數(shù)學(xué)論文英文翻譯

1、畢業(yè)設(shè)計(論文)附錄（翻譯）課 題 名 稱一些周期性的二階線性微分方程解的方法學(xué) 生 姓 名萬 益目錄1 畢業(yè)設(shè)計（論文）附錄（翻譯）英文2 畢業(yè)設(shè)計（論文）附錄（翻譯）中文2014年 5月25日Some Properties of Solutions of Periodic Second OrderLinear Differential Equations1. Introduction and main resultsIn this paper, we shall assume that the reader is familiar with the fundamental results

2、and the stardard notations of the Nevanlinna's value distribution theory of meromorphic functions 12, 14, 16. In addition, we will use the notation，and to denote respectively the order of growth, the lower order of growth and the exponent of convergence of the zeros of a meromorphic function ，（s

3、ee 8），the e-type order of f(z), is defined to be Similarly, ，the e-type exponent of convergence of the zeros of meromorphic function , is defined to beWe say thathas regular order of growth if a meromorphic functionsatisfiesWe consider the second order linear differential equationWhere is a periodic

4、 entire function with period . The complex oscillation theory of (1.1) was first investigated by Bank and Laine 6. Studies concerning (1.1) have een carried on and various oscillation theorems have been obtained 211, 13, 1719. Whenis rational in ，Bank and Laine 6 proved the following theoremTheorem

5、A Letbe a periodic entire function with period and rational in .Ifhas poles of odd order at both and , then for every solutionof (1.1), Bank 5 generalized this result: The above conclusion still holds if we just suppose that both and are poles of, and at least one is of odd order. In addition, the s

6、tronger conclusion (1.2)holds. Whenis transcendental in, Gao 10 proved the following theoremTheorem B Let ,whereis a transcendental entire function with, is an odd positive integer and，Let .Then any non-trivia solution of (1.1) must have. In fact, the stronger conclusion (1.2) holds.An example was g

7、iven in 10 showing that Theorem B does not hold when is any positive integer. If the order , but is not a positive integer, what can we say? Chiang and Gao 8 obtained the following theoremsTheorem C Let ,where,andare entire functionstranscendental andnot equal to a positive integer or infinity, anda

8、rbitrary.(i) Suppose . (a) If f is a non-trivial solution of (1.1) with; thenandare linearly dependent. (b) Ifandare any two linearly independent solutions of (1.1), then .(ii) Suppose (a) If f is a non-trivial solution of (1.1) with,andare linearly dependent. Ifandare any two linearly independent s

9、olutions of (1.1),then.Theorem D Letbe a transcendental entire function and its order be not a positive integer or infinity. Let; whereand p is an odd positive integer. Thenor each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.Examples were also given in 8 showing tha

10、t Theorem D is no longer valid whenis infinity.The main purpose of this paper is to improve above results in the case whenis transcendental. Specially, we find a condition under which Theorem D still holds in the case when is a positive integer or infinity. We will prove the following results in Sec

11、tion 3.Theorem 1 Let ,where,andare entire functions withtranscendental andnot equal to a positive integer or infinity, andarbitrary. If Some properties of solutions of periodic second order linear differential equations and are two linearly independent solutions of (1.1), thenOrWe remark that the co

12、nclusion of Theorem 1 remains valid if we assumeis not equal to a positive integer or infinity, andarbitrary and still assume,In the case whenis transcendental with its lower order not equal to an integer or infinity andis arbitrary, we need only to consider in,.Corollary 1 Let,where,andareentire fu

13、nctions with transcendental and no more than 1/2, and arbitrary.(a) If f is a non-trivial solution of (1.1) with,then and are linearly dependent.(b) Ifandare any two linearly independent solutions of (1.1), then.Theorem 2 Letbe a transcendental entire function and its lower order be no more than 1/2

14、. Let,whereand p is an odd positive integer, then for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds. We remark that the above conclusion remains valid ifWe note that Theorem 2 generalizes Theorem D whenis a positive integer or infinity but . Combining Theorem D w

15、ith Theorem 2, we haveCorollary 2 Letbe a transcendental entire function. Let where and p is an odd positive integer. Suppose that either (i) or (ii) below holds:(i) is not a positive integer or infinity;(ii) ;thenfor each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds

16、.2. Lemmas for the proofs of TheoremsLemma 1 (7) Suppose thatand thatare entire functions of period,and that f is a non-trivial solution ofSuppose further that f satisfies; that is non-constant and rational in，and that if，thenare constants. Then there exists an integer q with such that and are linea

