Rational Landen Transformations, Dynamical Systems, and Hurwitz Zeta Function

有理 Landen 变换、动力系统和 Hurwitz Zeta 函数

• 批准号: 0409968
• 负责人: Victor Moll
• 金额: --
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0409968&HistoricalAwards=false
• 关键词: Rational; Landen; Transformations; Dynamical; Systems

Moll0409968Many problems in physics and engineering require the exactevaluation of integrals in terms of parameters appearing in thoseintegrals. These problems come up in the study of particlephysics and classical mechanics. While it is not always possibleto find analytic expressions, the investigator and his colleaguesstudy questions that appear in the development of an efficientand robust symbolic software package that should give suchresults in closed form, or decide whether such an expression isachievable. The investigator's goal is to develop algorithmsthat expand upon the capabilities of existing software pacakagesthat are widely used in industry and universities. Theinvestigator's efforts are concentrated in two classes: rationalfunctions, and elementary functions related to the Hurwitz zetafunction. The study of this last class has produced a newapproach to conjectures on the so-called multiple zeta values andthe cloed-form evaluation of Tornheim-Zagier sums.The problem of definite integration of special functions startedin the eighteen century with the work of J. Bernoulli. Theknowledge developed from this problem has been central in manyparts of analysis and applied mathematics. The development ofhighly sophisticated symbolic languages during the last part ofthe 20th century has required further study of this old problem. The investigator and his colleagues are developing themathematical theory that is needed for the symbolic evaluation ofthese integrals. It is a surprising fact that this endevour hasconnections with many areas of mathematics. The investigatordevelops new algorithms that ensure the efficiency and robustnessof the current symbolic languages available to the scientificcommunity.

Moll0409968物理和工程中的许多问题都需要根据积分中出现的参数来精确计算积分。 这些问题在粒子物理学和经典力学的研究中出现。虽然并不总是能够找到分析表达式，但研究人员和他的同事研究了在开发高效且强大的符号软件包时出现的问题，该软件包应以封闭形式给出此类结果，或决定此类表达式是否可以实现。 研究人员的目标是开发算法，扩展工业界和大学广泛使用的现有软件包的功能。 研究人员的工作集中在两类：有理函数和与 Hurwitz zeta 函数相关的初等函数。 最后一课的研究产生了一种新的方法来猜想所谓的多重 zeta 值和 Tornheim-Zagier 和的封闭形式评估。特殊函数的定积分问题始于 18 世纪 J. Bernoulli 的工作。 从这个问题中发展出来的知识一直是分析和应用数学许多部分的核心。 二十世纪后半叶高度复杂的符号语言的发展需要对这个老问题进行进一步的研究。研究人员和他的同事正在开发这些积分的符号评估所需的数学理论。 令人惊讶的是，这项工作与数学的许多领域都有联系。 研究人员开发了新的算法，以确保科学界当前可用的符号语言的效率和稳健性。