The spiral of Theodorus, numerical analysis, and special functions

[摘要] Theodorus of Cyrene (ca. 460-399 B.C.), teacher of Plato und Theaetetus, is known for his proof of the irrationality of root n, n = 2, 3, 5,..., 17. He may have known also of a discrete spiral, today named after him, whose construction is based on the square roots of the numbers n = 1, 2, 3,.... The subject of this lecture is the problem of interpolating this discrete, angular spiral by a smooth, if possible analytic, spiral. An interesting solution was proposed in 1993 by RI Davis, which is based on an infinite product. The computation of this product gives rise to problems of numerical analysis, in particular the summation of slowly convergent series, and the identification of the product raises questions regarding special functions. The former are solved by a method of integration, in particular Gaussian integration, the latter by means of Dawson's integral und the Bose-Einstein distribution. Number-theoretic questions also loom behind this work. (C) 2009 Elsevier B.V. All rights reserved.

[发布日期] 2010-12-15

[效力级别]  Proceedings Paper

[关键词] Spiral of Theodorus;Slowly convergent series;Gaussian quadrature;Bose-Einstein distribution