2k1l" =F

11"

+ arg

(-c)J

+ 0(1).

(12.9.13)

EXERCISE

Find the asymptotic form of the characteristic roots of each of the following equations: (a) u"(t) (b) u"(t)

+ au'(t - w) + bu(t - w) = O. + au(t) + bu"(t - w) + cu'(t -

w)

+ duet

- w)

= o.

12.10. Conditions That All Roots Be of Specified Type

In our discussion of the general system m

L

i=O

[Aiy'(t - w,)

+ Biy(t

- Wi)J

= 0

(l? 10.1)

418

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

in Chapter 6, we found important qualitative differences depending on whether the various root chains of h(s) = det H(s) were of retarded, neutral, or advanced type. In this section, we shall develop a few simple conditions which ensure that no chains are advanced, or all are retarded, etc. In §12.2 we found that

h (s) = det H (s) = exp ( - N wms) det G(s) = exp (- N wms) g (s) , (12.10.2)

where

G(s) = ewmsH(s) a; =

Wm -

m

L: (Am-is + Bm_i)e

Ui' ,

i=O

i

Wm-i,

= 0, 1, •.. , m.

(12.10.3) (12.10.4)

The zeros of h (s) are the same as the zeros of g (s) = det G (s). As shown in §12.2, g(s) has the form g(s)

n

L: pj(s)efJ;s,

(12.10.5)

j~O

where 0 = 130 < 131 < ... < 13n, each 13 is a combination of a's, and each pj(s) is a polynomial of degree at most N. Also,

pn(S) = det (Aos

+ B o).

(12.10.6)

Let m, denote the degree of Pj(s). Then the distribution diagram is constructed from the points (13;, mj). A necessary and sufficient condition that all chains be of retarded or neutral type is that the point (13n, m n) be at least as high as every other point, in other words, that m n 2:: m, for j = 0, ••• , n - 1. This is certainly the case if m; = N. From (12.10.6), we see that mn = N if det A o ~ O. Thus we have Theorem 12.11. A sufficient condition that all root chains of (12.10.1) be of retarded or neutral type is that det A o ;e O. It is, of course, possible to have det A o = 0 and yet to have all chains of retarded or neutral type. However, the notational difficulties involved in setting forth a general necessary and sufficient condition directly in terms of the coefficients in (12.10.1) would be great. We shall now obtain a simple sufficient condition for all mot chains to be retarded. 7heorem 12.12. A sufficient condition that all root chains of (12.10.1)

12.10

419

CONDITIONS THAT ALL ROOTS BE SPECIFIED

be of retarded type is that det A o matrices. Under the stated conditions, G(s) = B m + Bm_lealS

0, whereas AI, A 2 ,

¢

"',

Am are all zero

+ ... + B l earn-l' + (Aos + Bo)eams. (12.10.7)

If we now compute det G(s), we find that in the form obtained in (12.10.5), the coefficient det (Aos + B o) of ei3ns is a polynomial of degree N, whereas coefficients of all other ei3jS are of degree at most N - 1. Therefore, the point ({3n, m n ) on the distribution diagram is higher than every other point. Because of the convexity of this diagram, its segments must therefore all have positive slope, and all root chains must be retarded. Also we note that for the class of systems in which at least one polynomial Pi(S) is of degree N, the conditions in Theorems 12.11 and 12.12 are both necessary and sufficient. EXERCISES

1. Show that det Am ¢ 0, A o = Al = ... = A m - l = 0, is a sufficient condition that all root chains be of advanced type. 2. For the two-by-two system ur'(t) Ul'(t

+ Ul(t) + U2(t - w) = 0, - w) + U2(t) + Ul(t - w) + U2(t -

w) = 0,

for which det A o = 0, show that

G(s) =

(S [

_

+ l)e""

s

+1

Hence show that the system is neutral, and show that all roots are given by s = -1 and the double chain of roots of e2"'. + e" - 1 = O. See Exercise 1, §6.4. 3. Show that if A o = 0 and det B o ¢ 0, the system in (12.10.1) has at least one advanced root chain. 4. Consider the general two-by-two system of the form in (12.10.1), with one lag (m = 1), and det A o ¢ 0. Show that a necessary and sufficient condition that all root chains be retarded is that det Al = and

°

I

au

0

= 0,

420

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

where ail denotes the element in the ith row and jth column of A k (i, j = 1, 2; k = 0, 1). 5. Under what conditions are all characteristic roots of the scalar equation m

n

L L i=O

ai/u(j)(t - Wi) = 0

j~O

of neutral or retarded type, or all of retarded type? (See §6.9.) Assume that the conditions in (6.9.4) and (6.9.5) are satisfied. 12.11. Construction of Contours

(l

We shall now establish the existence of integration contours C, = 1, 2, •.. ) of the type used in Chapters 4-6.

Theorem 12.13. Let g (s) be an exponential polynomial of the form discussed in Theorem 12.9,

g(s) =

n

L

pj(s)el';".

(12.11.1)

j~O

For any sufficiently small number k, there exists a sequence of closed contours CI (l = 1, 2, ... ) and a positive integer l« such that (a) CI contains the origin in its interior. (b) c, C CI+I (l = 1,2, ... ). (c) The contours C I have a least distance, d, greater than zero, from the set of all zeros of g(s). (d) For l ~ lo, the contour CI lies along the circle I s I = kl ~f s does not lie in one of the strips Yr. If s lies in a strip V r, Ci lies between I s I = (l - l)k and I s I = (l l)k. (e) The total length of the parts of Cl within strips V r is bounded as

+

l ----7 00. (f) If l ~ lo, the number of zeros of g(s) between C, and C/+I is at most 2n.