17、rly dependent. The same conclusion holds ifis transcendental in，and f satisfies，and if ，then asthrough a setof infinite measure, we havefor.Lemma 2 (10) Letbe a periodic entire function with periodand be transcendental in, is transcendental and analytic on.Ifhas a pole of odd order at or(including t

18、hose which can be changed into this case by varying the period of and. (1.1) has a solutionwhich satisfies , then and are linearly independent.3. Proofs of main resultsThe proof of main results are based on 8 and 15.Proof of Theorem 1 Let us assume.Since and are linearly independent, Lemma 1 implies

19、 that and must be linearly dependent. Let,Thensatisfies the differential equation, (2.1)Where is the Wronskian ofand(see 12, p. 5 or 1, p. 354), andor some non-zero constant.Clearly, and are both periodic functions with period,whileis periodic by definition. Hence (2.1) shows thatis also periodic wi

20、th period .Thus we can find an analytic functionin,so thatSubstituting this expression into (2.1) yields (2.2)Since bothand are analytic in,the Valiron theory 21, p. 15 gives their representations as， （2.3）where，are some integers, andare functions that are analytic and non-vanishing on ，and are enti

21、re functions. Following the same arguments as used in 8, we have， （2.4）where.Furthermore, the following properties hold 8,Where (resp, ) is defined to be(resp, ),Some properties of solutions of periodic second order linear differential equationswhere(resp. denotes a counting function that only count

22、s the zeros of in the right-half plane (resp. in the left-half plane), is the exponent of convergence of the zeros of in, which is defined to beRecall the condition ,we obtain.Now substituting (2.3) into (2.2) yields （2.5）Proof of Corollary 1 We can easily deduce Corollary 1 (a) from Theorem 1 .Proo

23、f of Corollary 1 (b). Supposeandare linearly independent and,then,and .We deduce from the conclusion of Corollary 1 (a) thatand are linearly dependent, j = 1; 2. Let.Then we can find a non-zero constant such that.Repeating the same arguments as used in Theorem 1 by using the fact that is also period

24、ic, we obtain,a contradiction since .Hence .Proof of Theorem 2 Suppose there exists a non-trivial solution f of (1.1) that satisfies . We deduce , so and are linearly dependent by Corollary 1 (a). However, Lemma 2 implies that andare linearly independent. This is a contradiction. Hence holds for eac

25、h non-trivial solution f of (1.1). This completes the proof of Theorem 2.Acknowledgments The authors would like to thank the referees for helpful suggestions to improve this paper.References1 ARSCOTT F M. Periodic Di®erential Equations M. The Macmillan Co., New York, 1964.2 BAESCH A. On the exp

26、licit determination of certain solutions of periodic differential equations of higher order J. Results Math., 1996, 29(1-2): 4255.3 BAESCH A, STEINMETZ N. Exceptional solutions of nth order periodic linear differential equations J.Complex Variables Theory Appl., 1997, 34(1-2): 717.4 BANK S B. On the

27、 explicit determination of certain solutions of periodic differential equations J. Complex Variables Theory Appl., 1993, 23(1-2): 101121.5 BANK S B. Three results in the value-distribution theory of solutions of linear differential equations J.Kodai Math. J., 1986, 9(2): 225240.6 BANK S B, LAINE I.

28、Representations of solutions of periodic second order linear differential equations J. J. Reine Angew. Math., 1983, 344: 121.7 BANK S B, LANGLEY J K. Oscillation theorems for higher order linear differential equations with entire periodic coe±cients J. Comment. Math. Univ. St. Paul., 1992, 41(1

29、): 6585.8 CHIANG Y M, GAO Shi'an. On a problem in complex oscillation theory of periodic second order lineardifferential equations and some related perturbation results J. Ann. Acad. Sci. Fenn. Math., 2002, 27(2):273290.一些周期性的二階線性微分方程解的方法1 簡介和主要成果在本文中，我們假設(shè)讀者熟悉的函數(shù)的數(shù)值分布理論12，14，16的基本成果和數(shù)學(xué)符號。此外，我們將使

