古典数学名著《九章算术》刘徽注中的千古疑谜——刘徽级数

Liu Hui's Series—a Puzzle of Thousauds Years in His Commentaries on "Arithmetic in Nine Sections", the Classical Mathematical Masterpiece

世界古典数学名著《九章算术》代表着古代东方数学的特色,反映了数学应用的广泛性。公元263年中国杰出数学家刘徽注《九章算术》“方田章弧田术”时,创造性的运用数列极限思想和出入相补原理的知识提出了一个无穷级数,这是值得称道的。为了纪念刘徽在数学上的卓越贡献,本文命名为刘徽级数。1700多年来,《九章算术》与刘徽注一直都很受重视。特别近年来,许多学者进行了专题研究,把这项工作推向新的高潮。目前对于刘徽级数中“算数差繁”、“必欲有所寻究”尚未论及。本文获得刘徽级数的两种等价表达式,即初等函数表达式和幂级数表达式。

'Arithmetic in Nine Sestion', the famous classical mathematical book, represents the distinguished features of the ancient mathematics in the Orient, and reflests the wide application of mathematics. In A. D. 263, when Liu Hui, the great Chinese mathematician, made commentaries on the section 'On segment of a cicle ' in the chapter' On square' of the book 'Arithmetic in Nine Sections', he advanced an infinite series by applying the idea of the limit sequence of number and the principle of congruency by addition and substraction. To the memory of Liu Hui's great contribution in mathematics, this infinite series is named after him. For over 1700 years great attention has been paid to the 'Arithmetic in Nine Sections' and Liu's commentaries and , in recent years, monographic researches into them have been carried out by many scholars in particular, which have pushed the study to a climax. Being complicated for its calculation some other problems in Liu's Series still need to be solved . This paper obtains two new expressions in liu's Series, namely, the expression of elementary function and the expression of power serxies