Choose k smaller than 1r({3n - (30)-1. Suppose lo is so large that all zeros outside the circle I s I = lok lie in one of the strips Yr. We construct Ci along I s I = lk (l ~ lo), outside the strips Yr. Inside the strips v., it may not be possible to let C, lie along the circular are, which may be arbitrarily close to zeros. Instead, we make a small detour. This detour can be restricted to lie entirely within the curvilinear rectangles R,

I Re (s +

J.Lr log s)

I :::

CI,

I Im(s + J.Lr log s)

- lk =F J.Lr (1r/2)

I :::

k/8,

(12.11.2)

12.11

421

CONSTRUCTION OF CONTOURS

FIG. 12.4.

provided lo is large enough. In fact, we shall let C I follow the nearly vertical sides of the rectangle, together with a nearly horizontal path, * along Im(8 + /l-r log 8) = constant, through the rectangle. (See Fig. 12.4, in which R is the region, within the strip, bounded by dotted lines.) It is always possible to choose a path of this kind which is bounded away from all zeros by a distance independent of land r, since the number of zeros in the rectangle R is bounded uniformly in land r (and the strips V r are disjoint if lo is large enough). For 1 ::::; l ::::; lo, choose any convenient contours satisfying (a) and (b) of the theorem and not passing through any zeros. It is now clear that the first four statements of the theorem are correct. Statement (e) is correct, since the length of C, within R is less than the perimeter of R if lo is large, and the perimeter of R is asymptotically equal to 4CI + k/2. To prove statement (f), observe that within V r , C, and C I+I both lie within the curvilinear rectangles

I Re (8 + /l-r log 8) I ::::;

I Im(8 + /l-r log 8)

- (l

+ t)k T

CI,

/l-r

(11"/2)

I ::::;

k.

(12.11.3)

By Theorem 12.10, the number of zeros within each rectangle is no greater than and this is less than n; by choice of k; that is, the number of zeros between Ci and C I +I in the upper half-plane and within V r is strictly less than the

+

+

* On Im(s 1", log s) = c, we have y J1.r arg 8 = C, so that y = C 'F J1.r(7r,/2) The upper sign in (12.11.2) is used for the rectangle in the upper half-plane.

+ 0(1).

422

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

number of points on L; Hence the total number of zeros between C I and C 1+1 in the upper half-plane is less than the total number of points on the distribution diagram, and so is less than or equal to n. 12.12. Order Results for forI (s)

We shall now derive some results on the order of magnitude of H-1(S) as I s I - 7 00. In subsequent sections, we shall use these results to discuss the convergence properties of the integrals fe tsH-1(s) ds that were needed in proving the expansion theorems of Chapters 4-6. Theorem 12.14. A ssume that the expansion of det H (s) contains at least one term containing the Nth power of s, where N is the order (dimension) of H. If s lies in a left half-plane Re (s) ::; Cl and outside a suificiently large circle I s I = C2, and is uniformly bounded away from zeros of det H (s), then

II H-1(S) II

I

= O{ S-l exp [ - (N -

In a right half-plane Re(s)

~

If it is also assumed that det A o

~

0(1 S-leN w s l) . ffl

(12.12.1) (12.12.2)

0, the latter relation can be replaced by

II H-1(S) II = ~

IJ.

C1,

II H-l(S) II =

If it is assumed that det Am

l)wm s]

0(1 s

1-1) .

(12.12.3)

0, the relation in (12.12.1) can be replaced by (12.12.4)

By assumption, the distribution diagram for G(s) contains at least one point with ordinate N. Let this point be denoted by ({3r, N). From the equation in (12.8.18) we obtain

11/g(s)

I=

0(1 S-Ne- fJ r8 l) ,

(12.12.5)

in any region uniformly bounded away from zeros. In particular, if det A o ~ 0, (12.12.6) On the other hand, we can estimate the order of magnitude of the cofactors in G(s). Each of these is a determinant of order N - 1, and its expansion contains s to at most the power N - 1. In any left half-plane, the exponential factors are bounded. In any right half-plane, the exponential factor of greatest order is the one with the largest value of {3, at most (N - l)w m • Hence each cofactor is 0(1 s IN-1) in a left half-plane, and

12.13

01lsN-1 exp [(N ~ 1)w ms]

II G-1(S) 1I = II G-l(S) II =

423

ORDER RESULTS IN THE SCALAR CASE

0(ls-1e-13r
II in

a right half-plane. Thus

= O(/S-le- N wm< I),

Re(s)::;

Cl,

Olls-l exp [(N - 1)wms - .arS] II

= Olls-lexp [(N -

1)wm s]

I},

(12.12.7)

Re(s) ~

Cl.

Since H-1 (s) = e (s), we obtain the relations in (12.12.1) and (12.12.2) . If det A o ;= 0, we can take s, = NWm in the second part of (12.12.7) and obtain the improved estimate in (12.12.3). If det Am ;= 0, we can take = 0 in the first part of (12.12.7) and obtain the improved estimate in (12.12.4). W m
.ar

12.13. Order Results in the Scalar Case

The results of the preceding section can be considerably improved if the matrix H-1(S) arises from a scalar equation of the form m

n

i-O

i~O

L: L: ai!u(i) (t

- Wi) = 0,

t> wm •

(12.13.1)

Referring to Chapter 6, we find that in this case H (8) has the form H(s)

s

-1

o

0

o

s

o

0

o

0

s

-1

(each sum is taken from i = 0 to i = m). In this case we can obtain the following theorem.

