数学论文英文翻译.docx
《数学论文英文翻译.docx》由会员分享,可在线阅读,更多相关《数学论文英文翻译.docx(5页珍藏版)》请在咨信网上搜索。
1、毕业设计(论文)附录(翻译)课 题 名 称一些周期性的二阶线性微分方程解的方法学 生 姓 名万 益目录1 毕业设计(论文)附录(翻译)英文2 毕业设计(论文)附录(翻译)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 Nevanlinnas 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 ,(see 8)
3、,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 enti
4、re 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 A Let
5、be 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 strong
6、er 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 given
7、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, andarbitr
8、ary.(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 soluti
9、ons 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 that The
10、orem 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 Section
11、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 conclus
12、ion 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 functio
13、ns 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. Let
14、,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 with T
15、heorem 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.2. L
16、emmas 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 linearly d
17、ependent. 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 those
18、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 that
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 数学论文 英文翻译
1、咨信平台为文档C2C交易模式,即用户上传的文档直接被用户下载,收益归上传人(含作者)所有;本站仅是提供信息存储空间和展示预览,仅对用户上传内容的表现方式做保护处理,对上载内容不做任何修改或编辑。所展示的作品文档包括内容和图片全部来源于网络用户和作者上传投稿,我们不确定上传用户享有完全著作权,根据《信息网络传播权保护条例》,如果侵犯了您的版权、权益或隐私,请联系我们,核实后会尽快下架及时删除,并可随时和客服了解处理情况,尊重保护知识产权我们共同努力。
2、文档的总页数、文档格式和文档大小以系统显示为准(内容中显示的页数不一定正确),网站客服只以系统显示的页数、文件格式、文档大小作为仲裁依据,平台无法对文档的真实性、完整性、权威性、准确性、专业性及其观点立场做任何保证或承诺,下载前须认真查看,确认无误后再购买,务必慎重购买;若有违法违纪将进行移交司法处理,若涉侵权平台将进行基本处罚并下架。
3、本站所有内容均由用户上传,付费前请自行鉴别,如您付费,意味着您已接受本站规则且自行承担风险,本站不进行额外附加服务,虚拟产品一经售出概不退款(未进行购买下载可退充值款),文档一经付费(服务费)、不意味着购买了该文档的版权,仅供个人/单位学习、研究之用,不得用于商业用途,未经授权,严禁复制、发行、汇编、翻译或者网络传播等,侵权必究。
4、如你看到网页展示的文档有www.zixin.com.cn水印,是因预览和防盗链等技术需要对页面进行转换压缩成图而已,我们并不对上传的文档进行任何编辑或修改,文档下载后都不会有水印标识(原文档上传前个别存留的除外),下载后原文更清晰;试题试卷类文档,如果标题没有明确说明有答案则都视为没有答案,请知晓;PPT和DOC文档可被视为“模板”,允许上传人保留章节、目录结构的情况下删减部份的内容;PDF文档不管是原文档转换或图片扫描而得,本站不作要求视为允许,下载前自行私信或留言给上传者【可****】。
5、本文档所展示的图片、画像、字体、音乐的版权可能需版权方额外授权,请谨慎使用;网站提供的党政主题相关内容(国旗、国徽、党徽--等)目的在于配合国家政策宣传,仅限个人学习分享使用,禁止用于任何广告和商用目的。
6、文档遇到问题,请及时私信或留言给本站上传会员【可****】,需本站解决可联系【 微信客服】、【 QQ客服】,若有其他问题请点击或扫码反馈【 服务填表】;文档侵犯商业秘密、侵犯著作权、侵犯人身权等,请点击“【 版权申诉】”(推荐),意见反馈和侵权处理邮箱:1219186828@qq.com;也可以拔打客服电话:4008-655-100;投诉/维权电话:4009-655-100。