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

Transcription

1 lvmat. J. Math. Math. S. Vol. 4 No. 2 (1981) YET ANOTHER CHARACTERIZATION OF THE SINE FUNCTION ROBERT GERVAIS Dpartement de Mathmatiques Co118e Llitaire royal de Saint-Jean Saint-Jean, Quebec, Canada J0J 1R0 LEE A. RUBEL Department of Mathematics Un/versity of Illinois Urbana, Illinois 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 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

2 372 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 The "CeT( ).. C.err. 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.

3 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 sincz) Iz -Iz eiz+e-lz e -e and coscz) 21 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) e h(z) for some entire function h. We observe /@ 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)e h(z)

4 374 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 a 1,..., a n 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 lo+t log t fats1 an log t f 0 for t>l 0 fer O<t<l -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 2 0 o + l(ee)l e 0 lo- f(fete) de and we thus obtain

5 CHARACTERIZATION OF THE SINE FUNCTION 375 os-i(e)l 12r de log+l(rele) de I2 0 0 I2w log+(cetr) 0 < Clr d6 for a constant C 1. Finally, using the triangle inequality, we deduce the result s 2Clr. In the same fashion, we could demonstrate that, for another constant C 2, 2= on supposing also that If"(0) 1. 0 d8 S 2C2r 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 C3r for a constant C 3. Finally, we write the representation of h as a coplex Polsson

6 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 ]h(ri)l u(2rele) 2re ie d + re i@ 0 e2 2 0@2w where the constant D, independent of r, satisfies the inequality OO2r 02 2re io + re i$ re i 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. that Thus, if we summarize the present situation, we have, for every f e S such and If"(0) k i, (3) the identity P (z) (z) Az+B 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

7 CHARACTERIZATION OF THE SINE FUNCTION m 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 f2 has 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) where C is a constant. C ez(z) Let be the Taylor series of f. f (z) Z a z n n-0 We may estimate the coefficients as follows: I%! (n),co) Ce TE for all r > 0 and n > O. r n

8 378 R. GERVAIS AND L.A. RUBEL Let us choose r n and use Stirling s formula to deduce the estimate c" e r )E c" e 2"q < (2an) n (I + :t ) C for all n z 0 where we suppose the elementary fact that limn I/n i. Consequently we find sup 1 ]ann]] n p < ", which sinlfies that the series..ann. converges uniformly for Iwl > p > 0 and thus defines a function that is holomorphlc 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 n n Z 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

9 CHARACTERIZATION OF THE SINE FUNCTION 379 that the derivative of an entire function of exponentlal type is again of exponentlal type. We may apply this to do the following integration by parts C-) f C) e-vt: cst e- f(o) + f (t) dt w w 0 f (O) f f,, e "+ + (t) -wt w w2 w2 dr. 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 is holomorphic on the whole Itlemann sphere, and must therefore be a constant. This constant is actually zero, since (=) 0. Now since we have ann" f(z) Z a z n n and (w)-z n=0 n=0 wn+l 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.

10 380 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 Cle IA 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 -iz F(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) c I cos z + c 2 sin z where c I a + b and c 2 i(a b), and we remark that 2 2 c I + c 2 4ab c and C arc tan c-- to since a 0 and b O. Let us choose A /c I 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.

11 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), R. Gervats and Q.I. Rahman, An extension of Carlson s theorem for entire functions of exponential type, Trans. Amer. Hath. Soc (1978) R. Gervals and Q.I. Rahman, An extension of Carlson s theorem for entire functions of exponential type II, J. Math. Anal. App. (2) 69 (1979) M. Ozawa, A characterization of the cosine function by the value distribution, Kodai Math. J. 1 (1978) L.A. Rubel and C.C. Yang, Interpolation and unavoidable families of meromorphic functions, Michigan Math. J. 20 (1973) W. Rudln, R.e.a.l..and Complex Analysis., McGraw-Hill (1966). 8. H. Wittlch, Neuere Untersuchungen Uber eindeutlge analytlsche Funktlonen, Springer-Verlag (1968).

12 Mathematical Problems in Engineering Special Issue on Time-Dependent Billiards Call for Papers This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome. To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors Edson Denis Leonel, Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, Rio Claro, SP, Brazil ; edleonel@rc.unesp.br Alexander Loskutov, Physics Faculty, Moscow State University, Vorob evy Gory, Moscow , Russia; loskutov@chaos.phys.msu.ru Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Hindawi Publishing Corporation