(12.13.2)

Theorem 12.15. Assume that in (12.13.1) at least one of the numbers ami (} = 0, "', n) is not zero, at least one of the numbers o « (i = 0, "', m) is not zero, and m > 0, n > O. Let II be the smallest value of i for which a« ;= O. If e lies in a half-plane Re(s) ::; C1 and outside a sufficiently large

424

circle I s

12.

I=

C2,

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

and is uniformly bounded away from zeros of det H (s), then \1

H-I(S)

In a right half-plane Re(s) ::::

II =

O(\sl-I).

(12.13.3)

0(1 S-Iew,. \).

(12.13.4)

CI,

II H-I(S) II =

If aOn ,e 0, the relation in (12.13.3) holds for Re (s) :::: CI also.* In a left half-plane, Re(s) ~ CI, the dominant terms in the cofactors of H(s) are sn-\ multiples of sn- 2e- wm', or terms ainsn-Ie-Wi8. But det H(s) certainly contains a term of order at least Sn-Ie-w m., since not all amj are zero, and also terms ainsne-WiS. Hence, for s uniformly bounded from all characteristic roots,

II H-I(S) II

= 0(1 s 1-1),

Re(s)

~

CI.

(12.13.5)

In a right half-plane, the dominant term in the cofactors of H(s) is sn-I. Let v be the smallest value of i for which ain ,e 0. Then [det H (s) J-I = o (! s-neW" I). Hence

II H-l(S) II =

0(1 S-IeW,s!).

Re(s)::::

CI.

(12.13.6)

In particular, if aOn ,e 0, II H-I(S') II = O(lsl-I) for all s uniformly bounded from all characteristic roots and such that I s I :::: cz. 12.14. Convergence of Integrals over the Contours

In this section, we shall assemble and prove various results on contour integrals that we have used in previous chapters. The first theorem deals with the scalar equation m

n

i=O

j=O

L L

aiju(j) (t - Wi)

=

0.

(12.14.1)

As usual, we assume that at least one of the numbers amj (j = 0, "', n) is not zero, at least one of the numbers ain (i = 0, "', m) is not zero, and m > 0, n > 0. Theorem 12.16. Let {Cd (l = 1,2, ... ) be the sequence of contours constructed in §12.11, and let H (s) denote the matrix function associated with the equation in (12.14.1). t Let Ci: denote the intersection of Ci with a

°

* The assumption amn 7'" does not lead to an improved estimate in Re(s) ~ CI. At first glance, this seems contrary to the assertion in (12.12.4), but since u'nn 7'" does not imply det Am 7'" 0, there is no contradiction here. t The matrix H(s) is given explicitly in (12.13.2).

°

12.14 half-plane Re (s) Then

c, and let CZ+ denote the intersection of C, with Re (s) ;::: c.

~

lim l-+oo

425

CONVERGENCE OF INTEGRALS OVER CONTOURS

f

II etsH-I(s) III ds[ =

C l-

0,

(12.14.2)

t> O.

°

The convergence is absolute and uniform in every finite interval to ~ t ~ to' (to> 0), and bounded in < t ~ to'. It is uniform in to ~ t < 00 (to> 0) provided c ~ O. Also

lim I~ro

f

Gl+

II etsH-I(s) III ds I =

0,

t <

(12.14.3)

-W p ,

where 1I denotes the smallest value of i for which ain ~ 0. Here the convergence is uniform in to ~ t ~ to' < -W p, and in - 00 < t ~ to' < -W p if c ;::: 0. Consider first the integral in (12.14.2). Choose any fJo > 0 and let II denote the integral over the portion of C 1- on which Re (s) ~ - fJo log I s I, and 12 the integral over the portion on which Re(s) > -fJo log I s ]. Because of the manner in which the contours C, were constructed, they are uniformly bounded from the zeros, and II H-I(S) II = O(lsl-I), as stated in (12.13.3). Moreover, the length of C,_ in Re(s) ~ - fJolog I s I is O(ls [), and in -fJolog lsi < Re(s) ~ 'c is o (log Is [). Hence

111111 = O(etRe(s)/!s!)O(lsl)· Since Re(s) --7 - 00 as lsi --7 00 in Re(s) ~ -fJo log lsi, II --7 0 if t > O. It is clear that the convergence is absolute and uniform in to ~ t < 00 (to> 0), and bounded in t > O. Also

etRe(S»)

111211 = 0 (-

Is I

0 (log lsi) = 0

(eet log lsi

lsi)

,

t

>:

O.

Thus 12 --7 0 if t ;::: 0, absolutely and uniformly in 0 ~ t ~ to', and in t < 00 if c ~ O. Combining the statements about II and h we obtain the stated result concerning the limit in (12.14.2). The proof of the relation in (12.14.3) proceeds along the same lines, using the order estimate in (12.13.4). We shall now derive corresponding results for the general system

o~

m

L

i=O

[Aiy'(t - Wi)

+ Biy(t

- Wi)]

= 0

(12.14.4)

of dimension N. Theorem 12.17. Let

I Czl

(l

1, 2, ... ) be the sequence of contours

426

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

constructed in §12.1l, and let

L m

H(s) =

i=O

(A;s

+ B;)e-w;s.

