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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Vydáno v:	Journal of computational and applied mathematics Ročník 235; číslo 4; s. 1042 - 1052
Hlavní autor:	Gautschi, Walter
Médium:	Journal Article Konferenční příspěvek
Jazyk:	angličtina
Vydáno:	Kidlington Elsevier B.V 15.12.2010 Elsevier
Témata:	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
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Abstract	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.
AbstractList	Theodorus of Cyrene (ca.aa460-399 B.C.), teacher of Plato und Theaetetus, is known for his proof of the irrationality of n , n = 2 , 3 , 5 , a[brvbar] , 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 , a[brvbar] . 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. 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.
Author	Gautschi, Walter
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Cites_doi	10.1007/BF01571563 10.1090/S0025-5718-1991-1079017-5 10.1090/S0025-5718-1985-0771039-1 10.1090/S0025-5718-69-99647-1 10.1007/s11075-009-9283-2 10.2307/4145130 10.2307/2309786
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Keywords	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
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References	Gautschi, Milovanović (b9) 1985; 44 Hlawka (b6) 1984 Heuvers, Moak, Boursaw (b15) 2000; vol. 518 Gautschi (b11) 2004 Gronau (b5) 2004; 111 Gautschi (b12) 2009; 52 Golub, Welsch, Golub, Welsch (b10) 1969; 23 Gautschi (b14) 1991; 57 Davis (b1) 1993 (b8) 1970; vol. II Jörg Waldvogel, Analytic continuation of the Theodorus spiral (in preparation); see Pólya, Szegö (b7) 1978 Hlawka (b2) 1980; 89 Davis (b3) 1959; 66 Artin (b4) 1964 10.1016/j.cam.2009.11.054_b13 Hlawka (10.1016/j.cam.2009.11.054_b6) 1984 Golub (10.1016/j.cam.2009.11.054_b10_2) 1969; 23 Pólya (10.1016/j.cam.2009.11.054_b7) 1978 Gautschi (10.1016/j.cam.2009.11.054_b14) 1991; 57 Hlawka (10.1016/j.cam.2009.11.054_b2) 1980; 89 Golub (10.1016/j.cam.2009.11.054_b10_1) 1969; 23 Heuvers (10.1016/j.cam.2009.11.054_b15) 2000; vol. 518 Davis (10.1016/j.cam.2009.11.054_b3) 1959; 66 Gronau (10.1016/j.cam.2009.11.054_b5) 2004; 111 Artin (10.1016/j.cam.2009.11.054_b4) 1964 (10.1016/j.cam.2009.11.054_b8) 1970; vol. II Davis (10.1016/j.cam.2009.11.054_b1) 1993 Gautschi (10.1016/j.cam.2009.11.054_b9) 1985; 44 Gautschi (10.1016/j.cam.2009.11.054_b12) 2009; 52 Gautschi (10.1016/j.cam.2009.11.054_b11) 2004
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SubjectTerms	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
Title	The spiral of Theodorus, numerical analysis, and special functions
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