OFFSET
1,3
COMMENTS
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.
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
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.
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)
REFERENCES
H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
C. E. Love, Differential and Integral Calculus, 4th ed., Macmillan, 1950, pp. 286-288.
C. S. Ogilvy, Excursions in Geometry, Oxford Univ. Press, 1969, p. 84.
S. Reese, A universal parabolic constant, 2004, preprint.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
J. L. Diaz-Barrero and W. Seaman, A limit computed by integration, Problem 810 and Solution, College Math. J., 37 (2006), 316-318, equation (5).
S. R. Finch, Mathematical Constants, Errata and Addenda, 2012, section 8.1.
M. Hajja, Review Zbl 1291.51018, zbMATH 2015.
M. Hajja, Review Zbl 1291.51016, zbMATH 2015.
H. Khelif, L’arbelos, Partie II, Généralisations de l’arbelos, Images des Mathématiques, CNRS, 2014.
J. Pahikkala, Arc Length Of Parabola, PlanetMath.
S. Reese, J. Sondow, Universal Parabolic Constant, MathWorld
J. Sondow, The parbelos, a parabolic analog of the arbelos, arXiv 2012, Amer. Math. Monthly, 120 (2013), 929-935.
E. Tsukerman, Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos, arXiv 2012, Amer. Math. Monthly, 121 (2014), 438-443.
Eric Weisstein's World of Mathematics, Universal Parabolic Constant
Wikipedia, Universal parabolic constant
FORMULA
Equals Integral_{x = 0..1} sqrt(1 + x^2) dx. - Peter Bala, Feb 28 2019
EXAMPLE
1.14779357469631903701714902459474519379891610181929174196498767332...
MATHEMATICA
RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/2, 10, 111][[1]] (* Robert G. Wilson v, Feb 14 2005 *)
N[Integrate[Sqrt[1 + x^2], {x, 0, 1}], 120] (* Clark Kimberling, Jan 06 2014 *)
CROSSREFS
KEYWORD
AUTHOR
Sylvester Reese and Jonathan Sondow, Feb 13 2005
STATUS
approved