cp's OEIS Frontend

A103710 Decimal expansion of the ratio of the length of the latus rectum arc of any parabola to its semi latus rectum: sqrt(2) + log(1 + sqrt(2)).

%I A103710 #101 Mar 08 2025 19:48:02
%S A103710 2,2,9,5,5,8,7,1,4,9,3,9,2,6,3,8,0,7,4,0,3,4,2,9,8,0,4,9,1,8,9,4,9,0,
%T A103710 3,8,7,5,9,7,8,3,2,2,0,3,6,3,8,5,8,3,4,8,3,9,2,9,9,7,5,3,4,6,6,4,4,1,
%U A103710 0,9,6,6,2,6,8,4,1,3,3,1,2,6,6,8,4,0,9,4,4,2,6,2,3,7,8,9,7,6,1,5,5,9,1,7,5
%N A103710 Decimal expansion of the ratio of the length of the latus rectum arc of any parabola to its semi latus rectum: sqrt(2) + log(1 + sqrt(2)).
%C A103710 The universal parabolic constant, equal to the ratio of the latus rectum arc of any parabola to its focal parameter. Like Pi, it is transcendental.
%C A103710 Just as all circles are similar, all parabolas are similar. Just as the ratio of a semicircle to its radius is always Pi, the ratio of the latus rectum arc of any parabola to its semi latus rectum is sqrt(2) + log(1 + sqrt(2)).
%C A103710 Note the remarkable similarity to sqrt(2) - log(1 + sqrt(2)), the universal equilateral hyperbolic constant A222362, which is a ratio of areas rather than of arc lengths. Lockhart (2012) says "the arc length integral for the parabola .. is intimately connected to the hyperbolic area integral ... I think it is surprising and wonderful that the length of one conic section is related to the area of another."
%C A103710 Is it a coincidence that the universal parabolic constant is equal to 6 times the expected distance A103712 from a randomly selected point in the unit square to its center? (Reese, 2004; Finch, 2012)
%D A103710 H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
%D A103710 P. Lockhart, Measurement, Harvard University Press, 2012, p. 369.
%D A103710 C. E. Love, Differential and Integral Calculus, 4th ed., Macmillan, 1950, pp. 286-288.
%D A103710 C. S. Ogilvy, Excursions in Geometry, Oxford Univ. Press, 1969, p. 84.
%D A103710 S. Reese, A universal parabolic constant, 2004, preprint.
%H A103710 Vincenzo Librandi, <a href="/A103710/b103710.txt">Table of n, a(n) for n = 1..10000</a>
%H A103710 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 A103710 S. R. Finch, <a href="http://arxiv.org/abs/2001.00578">Mathematical Constants, Errata and Addenda</a>, 2012, section 8.1.
%H A103710 M. Hajja, <a href="https://zbmath.org/?q=an:1291.51018">Review Zbl 1291.51018</a>, zbMATH 2015.
%H A103710 M. Hajja, <a href="https://zbmath.org/?q=an:1291.51016">Review Zbl 1291.51016</a>, zbMATH 2015.
%H A103710 H. Khelif, <a href="http://images-archive.math.cnrs.fr/L-arbelos-Partie-II.html#nb4">L’arbelos, Partie II, Généralisations de l’arbelos</a>, Images des Mathématiques, CNRS, 2014.
%H A103710 J. Pahikkala, <a href="http://planetmath.org/arbelosandparbelos">Arc Length Of Parabola</a>, PlanetMath.
%H A103710 S. Reese, <a href="https://adelphi.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=036c3e0a-935d-43b7-9042-1e8676a908fc">Pohle Colloquium Video Lecture: The universal parabolic constant, Feb 02 2005</a>
%H A103710 S. Reese, Jonathan Sondow, Eric W. Weisstein, <a href="https://mathworld.wolfram.com/UniversalParabolicConstant.html">MathWorld: Universal Parabolic Constant</a>
%H A103710 Manas Shetty, Prajwal DSouza, Sparsha Kumari, Vinton Adrian Rebello, <a href="https://prajwalsouza.github.io/universal-parabolic-constant.html">Where is the Parabola? The Parabolic Constant Mystery</a>
%H A103710 Jonathan 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 A103710 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 A103710 Wikipedia, <a href="http://en.wikipedia.org/wiki/Universal_parabolic_constant">Universal parabolic constant</a>
%H A103710 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A103710 Equals 2*Integral_{x = 0..1} sqrt(1 + x^2) dx. - _Peter Bala_, Feb 28 2019
%e A103710 2.29558714939263807403429804918949038759783220363858348392997534664...
%t A103710 RealDigits[ Sqrt[2] + Log[1 + Sqrt[2]], 10, 111][[1]] (* _Robert G. Wilson v_ Feb 14 2005 *)
%o A103710 (Maxima) fpprec: 100$ ev(bfloat(sqrt(2) + log(1 + sqrt(2)))); /* _Martin Ettl_, Oct 17 2012 */
%o A103710 (PARI) sqrt(2)+log(1+sqrt(2)) \\ _Charles R Greathouse IV_, Mar 08 2013
%Y A103710 A002193 + A091648.
%Y A103710 Cf. A103711, A103712, A222362, A232716, A232717.
%K A103710 cons,easy,nonn
%O A103710 1,1
%A A103710 Sylvester Reese and _Jonathan Sondow_, Feb 13 2005