(12.14.5)

Assume that the expansion of det H (s) contains at least one term containing the Nth power of s. Let Ci: be the intersection of C, with Re(s) :::; c, and let C z+ be the intersection with Re (s) :2: c. Then

lim

l_co

f

Cl-

II e tsH-1(s) II

[dsl

t

= 0,

>

(12.14.6)

(N - l)wm •

The convergence is absolute and uniform in every finite interval to :::; t :::; to', to > (N - l)w m, and bounded in (N - l)wm < t :::; to'. It is uniform in to :::; t < 00, to > (N - 1) W m, provided c :::; O. Also

lim

L-e- co

f

Cl+

II etsH-l(s) III dsl

= 0,

t <

-v«;

(12.14.7)

The convergence is uniform in to :::; t :::; to' - N Wm , and in - 00 < t :::; to' < -tt»; if c :2: o. If det A o .,t. 0, the relation in (12.14.7) is valid for t < 0, and if det Am .,t. 0, the relation in (12.14.6) is valid for t > -w m • The proof is like that of Theorem 12.16. In dealing with the integral over C 1_ , we again choose any J./, > 0 and let II denote the integral over the part of C 1- on which Re(s) :::; -J./, log Is I, and 1 2 the integral over the. part on which Re(s) > -J./, log lsi. Using Equation (12.12.1), we obtain

111111 = 0 (exp {[t -

(N -

l)wm J Re(s)}),

111211 = 0 ( exp {[t -

(N -

l)wm J Re(s)

I

log [ s

I)

-I-s-' .

The result in (12.14.6) follows. The proofs of the other results are similar but employ the order estimates given in (12.12.2), (12.12.3), and (12.12.4). In the next section, we shall consider integrals of e tsH-1(s) over lines Re(s) = c. EXERCISES

1. Write out the proof of the relation in (12.14.3).

2. Write out the proof of the relation in (12.14.7).

12.15

427

INTEGRALS ALONG VERTICAL LINES

12.15. Integrals along Vertical Lines

The purpose of this section is to establish the results on integrals of the form

f

K(t) =

(c)

1

eI 8H-l(s) ds = lim - . T_oo 2n

fC+iT

eI 8H-l(s) ds,

c-iT

(12.15.1)

which were used in Chapters 3-6. We begin with the general system, so that

H(s) =

m

L: (Ais + Bi)e-

wiS

(12.15.2)

,

i~

but assume that det A o ~ 0, since we used integrals of the form in (12.15.1) only in the retarded-neutral cases. In Chapter 6, we proved convergence of the integral in (12.15.1) by an indirect method. Here we shall use some theorems from mathematical analysis to provide a more direct proof, and to supply information on the magnitude of K (t) . First we consider the case in which all root chains are retarded. That is, we suppose that det A o ~ and that the function h(s) = det H(s) contains but one term in SN. Restricting s to lie on a vertical line Re (s) = C, we have

°

h(s) = (det Ao)SN

+ 0(1 s IN-l),

I Im ts)

[~

'Xi.

(12.15.3)

On the other hand, each subdeterminant of H (s) of order N - 1 will be of the form (12.15.4 ) where d, =

m

L: i~

m

nijWi,

""n··=N-1 £..., 1,3 ,

nij ;:::

0,

(12.15.5)

i~

and where each qj(s) is a polynomial of degree N - 1 at most. Therefore each element in H-l(S) can be written as a sum of terms of the form cis-le-dj8, and of terms that are 0([ s 1-2) as I Im(s) I ~ 'Xi, where c, is a constant and d.is a number in the set Sodefined by the relations in (12.15.5). The integral in (12.15.1) can therefore be written as a sum of integrals of the forms

f

(c)

exp [(t - dj)s] - - - - - - ds s

and

f

el80 ([ s [-2) ds.

(12.15.6)

(c)

Provided C ~ 0, and provided no zeros of h(s) lie on the line Re(s) = c, the latter are evidently uniformly convergent for t in any finite interval,

428

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

+

bounded by a multiple of ec t for all t, and uniformly convergent as t -+ 00 if c < 0. The first integral in (12.15.6), as is well known, is uniformly convergent over any finite interval except in the neighborhood of the point t = dj, boundedly convergent over any finite interval, and uniformly convergent as t -+ 00 if c < 0. It is also bounded by a multiple of eel. If c = 0, and no zeros of h(s) lie on the line Re(s) = c = 0, the same conclusions hold. Indeed, the contour can be shifted slightly to a line Re (s) = c', without affecting the value of the integral, since the integrals over horizontal crossbars

+

approach zero as 11'[ -+ 00 by virtue of the fact that H-l(S) = 0(1 S 1-1) when c' ::; Re(s) ::; c, I Im(s) 1-+ 00. It is evident that the introduction of an extra factor s into the denominators in (12.15.6) will guarantee uniform convergence over any finite interval. Thus we have proved the following result.

°

Theorem 12.18. Assume that det A o ~ and that all root chains of det H(s) are retarded. If no characteristic roots lie on the line Re(s) = c, the integral

K(t) =

f

et'H-l(s) ds

(c)

converges for all t. The integral converges uniformly on any finite interval except in the neighborhood of points of So, and boundedly on any finite interval. If c ::; 0, the convergence is uniform as t -+ + 00 • Moreover, there is a constant Cl such that

(12.15.7) The integral

f

(12.15.8)

e tsH-l(s)s-l de

(c)

+

converges uniformly for t in any finite interval, and for t -+ 00 if c ::; 0. If we assume that there are neutral root chains, the discussion becomes considerably more difficult. Instead of (12.15.3), we now have a relation h(s) = SN

L 71

j-O

hje-k;s

+ O(IS!N-l),

I Im(s) I -+

00,

(12.15.9)

12.15

429

INTEGRALS ALONG VERTICAL LINES

where each k, has the form kj

m

= L nijWi,

m

nij 2: 0,

i=O

(12.15.10)

Lnij = N. i~O

The li, are constants, at least one of which is not zero. Let hl(s) =

L n

h je- kj8.

