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On Value Distribution Theory of Second Order Non-homogeneous Periodic ODEs and the Lommel Functions.

机译：二阶非齐次周期ODE的价值分布理论和Lommel函数。

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摘要

There should be no great disagreement to say that linear differential equations are important both within mathematics or in their applications to other disciplines. Historically, differential equations were first written down by Newton ever since he invented the differential calculus in his search for the understanding of physical world. Thus, it is natural to treat both the independent and dependent variables of the differential equation and its solutions to be real. However, experience shows us that many important DEs and their solutions have their natural domain on the complex plane C and that they assume complex values in general. For example, the Bessel equation, Hermite equation, Laguerre equation, etc, are all equations that have their natural domains on or part of C. Apart from these familiar examples, there are also less familiar but by no means less important differential equations that have not been well-understood. Prominent examples are the Mathieu equation, Lame equation, and Hill's equation, see [3]. Each of them is at least one and a half century old. These equations sit at a higher level than the previously mentioned equations and so they are much harder to treat and with less tools available. Bank and Laine discovered in the 1980s that the Nevanlinna value distribution theory could be applied to treat special cases of the Hill's equation which have periodic potentials. They discovered that the zero-distribution of solutions have close relation with the quantization of the corresponding DEs. Chiang and Ismail were able to show that this Hill's equation can be solved by special functions of the confluent hypergeometric type in 2006. They also unified the Nevanlinna approach and the classical special function approach.;The main focus of this thesis is to consider special cases of certain non-homogeneous Hill's equation and to show that one can use Nevanlinna theory viewpoint (subnormality and oscillatory) to discover new "function-theoretic quantization" criteria of the equations. New facts about the corresponding special function, namely the Lommel functions, are also established. *Please refer to dissertation for footnotes.
机译：可以说，线性微分方程在数学中或在其他学科中的应用都是重要的，这一点不应有很大的分歧。从历史上看，自从牛顿在寻求对物理世界的理解中发明微分学以来，他就首先写下了微分方程。因此，将微分方程及其解的自变量和因变量都视为实数是自然的。但是，经验表明，许多重要的DE及其解决方案在复平面C上具有其自然域，并且通常具有复杂的值。例如，贝塞尔（Bessel）方程，埃尔米特（Hermite）方程，拉盖尔（Laguerre）方程等都是在C或C上具有其自然域的方程。除了这些熟悉的示例外，还不那么熟悉，但绝不那么重要没有被很好的理解。著名的例子是Mathieu方程，Lame方程和Hill方程，请参见[3]。他们每个人都至少有一个半世纪的历史。这些方程式处于比前面提到的方程式更高的水平，因此它们更难以处理且使用的工具更少。班克斯和莱恩在1980年代发现，Nevanlinna值分布理论可以用于处理具有周期性潜力的希尔方程的特殊情况。他们发现解决方案的零分布与相应DE的量化密切相关。 Chiang和Ismail能够证明该希尔方程可以在2006年通过融合的超几何类型的特殊函数来求解。他们还把Nevanlinna方法和经典特殊函数方法统一了起来；本论文的主要重点是考虑特殊情况某些非齐次Hill方程的公式，并表明可以使用Nevanlinna理论观点（次正规性和振荡）发现方程的新“函数理论量化”准则。还建立了有关相应特殊功能（即Lommel函数）的新事实。 *请参阅论文的脚注。

著录项

  • 作者

    Yu, Kit-Wing.;

  • 作者单位

    Hong Kong University of Science and Technology (Hong Kong).;

  • 授予单位 Hong Kong University of Science and Technology (Hong Kong).;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2010
  • 页码 198 p.
  • 总页数 198
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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