The spiral of Theodorus, numerical analysis, and special functions

Theodorus of Cyrene (ca. 460–399 B.C.), teacher of Plato und Theaetetus, is known for his proof of the irrationality of 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 , … ....

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Bibliographic Details
Published in:Journal of computational and applied mathematics Vol. 235; no. 4; pp. 1042 - 1052
Main Author: Gautschi, Walter
Format: Journal Article Conference Proceeding
Language:English
Published: Kidlington Elsevier B.V 15.12.2010
Elsevier
Subjects:
Acceleration of convergence
Bose–Einstein distribution
Computation
Exact sciences and technology
Gaussian
Gaussian quadrature
Irrationality
Mathematical analysis
Mathematical models
Mathematics
Measure and integration
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Sciences and techniques of general use
Slowly convergent series
Special functions
Spiral of Theodorus
Spirals
Teachers
33C47 51-03
65B10
65D30
Bose–Einstein distribution
Spiral of Theodorus
01A20
Slowly convergent series
65D32
30E05
Gaussian quadrature
Numerical integration
Convergence acceleration
Bose-Einstein distribution
Special function
Series summation
Numerical analysis
Applied mathematics
Distribution function
Abstract space
ISSN:0377-0427, 1879-1778
Online Access:Get full text
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Summary:Theodorus of Cyrene (ca. 460–399 B.C.), teacher of Plato und Theaetetus, is known for his proof of the irrationality of 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 P.J. 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.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
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ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2009.11.054