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%I #80 Jan 17 2020 03:28:43
%S 1,1,4,7,7,9,3,5,7,4,6,9,6,3,1,9,0,3,7,0,1,7,1,4,9,0,2,4,5,9,4,7,4,5,
%T 1,9,3,7,9,8,9,1,6,1,0,1,8,1,9,2,9,1,7,4,1,9,6,4,9,8,7,6,7,3,3,2,2,0,
%U 5,4,8,3,1,3,4,2,0,6,6,5,6,3,3,4,2,0,4,7,2,1,3,1,1,8,9,4,8,8,0,7,7,9,5,8,7
%N Decimal expansion of the ratio of the length of the latus rectum arc of any parabola to its latus rectum: (sqrt(2) + log(1 + sqrt(2)))/2.
%C Just as all circles are similar, all parabolas are similar. Just as the ratio of a semicircle to its diameter is always Pi/2, the ratio of the length of the latus rectum arc of any parabola to its latus rectum is (sqrt(2) + log(1 + sqrt(2)))/2.
%C Let c = this constant and a = e - exp((c+Pi)/2 - log(Pi)), then a = .0000999540234051652627... and c - 10*(-log(exp(a) - a - 1) - 19) = .000650078964115564700067717... - _Gerald McGarvey_, Feb 21 2005
%C Half the universal parabolic constant A103710 (the ratio of the length of the latus rectum arc of any parabola to its focal parameter). Like Pi, it is transcendental.
%C Is it a coincidence that this constant is equal to 3 times the expected distance A103712 from a randomly selected point in the unit square to its center? (Reese, 2004; Finch, 2012)
%D H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
%D C. E. Love, Differential and Integral Calculus, 4th ed., Macmillan, 1950, pp. 286-288.
%D C. S. Ogilvy, Excursions in Geometry, Oxford Univ. Press, 1969, p. 84.
%D S. Reese, A universal parabolic constant, 2004, preprint.
%H Vincenzo Librandi, <a href="/A103711/b103711.txt">Table of n, a(n) for n = 1..10000</a>
%H J. L. Diaz-Barrero and W. Seaman, <a href="http://www.jstor.org/stable/pdfplus/27646363.pdf?acceptTC=true">A limit computed by integration</a>, Problem 810 and Solution, College Math. J., 37 (2006), 316-318, equation (5).
%H S. R. Finch, <a href="http://arxiv.org/abs/2001.00578">Mathematical Constants, Errata and Addenda</a>, 2012, section 8.1.
%H M. Hajja, <a href="https://zbmath.org/?q=an:1291.51018">Review Zbl 1291.51018</a>, zbMATH 2015.
%H M. Hajja, <a href="https://zbmath.org/?q=an:1291.51016">Review Zbl 1291.51016</a>, zbMATH 2015.
%H H. Khelif, <a href="http://images.math.cnrs.fr/L-arbelos-Partie-II.html#nb4">L’arbelos, Partie II, Généralisations de l’arbelos</a>, Images des Mathématiques, CNRS, 2014.
%H J. Pahikkala, <a href="http://planetmath.org/arclengthofparabola">Arc Length Of Parabola</a>, PlanetMath.
%H S. Reese, <a href="http://gaia.adelphi.edu/cgi-bin/makehtmlmov-css.pl?rtsp://gaia.adelphi.edu:554/General_Lectures/Pohle_Colloquiums/pohle200502.mov,pohle200502.mov,256,200">Pohle Colloquium Video Lecture: The universal parabolic constant, Feb 02 2005</a>
%H S. Reese, J. Sondow, <a href="http://mathworld.wolfram.com/UniversalParabolicConstant.html">Universal Parabolic Constant</a>, MathWorld
%H J. Sondow, <a href="http://arxiv.org/abs/1210.2279">The parbelos, a parabolic analog of the arbelos</a>, arXiv 2012, Amer. Math. Monthly, 120 (2013), 929-935.
%H E. Tsukerman, <a href="http://arxiv.org/abs/1210.5580">Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos</a>, arXiv 2012, Amer. Math. Monthly, 121 (2014), 438-443.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UniversalParabolicConstant.html">Universal Parabolic Constant</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Universal_parabolic_constant">Universal parabolic constant</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals Integral_{x = 0..1} sqrt(1 + x^2) dx. - _Peter Bala_, Feb 28 2019
%e 1.14779357469631903701714902459474519379891610181929174196498767332...
%t RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/2, 10, 111][[1]] (* _Robert G. Wilson v_, Feb 14 2005 *)
%t N[Integrate[Sqrt[1 + x^2], {x, 0, 1}], 120] (* _Clark Kimberling_, Jan 06 2014 *)
%Y Equal to (A103710)/2 = (A002193 + A091648)/2 = 3*(A103712).
%Y Cf. A103711, A222362, A232716, A232717.
%K cons,easy,nonn
%O 1,3
%A Sylvester Reese and _Jonathan Sondow_, Feb 13 2005
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