(12.15.11)

j~O

This function is the comparison function for h(s) in the neutral strip. Denote by ;ml the set of all real parts of zeros of h (s), and by ;m the union of ;ml with the set of all limit points of ;mI. In the neutral-retarded case under consideration, there is at least one finite limit point. The complementary set of ;m on the real line consists of a countable set of open intervals, one of which has the form (x, + 00 ), where x is the least upper bound of the set ;m. We now assume that c EE ;m. Let fI denote the open interval to which c belongs. Then no zeros of h(s) and at most a finite number of zeros of hl(s) lie within fI. We suppose for the moment that no zero of hl(s) lies on the line Re (s) = c, and let fI I denote a subinterval of fI such that the strip Re (s) E fI l contains no zeros of hi (s) . The function hl-I(s) is in fI l an analytic, almost periodic function and possesses an absolutely convergent generalized Dirichlet series of the form hl-I(S) =

L OJ

k=-co

h/e- lkS,

(12.15.12)

where the sequence {lk1 contains all numbers in the set

S =

E

{t I t =

n.,

niWi,

(12.15.13)

integer}.

=

We can suppose the lk are arranged in a monotone increasing sequence, with lo = O. Therefore, within fI l , each element of H-I(S) can be written as S-1

L

L OJ

qj

dj

h/ exp [- (d j

k~-OJ

+ lk)S] + 0(1 S 1-

2

dj

qj

f

L OJ

(e) k~-OJ

h/

exp [(t - d·J - lk)S] S

ds

I Im(s) I --->

00.

(12.15.14)

Hence each element of K (t) has the form

L

) ,

+

f (e)

etsO(1 s C

E

1-

2

)

fI l .

Since the series in (12.15.12) converges absolutely for s E

fI l ,

ds, (12.15.15)

it converges

430

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

uniformly in y [y = 1m (s)]. Hence

f

L

C- H ~ 2 C+i~l

co

h/

exp [(t - d· - lle)s]

ds

1

S

k~co

L co

=

k=-co

hie'

. :

a, -

exp [(t -

lle)s]

8

c+i111

ds

(12.15.16)

for any '171 and '172. Consequently, using the uniform-bounded convergence properties of the integrals

a, -

f

exp [(t lle)s] - - - - - - - - de,

we get

f

L CC

h/

(c) Ie=-cc

=

L cc

Ie--cc

exp [(t - d· - lle)s] 1

S

'f

hie

exp [(t - d j

ds lle)s]

-

S"

(c)

(12.15.17)

s

(c)

c

ds,

~

O. (12.15.18)

Also, the absolute value of this expression is no greater than co

L I hie' I exp [c(t

-

Ie~co

a, -

lie)] = D(e ct ) .

Once again, the conclusion is valid for all c E £I, since we can shift the contour slightly if it contains a zero of hI (s) . We accordingly have the following theorem: Theorem 12.19. Assume that det A o ~ 0 and that h(s) has a neutral root chain. Let mI. denote the set of real parts of zeros of h (s), together with their limit points, and suppose that c ~ mI.. Let S denote the set of points

S =

{t I t =

i:

niWi, ni =

t=O

Then the integral K(t) =

f

integer}.

(12.15.19)

e t 8H-l(s) ds

(c)

converges for all t. The integral converges uniformly on any finite interval except in the neighborhood of the points of S, and boundedly on any finite

12.15

431

INTEGRALS ALONG VERTICAL LINES

interval. There is a constant

CI

such that

( 12.15.20) The integral

f

e

t8H-I(s)s-1 ds

(e)

converges uniformly for remains uniform as i ->

t in any finite interval. If c :::; 0, the convergence

+

00.

We also need to discuss convergence of integrals of the form

f

e

(e)

8tH-I(s) [,m-Wi

exp [- (t l

0

+ Wi)S]g(t l) .u, ds.

(12.15.21)

If g(tl ) is of bounded variation, we can integrate by parts with respect to u, and the extra factor s in the denominator will guarantee uniform convergence. However, the same result can be derived if we merely assume that g is continuous for 0 :::; tl :::; W m - Wi. To prove uniform convergence for all t, it is enough to prove convergence of

foo I H-I(e + iy) jWm-Wi ~xp

[ - (il

0

-00

+ Wi) (e + iy) ]g(t l)

(12.15.22)

or, since H-I(S) = 0(1 S 1-1) on Re(s) = c, of

i

I gl(e + ~y) Idy, el

-00

where gl(e

+ iy)

=

t: o

exp [ - (t l

dtll dy,

(12.15.23)

+ 1,y

+ Wi) (c + iy) ]g(tl)

(12.15.24)

dt l.

Furthermore, since

[f OO I gl(e e -00

+ .iy) I d y J2 -< + 1,y

foo

1

-00

gl

(e

+ i Y) 2dY fOO 1

-00

e

2

+y

dy

2'

(12.15.25)

it suffices to prove convergence of

fOO

-00

I

gl(e

+ iy)

2

1

(12.15.26)

dy.

But glee

+ iy)

=

fw

Wi

m

exp [ -t2(e

+ iy) ]g(t

2 -

Wi) di«.