30、用的符號，and ，表示的順序分別增長，低增長的一個純函數(shù)的零點收斂指數(shù)，（8），E型的f(z)，被定義為同樣，E型的亞純函數(shù)的零點收斂指數(shù)，被定義為我們說，如果一個亞純函數(shù)滿足增長的正常秩序我們考慮的二階線性微分方程在是一個整函數(shù)在。在（1.1）的反復(fù)波動理論的第一次探討中由銀行和萊恩6。已經(jīng)進行了研究在（1.1）中，并已取得各種波動定理在211，13，1719。在函數(shù)中正確的，銀行和萊恩6證明了如下定理定理A設(shè)這函數(shù)是一個周期性函數(shù)，周期為在整個函數(shù)存在。如果有奇數(shù)階極點在和，然后對于任何一個結(jié)果答案在(1.1)中廣義這樣的結(jié)果：上述結(jié)論仍然認為，如果我們只是假設(shè)，既和的極點，并且至少有一

31、個是奇數(shù)階。此外，較強的結(jié)論 （1.2）認為。當(dāng)是超越在，高10證明了如下定理定理B設(shè)，其中是一個超越整函數(shù)與，是奇正整并且，設(shè)，那么任何微分解在（1.1）的函數(shù)必須有。事實上，在（1.2）已經(jīng)有證明的結(jié)論。是在10 一個例子表明當(dāng)定理B不成立時，是任意正整數(shù)。如果在另一方面，但如果沒有一個正整數(shù)，我們可以說些什么呢？蔣和高8得到以下定理定理C 設(shè)，其中，函數(shù)和函數(shù)是整函數(shù)先驗和不等于一個正整數(shù)或無窮大，并函數(shù)任意。（一） 假設(shè)（a）如果函數(shù)f是一個非平凡解在（1.1），那么和是線性相關(guān)。（b）如果函數(shù)和函數(shù)在（1.1）是兩個線性無關(guān)函數(shù)，則存在這樣一個條件。（二） 假設(shè)（a）如果函數(shù)f有一個

32、非平凡解在（1.1）且，和是線性相關(guān)的。如果函數(shù)和函數(shù)在（1.1）在（1.1）是兩個線性無關(guān)函數(shù)，則存在這樣一個條件。定理 D 讓是一個超越整函數(shù)和它的秩序是正整數(shù)或無窮大。設(shè)，和p是一個奇正整數(shù)。然后或F得到每一個非平凡解在（1.1）。事實上，在（1.2）中已經(jīng)有證明的結(jié)論。例子表明在高8定理D不再成立，當(dāng)是無窮的。本文的主要目的是改善上述結(jié)果的情況下，當(dāng)是超越。特別地，我們找到的條件下定理D仍然成立的情況下，當(dāng)是一個正整數(shù)或無窮大。我們將證明在第3節(jié)的結(jié)果如下：定理1設(shè)，其中，和先驗和不等于一個正整數(shù)或無窮，任意整函數(shù)。如果定期二階線性微分方程和的解不是一些屬性是兩個線性無關(guān)的解在（1.1

33、），然后或者我們的說法，定理1的結(jié)論仍然有效，如果我們假設(shè)函數(shù)不等于一個正整數(shù)或無窮大，任意和承擔(dān)的情況下，當(dāng)其低階不等于一個整數(shù)或無窮超然是任意的，我們只需要考慮在，。推論1設(shè)，其中，函數(shù)和函數(shù)是整個先驗和不超過1 / 2，并且任意的。（一） 如果函數(shù)f是一個非平凡解在（1.1）中，那么和是線性相關(guān)。（二） 如果和是兩個線性無關(guān)解在（1.1）中，那么。定理2設(shè)是一個超越整函數(shù)及其低階不超過1 / 2。設(shè)，其中和p是一個奇正整數(shù)，則為每個非平凡解F到在（1.1）中。事實上，在（1.2）中證明正確的結(jié)論。  我們注意到，上述結(jié)論仍然有效的假設(shè)我們注意到，我們得出定理2推廣定理D，當(dāng)是一個正整數(shù)或無窮，但結(jié)合定理2定理的研究。推論2設(shè)是一個超越整函數(shù)。設(shè)，其中和p是一個奇正整數(shù)。假設(shè)要么（一）或（二）中認為：（一）不是正整數(shù)或無窮;（二）然后為每一個非平凡解在（1.1）中函數(shù)f對于。事實上，在（1.2）中已經(jīng)有證明的結(jié)論。2 引理為定理的證明引理1（7），和的假設(shè)是整個周期，并且函數(shù)f是有一個非平凡解進一步假設(shè)函數(shù)f滿足;，是在非恒定和理性的，而且，如果，且是常數(shù)。則存在一個整數(shù)q與 ，和是線性相關(guān)。相同的結(jié)論認為，如果是超越，和f滿足，如果，然后通過一個無限措施的集合為，且引理2（10）設(shè)是一個周期為在（包括那些可以改變這種情況下極奇數(shù)