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物理和工程中经常的问题需要在这些积分中出现的参数方面对积分进行精确评估。 这些问题出现在粒子物理学和经典力学的研究中。尽管并非总是可能会找到分析表达式，但研究者及其共同的问题出现在开发有效的和强大的符号软件软件包中，该软件包应该以封闭形式赋予这样的这种恢复，或者决定是否可以表达这种表达方式。 研究者的目标是开发算法，扩展了现有软件Pacakagesthat的功能，广泛用于行业和大学。 评估者的努力集中在两个类别中：合理功能和与Hurwitz Zetafunction相关的基本功能。 对上一类的研究为所谓的多个Zeta值以及对Tornheim-Zagier总和的悬崖形式评估产生了新的猜测。特殊功能的确定整合的问题与J. Bernoulli的工作始于18世纪。 从这个问题中发展的知识在分析和应用数学的许多方面都是核心。 在20世纪的最后一部分中，高复杂的象征性语言的发展需要进一步研究这个旧问题。研究者及其同事正在发展这些积分符号评估所需的含义理论。 令人惊讶的事实是，这种终结者具有许多数学领域的结合。 调查开发了新算法，以确保科学社区可用的当前符号语言的效率和鲁棒性。