lvmat. J. Math. Math. S.Vol. 4 No. 2 (1981) 311-381

371

YET ANOTHER CHARACTERIZATIONOF THE SINE FUNCTION

ROBERT GERVAIS

Dpartement de MathmatiquesCo118e Llitaire royal de Saint-JeanSaint-Jean, Quebec, Canada J0J 1R0

LEE A. RUBEL

Department of MathematicsUn/versity of Illinois

Urbana, Illinois 61801 U.S.A.

(Received February 14, 1980)

ABS..TACT. In this expository paper, it is shown that if an entire function of

exponential type van/shes at least once in the complex plane and if it has exactly

the same number of zeros (counting multiplicities) as its second derivative, then

this function must take the form Asin(Bz + C).

KEY WORDS AND. P..HASES. Entire function of exponential type, Jensen’s formula,Lioavlle’s theorem, Poisson intesral formula.

1980 MATHEMATICS SUBJECT CLASSIFICATION CODES: 30C15, 30D15, 301)35, 33AI0.

We give here a characterization of the sine function, and present a proof

that uses several of the standard results of the elementary theory of functions

of one complex variable. We make no claim to depth or originality of method.

Our intention is mostly expository- to provide an illustration of the elementary

theory in action. We have taken pains to keep the exposition elementary and

complete. Since more advanced methods (see, e.g. Wittich [8]) can be used to

372 R. GERVAIS AND L.A. RUBEL

get stronger results in uch less space, the reader could consider this artlcle

as an Invtatlon to the use of Nevanllnna theory in the study of differential

equations.

In the set of entire functions, it is customary to classlfy functions

according to the growth of their modulus. In this spirit, we give the follown8

deflntlon: an entire function f is of exponential type if there exist two real

positive constants C and such that

where designates the complex plane.

If f is a function of exponential type, then for every z E such that

zl r > o, we my .rite

in order to get the estimate

"CeT(’).. C’.err.’The last inequality is obtained va the preceding definition of a function of

exponential type, and we may deduce from this inequality that the derivative of a

function of exponential type is itself a function of exponential type.

The theorem we are about to establish may be formulated in the followng way:

THEOREM A. Let f be an entire function of exponential type, possessing at

least one zero. If f is such that z is a zero of multlpllclty m of f if and only

if z is also a zero of multiplicity m of f", the second derivative of f, then f

necessarily has the form

where A, B and C are three complex constants.

CHARACTERIZATION OF THE SINE FUNCTION 373

To facilltate the exposition, we introduce the claas $ of entire funetns-f

of exponential type that have at least one zero in the complex plane and that

have the followlng propertT: z is a zero of f if and only if z is a zero of f",

counting multipl.ic.itie.s. For convenience, we shall eliminate the function

constantly 0 from S. The functions

Iz -Iz Iz+e-lzsinCz) e -e and cosCz) e21 2

are examples of memSers of S. More generally, f(z) A sin (Bz + C) is a function

in S, and the preceding theorem asserts that every element of S is of this form.

We turn now to the proof of theorem k.

Let f be a function in S. Then the function

Z" (z)

is an entire function wthout zeros, and we shall show that, in this case, it

must take the form (z) eh(z) for some entire function h. We observe that

@’/@ is itself an entire function that must be the derivative of an entire

function #, i.e. ’ #’/. Consider now the new function

(z) (z) e-(z).If we calculate the dervative of H, we get

H’ (z) e-#(z) {#’ (z) #(z)#’ (z))

e-(z) {’(z) (z) .’..../l.z}

and we may conclude that

H(z) (Z) e-@(z) C.

Here, C is a constant. Hence each element f S satisfies the differential

equation

f"Cz) fCz)eh(z)

374 R. GERVAIS AND L.A. RUEL

for some entire function h.

We show now that in fact the function h must be a polynonal of degree at

most one. To do this, we shall use Jensen’s formula (cf. Convay [1], p. 283):

xo Io1 ./ log If(rete)J de : log (it|)"

Here it is supposed Chat f is holomorphtc in [zJ r, and that a1, ..., an are

the zeros of f contained in [z[ < r, repeated as many times as their multiplicity

indicates. From this inequality, we deduce that

With the notation

log t fatS1

lo+t an

0 fer O<t<l

f 0 for t>l

log t

-log t for 0<1

we may write

log t los+t log-t.