(12.15.27)

432

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

+

Thus gl(c iy) is the Fourier transform of a function fl (t 2 ) , which is zero outside the interval (Wi) wm ) and equal to e-e2 tg (t2 - Wi) within it. According to the Plancherel-Parseval theorem, §1.15, we therefore have

r;o

I

gl(c

-00

+ iy)

12

dy = 27r

fCC

I

-ro

fl(t 2 )

12

dt 2

This establishes the following theorem. Theorem 12.20. Suppose the conditions of Theorem 12.18 or Theorem :::; Wm - Wi. Then the integral in (12.15.21) is uniformly convergent for t in any finite interval, and is 0 (eet) as t ~ 00 • Turning to the scalar equation 12.19 are satisfied, and suppose that gCtI) is continuous/or 0 :::; t 1

+

m

n

i=O

;"=0

LL

aijUU) (t -

Wi)

=

(12.15.28)

0,

we find that an improvement of Theorem 12.18 is possible. Looking back at the form of the matrix H (s), in this case in which aO n ~ 0 but ain = 0 (i = 1, ... , m), we see that h(s) = aons n

+ O([sln-I),

I Im(s) I ~

00.

(12.15.29)

The subdeterminants of order n - 1 in H(s) contain constant multiples of sn-l and terms that are 0(ls!n-2) as [ Im(s) I ~ 00, but no terms of the form Sn-le-w i (Wi> 0). Hence 8

H-I(S) = cs- I

+

0([sl-2),

I Im(s) ! ~

00.

(12.15.30)

Consequently we can assert the following: Theorem 12.21. For the scalar equation in (12.15.21) in which aO n ~ 0, ai; = 0 (i = 1, ... , m), the statements of Theorem 12.18 remain correct if the set So is replaced by the set containing the single point t = O. Miscellaneous Exercises and Research Problems 1. Suppose that A(z) and B(z) are exponential polynomials, A(z) aoe"'OZ ale"'!Z ame amz and B(z) = boe{3Oz ble{3jZ

+

+ ... +

bne{3nz, where the a., b i ,

+

Oli,

=

+ ... +

{3i are complex constants. Prove that if

MISCELLANEOUS EXERCISES AND RESEARCH PROBLEMS

433

B(z) is not identically zero and if A(z)IB(z) is an entire function, then A (z) I B (z) is itself an exponential polynomial. (J. F. Ritt, "On the Zeros of Exponential Polynomials," Trans. Amer. Math. Soc.,

Vol. 31, 1929, pp. 680-686.)

Other proofs are given by H. Selberg, "Uber einige transzendente Gleichungen," Avh. Norske Vid. Akad. Oslo. I, No. 10, 1931. P. D. Lax, "The Quotient of Exponential Polynomials," Duke Math. J., Vol. 15, 1948, pp. 967-970.

2. The conclusion of the preceding exercise remains true if A(z)IB(z) is merely analytic in a sector of opening greater than 71". (J. F. Ritt, "Algebraic Combinations of Exponentials," Trans. Amer. Math. Soc.,

Vol. 31, 1929, pp. 654-679.)

+ ...

3. Let fez) = 1 + ale"lZ + ame"m z, where 0 < al < ... < am, am .,t- O. Let R (u, v) be the sum of the real parts of those zeros of fez) for which u < y < v, where u, v are any real numbers with v > u. Then

R (u, v) = _ (v - u) log I am I + 0 (1) , 271"

where 0(1) is bounded for all u, v. (J. F. Ritt, "Algebraic Combinations of Exponentials," op. cit.)

+

4. Let n be a positive integer and let A (x) = 1 L'I a.e»>, B(x) 1 + L1 b i e13 ix, with no b, equal to zero, where the ai, b i , a i, (3i are complex constants. If A(x) is divisible by B(x), then every (3 is a linear combination of aI, " ' , an with rational coefficients. (J. F. Ritt, "A Factorization Theory for Functions ~aie"iX,"

Soc., Vol. 29, 1927, pp. 584-596.)

Trans. Amer. Moth,

5. Let EP denote the class of exponential polynomials with polynomial coefficients, fez) = LPi(z)e a i z with the Pi(z) polynomials in z. Let E denote the class of exponential polynomials with constant coefficients, P i(Z) = constant. Show that if f is in EP, g is in E, and h = fig is entire, then h is in EP. If f is in E and g is in EP, h need not be in EP. (A. L. Shields, "On Quotients of Exponential Polynomials," Notices Amer. Math. Soc., Vol. 6, No.7, 1959, Abstr. 564-184.)

6. Consider the system of two equations (a) y(x (b) y(x

+ 1)y'(x)2 + 71"2ehixy(X) + 2)

- y(x)

=

O.

= 0,

434

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

Show from the second equation that y(x) has period 2 and that y(x)y(x 1) has period 1, and deduce from the first equation that y(x + I)2y'(x)2 has period 1. Hence show that for every solution either

+

(c) y(x)y'(x

+ 1)

- y(x

+ I)y'(x)

+

+ y(x +

= 0

or (d) y(x)y'(x

1)

l)y'(x)

= O.

Show from (b) and (c) that, except for the zero solution, y(x

+

l)jy(x)

= ±I,

and, using (a), that y (x) = ±ie"ix. From (d) deduce that y (x) y (x is a constant, say, c2, and from (a) obtain

+ 1)

= c exp (± e"ixj c) .

y (x)

This example was given by Fritz Herzog in a paper devoted to a decomposition theory for algebraic differential-difference equations patterned after the decomposition theory for algebraic differential equations due to J. F. Ritt. (Fritz Herzog, "Systems of Algebraic Mixed Difference Equations," Trans. Amer. Math. Soc., Vol. 37, 1935, pp. 286-300. J. F. Ritt, Differential Equations from the Algebraic Standpoint, Amer. Math. Soc. Colloq. Publ., Vol. 14, 1932. J. F. Ritt and J. L. Doob, "Systems of Algebraic Difference Equations," Amer. J. Math., Vol. 55, .1933, pp. 505-514.)

Other papers on exponential polynomials are W. Bouwsma, "The Greatest Common Divisor Property for Exponential Polynomials," Notices Amer. Math. Soc., Vol. 6, 1959, Abstr. 564-3. F. W. Carroll, "Difference Properties for Polynomials and Exponential Polynomials 011 Topological Groups," Notices Amer. Math. Soc., Vol. 6, 1959, Abstr.564-179.

7. Consider the characteristic equation ee: + a = O. Establish the following results. If a > e-l, the roots of the equation occur in conjugate, complex pairs, Sp, sp, where 2p7r

< tp <

(2p

+

If a < cl, the same is true except that real roots, (TO', (To, with

o>

(TO'

>

-1

>

log

p

I)7r,

a

80,

>

= 0, 1,2, ....

So are replaced by two

0"0.

435

MISCELLANEOUS EXERCISES AND RESEARCH PROBLEMS

If a

= e-l, then

!TO, !TO'

are replaced by a double root at

8

= -1.

(E. M. Lemeray, "Sur les racines de l'equation x = a"'," Nouv. Ann. Math.,

(3), Vol. 15, 1896, pp. 548-556; Vol. 16, 1896, pp. 54-6l. E. M. Lemeray,·"Le quatrieme algorithme naturel," Proc. Edinburgh Math, 80('., Vol. 16, 1897, pp. 13-35.)

8. If a

<

7r/2, every root has its real part negative. If a ~ 7r/2, then as p ~ (2a - 11'")/411'". In particular, there are two roots with real part zero if and only if a = 2k7r + 7r/2 for some nonnegative integer k. Also !To = 0 for a = 7r/2, and !To > 0 for a < 11'"/2. v»

Z0 according

(E. M. Wright, "A Non-linear Difference-differential Equation," J. Reine Angew. Math., Vol. 194, 1955, pp. 66-87.)

9. The quantity !Tl' decreases steadily as p increases, so that for all p ~ O. For large p, !Tp

2P7r ) 1 = -log ( -;; - 4p

tp = 2p7r

+ -7r - .log 2

+0

(2p7r/a) 2p7r

!T1'+1

<

!T l'

P)2] , --:;-

[(log

+

0

P)2] .

[(log -P

(E. M. Wright, "A Non-linear Difference-differential Equation," op. cit.)

10. Let z = a + tg(z), where g(z) is analytic in the neighborhood of a, and I t I is sufficiently small. Establish the Lagrange expansion

for h(z) an analytic function in the neighborhood of z = a. 11. Apply this result to the equation z

=a

+ be:»,

12. Improve the asymptotic formulas in (12.9.12) and (12.9.13) by using these formulas in (12.9.10). 13. Suppose that f and g are in E (in the notation of Exercise 5), and let h = f/g. Suppose that the number of poles of h in the circle I z I < r is o(r) as r ~ 00. Then h is in E. (A. L. Shields, "On the Quotient of Exponential Polynomials," Notices Amer. Math. Soc., Vol. 9, No.1, 1962, Abstr, 588-27.)

436

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

14. Discuss the distribution of the zeros of generalized exponential polynomials of the form n

L

Pi(S) exp [qi(S)],

j~O

where the Pi(S) and qi(S) are ordinary polynomials. (L. A. MacCoIl, "On the Distribution of the Zeros of Sums of Exponentials of Polynomials," Trans. Amer. Math. Soc., Vol. 36, 1934, pp. 341-360. H. L. Turrittin, "Asymptotic Distribution of Zeros for Certain Exponential Sums," Amer. J. Math., Vol. 66, 1944, pp. 199-228.)

< !, v >

15. Let 0 and v be constants, 0 1/;(t)

= =

(-a)k exp (-v k) , Ok -

°

a-k

2. Let a = v/O. Let

<

t

~

Ok,

k

=

1,2,3,

elsewhere on (0, 1).

Then 1/;(t) is integrable (absolutely) in the sense of Lebesque. The function

t

o

1/;(t)e8t dt -

has at least one real zero in each interval [-an+r, -an]. (E. C. Titchmarsh, "The Zeros of Certain Integral Functions," Proc. London

Math. Soc., Vol. 25, 1926, p. 283.)

16. Let 1j; (t) be called exceptional if it is a step function with a finite number of discontinuities. If 1/;(t) is not exceptional, and is positive and nondecreasing on (0, 1), then the zeros of the function

t

all satisfy Re(s)

<

o

1/;(t) est dt

0.

(G. P6lya, "Uber die Nullstellen gewisser ganzer Funktionen," Math. Z., Vol. 2 1918, pp. 352-383.)

17. If 1/;(t) is positive and continuous, and if 1/;' (t) exists, and a

<

-

_1/;'(t) < (3 1/;(t) - ,

then the zeros of the integral in the preceding problem satisfy a

<

Re(s)

[The trivial case 1j;(t) = exp (ct (P6lya, op. cit.)

< (3.

+ d)

is excepted.]

MISCELLANEOUS EXERCISES AND RESEARCH PROBLEMS

437

18. If jet) is positive and nondecreasing on (0, 1), the zeros of

t t

D

D

jet) sin st dt,

j(t) cos st dt

are all real and simple. The zeros of the former integral occur (except for e = 0) singly in each of the intervals [m7l', (m + I)7l'], m ;t. 0, -1. The zeros of the latter occur singly in each of the intervals [( m - t)7l', (m + t)7l'], m ;t. O. (Polya, op. cit.)