Thus, if f is in S, and if we moreover suppose that [f(0)l I, then we have

o+ l(ee)l e2 0

lo- f(fete) de0

and we thus obtain

CHARACTERIZATION OF THE SINE FUNCTION 375

I2 12ros-I(e)l de log+l(rele) de0 0

I2w log+(CeTr) d6

0

< Clr

for a constant C1. Finally, using the triangle inequality, we deduce the result

0 0 0

s 2Clr.

In the same fashion, we could demonstrate that, for another constant C2,

2=d8 S 2C2r

0

on supposing also that If"(0) 1.

Eeturnln8 to the function h of the identity (2) and writing

h(z) u(z) + Iv(z)

we may use equation (2) to write

logl f"(ree) logl f(reis) + u(rele)

This leads us (tak/ng account of the preceding inequalities) to the inequality

0 0 0

C3r

for a constant C3. Finally, we write the representation of h as a coplex Polsson

3?6 R. GERVAIS AND L.A. RUBEL

ntegral (cf. Rudin [7], p.228)

h(rei)0 2rei rei

and use the last inequality to obtain

u(2rele) d 2reie + rei@]h(ri)l0 e2 2

0@2w

where the constant D, independent of r, satisfies the inequality

OO2r02

2reiO + rei$

rei rei$

Since h is an entire function that grows no faster than a constant multiple of the

independent variable, we may use a direct consequence of Liouvtlle’s theorem to

conclude that h is a polynomial of degree at most 1.

Thus, if we summarize the present situation, we have, for every f e S such

that

the identity

and If"(0) k i, (3)

Az+BP’(z) (z) e

or equivalently

"(z) ceZ(z) (4)

for two possibly complex constants A and C. We now show directly that the wo

hypotheses in (3) only onstitute a simple normalization. In the first place, if

we had f(O) 0 (and hence f"(O) 0 since f S), the trouble would be that

If(O) < 1 or [f"(O)l < I, so we could take fl(z) af(z) where

CHARACTERIZATION OF THE SINE FUNCTION 377

1m I’(o)l

The function fl belongs to S and satisfies (3). If fl takes the form indicated in

theorem A, then f also does. In case f(O) 0 (and consequently f"(0) 0) then

we perform the translatlon

where is a constant chosen so that f2(0) O. Now one proceeds to show that f2has the required form, and hence that f does.

In the sequel, we shall simplify the exposition by supposing, wthout loss of

generality, that f e S, and that f(0)[ > 1 and [f"(O)[ > 1, so that f satisfies

().

Our aim, at this point, is to show that the constant A in (4) must be zero,

so let us suppose otherwise. For s/mpltcity, we shall suppose A 1 in (4) since

otherwise we could consider the function

z(z) ()’which also belongs to S and satisfies the dlfferentlal equation

F"(z) C’ eZ(z)

where C’ is a constant.

Let

f (z) Z a znn-0

be the Taylor series of f. We may estimate the coefficients as follows:

(n)

I%! ,co)

CeTEfor all r > 0 and n > O.

rn

378 R. GERVAIS AND L.A. RUBEL

Let us choose r n and use Stirling’s formula to deduce the estimate11 1

c" e’r )E

1 1 1c" e 2"q n :t< (2an) (I + )

C’ for all n z 0

where we suppose the elementary fact that limnI/n

i. Consequently we find

1

sup ]ann]] n p < ",

which sinlfies that the series

..ann’.

converges uniformly for Iwl > p’ > 0 and thus defines a function that is holomor-

phlc in a neighborhood of infinity and that vanishes at .Now consider, for Re(w) > max {0,T’}, the integral

n-0z an f0 tn e-we dt

a nnZ n+l

n=0 w

The interchange of the integration and the summation is Justified by (5) and its

consequences. By the remark of the preceding paragraph, the function thus

defined is holomorphic in a right half-plane. On the other hand, we have remarked

CHARACTERIZATION OF THE SINE FUNCTION 379

that the derivative of an entire function of exponentlal type is again of expo-

nentlal type. We may apply this to do the following integration by parts

C-) f C) e-vt: cSt

e-f(O) + f’ (t) dtw 0

w

f’ (O) f f,, e-wt"+ + (t) w2 ’dr.w w2 0

Finally, since f" must satisfy the equation (4) with A I, the function

satisfies the following relation:

(.) -(o) ’(o) -c(.- )

where C is the same constant as in (4). Now we have remarked above that since

is holomorphlc in a right half-plane, (and reca11ing that C O because we have

ruled out f _= O), the last Inequallty allows us to continue aualytlcally to the

whole complex plane, as follows. We know that is holomorphlc for Re(w) > B >

max {p, T}, and the preceding equation allows us to continue analytically to

Re(w) > B 1, then to Re(w) > B 2, and so on, untll the whole complex plane is

covered, moving to the left by a band of wdth 1 each time. But we know also that

is holomorphlc in a neighborhood of infinity. Hence the analytlc continuation

of’@ is holomorphic on the whole Itlemann sphere, and must therefore be a constant.

This constant is actually zero, since (=) 0. Now since we have

n ann"f(z) Z a z and (w)-Z wn+ln=0

nn=0

where the coefficients an, n 0, I, 2, ..., that appear in the two developments

are the same, and since we have shown that O, we have f O, which contradicts

our excluslon of 0 from S. Hence the constant A of equation (4) must be zero.

380 R. GERVAIS AND L.A. RUBEL

All these eonslderatlons lead to the following situation: if f e S and if

If(O)[ > 1 and [f"(O)[ > 1 then f must satisfy the differential equation

P’(z) c

for a (possibly complex) constant C O. Write C CleIA and consider the new

function

where

This function F also belongs to S and satisfies the differential equation

"(-) -(z).

Now in the elementary theory of dlfferntlal equations, it is shown that all

solutions of this equation must be of the form

:[.z -izF(z) ae + be

for wo complex numbers a and b. Since F S, it has at least one zero. This

implies that a 0 and b O. We may rewrite this equation in the form

F(z) cI cos z + c2 sin z

where cI a + b and c2 i(a b), and we remark that

2 2cI + c2 4ab 0

2 + c and C arc tan c-- tosince a 0 and b O. Let us choose A /cI

deduce from elementary trigonometry that

F(z) A sin (z + C)

and hence that

f(z) A sin (Bz + C)

where A, B and C are three complex constants. This concludes the proof of

theorem A.

CHARACTERIZATION OF THE SINE FUNCTION 381

Before ending, we remark that once it is established that a function f of

the class S must satisfy a differential equation of the form (4), there are

alternative elementary proofs at hand. The one we have chosen has the advantage

of remaining in the field of functions of a complex variable, but one could

alternatively proceed directly from the solution of (4) obtained by the classical

methods of the theory of differential equations and a detailed examination of

the solution to derive the conclusion of theorem A.

REFERENCES

i. J.B. Conway, Functions of One Complex Variable, Sprlnger-Verlag (1973).

2. H. Delange, Caractrisations des fonctlons clrculalres, Bull. Sc!._ath..(2)91 (1967), 65-73.

3. R. Gervats and Q.I. Rahman, An extension of Carlson’s theorem for entirefunctions of exponential type, Trans. Amer. Hath. Soc. 23__5(1978) 387-394.

4. R. Gervals and Q.I. Rahman, An extension of Carlson’s theorem for entirefunctions of exponential type II, J. Math. Anal. App. (2) 69(1979) 585-602.

5. M. Ozawa, A characterization of the cosine function by the value distribution,Kodai Math. J. 1 (1978) 213-218.

6. L.A. Rubel and C.C. Yang, Interpolation and unavoidable families of mero-morphic functions, Michigan Math. J. 20 (1973) 289-296.

7. W. Rudln, R.e.a.l..and Complex Analysis., McGraw-Hill (1966).

8. H. Wittlch, Neuere Untersuchungen Uber eindeutlge analytlsche Funktlonen,Springer-Verlag (1968).

Mathematical Problems in Engineering

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Call for PapersThis subject has been extensively studied in the past yearsfor one-, two-, and three-dimensional space. Additionally,such dynamical systems can exhibit a very important and stillunexplained phenomenon, called as the Fermi accelerationphenomenon. Basically, the phenomenon of Fermi accelera-tion (FA) is a process in which a classical particle can acquireunbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermiin 1949 as a possible explanation of the origin of the largeenergies of the cosmic particles. His original model wasthen modified and considered under different approachesand using many versions. Moreover, applications of FAhave been of a large broad interest in many different fieldsof science including plasma physics, astrophysics, atomicphysics, optics, and time-dependent billiard problems andthey are useful for controlling chaos in Engineering anddynamical systems exhibiting chaos (both conservative anddissipative chaos).

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