19. If if;(t) is integrable, x = x t

a

+ iy, then

Ht)e st dt

= a(e'zl),

and for any 8 > O.

tif;(t) est dt ;t. 0(elzHzI6), a

(Titchmarsh, "The Zeros of Certain Integral Functions," lac. cit.)

20. If if; (t) is an integral, if it is continuous at t = 1 and t = -1, and if if;(I) = if; ( -1) = 1, then the zeros, Sm, of the integral /

-1

if;(t)e8 t dt

are given by the asymptotic formula

Sm '"

msri.

(M. L. Cartwright, "The Zeros of Certain Integral Functions," Quart. J. Math., Vol. 1, 1930, p. 38.)

21. If if;(t) is of bounded variation, is continuous at t = 1 and t = -1, and if if;(I) = if; ( -1) = 1, then the zeros of the integral in the preceding problem satisfy I Re(s) I ~ c, for some constant c. The number of zeros less than r in absolute value, nir), satisfies the relation nCr) (Cartwright, op. cit.)

2r

-n: + 0(1),

438

12.

ASYMPTOTIC LOCATION OF ZEROS OF POLYNOMIALS

t

22. If 1/1 (t) is absolutely integrable, the zeros of the integral o

1/1 (t)

dt

est

are such that the series 1

00

2 Sm: \1+£ -

m~l

converges for every

E

1

> 0, and diverges for E =

0.

(Titchmarsh, "The Zeros of Certain Integral Functions," loco cit.)

°

+

23. All the roots of (1 2s)e s - S = lie in the strip -1 There is one and only one root in each rectangle

-1

<

2k'Tr

< Im(s) < 2k'Tr + -,

<

Re(s)

< 0.

< 0,

Re(s)

'Tr

= 0,1,2, """.

k

2

To each root there is a corresponding conjugate; otherwise no root lies outside these rectangles. (J. Pierpont, "On the Complex Roots of a Transcendental Equation Occurring in the Electron Theory," Ann. Math., Ser. 2, Vol. 30, 1928-29, pp. 81-91.)

24. Consider the matrix

2: Ais exp ~

H(s) =

(-ii'S)

i~O

n

+ 2: B, exp

(-ii S ) ,

i~O

which is associated with the equation n'

dy

i=O

dt

2: Ai -

(i - ii')

n

+ 2: Eiy(t i~O

ti)

-

=

fey, i).

Show that the condition inf i~O,

... ,n'

inf

ti':::;; i~O,

... ,n

ii

is not sufficient, in general, to ensure that all root chains are of neutral or retarded type. Prove that if det A o ~ 0, and if the inequality is satisfied, there is a constant c such that all zeros of det H (s) lie in the half-plane Re(s) :::;; c, and conversely that if inf ii' > inf ti, and if det Eo ~ 0, there is a sequence of roots whose real parts approach

+

00.

(K. L. Cooke, "On Transcendental Equations Related to Differential-difference Equations," J. Math. Anal. Appl., Vol. 4, 1962, pp. 65-71.)

BIBLIOGRAPHY AND COMMENTS

439

n:». For N = 2, 3, the 25. Consider the exponential polynomial I:;;~1 zeros have negative real parts. For N ~ 4, does this situation hold?

BIBLIOGRAPHY AND COMMENTS

§12.2. In this chapter, we follow the procedure in an excellent expository paper: R. E. Langer, "On the Zeros of Exponential Sums and Integrals," Bull. Amer. Math. Soc., Vol. 37, 1931, pp. 213-239.

Langer gives a number of references. Also see the recent thorough discussion in D. G. Dickson, "Expansions in Series of Solutions of Linear Difference-differential and Infinite Order Differential Equations with Constant Coefficients," 111em. Amer. Math. Soc., No. 23,1957.

§12.3. For these and further results, see E. C. Titchmarsh, The Theory of Functions, Oxford University Press, London, 1939.

§12.5. See page 18 of the paper by D. G. Dickson referred to above for the definition of a simple funct·ion. §12.7. See §6.4 of the book by Titchmarsh referred to above. §12.9. For further discussion of the control processes, see L. Collatz, "Uber Stabilitiit von Reglern mit Nachlaufzeit," Z. Angew. Math. Mech, Vols. 25-27, 1947, pp. 60-63. H. 1. Ansoff and J. A. Krumhansl, "A General Stability Criterion for Linear Oscillating Systems with Constant Time Lag," Quart. Appl. Math., Vol. 6, 1948, pp. 337-341.

E. lVI. Wright has done extensive work on the problem of precisely calculating the roots, including those of small modulus, of equations of the form (as + b) e = cs + d. A few of his results are stated in the Miscellaneous Exercises in this chapter. For further results, see 8

E. M. Wright, "Solution of the Equation ze" = a," Bull. Amer. Math. Soc., Vol. 65, 1959, pp. 89-93, and Proc. Roy. Soc. Edinburgh, Sect. A, Vol. 65, 1959, PP. 192-203. E. M. Wright, "Solution of the Equation (pz + q)eZ = rz + s," Bull. Amer. Math. Soc., Vol. 66, 1960, pp. 277-281. E. M. Wright, "Stability Criteria and the Real Roots of a Transcendental Equation," J. Soc. Indust. Appl. Math., Vol. 9, 1961, pp. 136-148.

§12.15. For further details and references, see S. Bochner, "Allgemeine lineare Differenzgleichungen mit asymptotisch Konstanten Koeffizienten," Math. Z., Vol. 33, 1931, pp. 426--450.