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)).

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, 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, 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

Offset: 1

Author

Sylvester Reese and Jonathan Sondow, Feb 13 2005

Keywords

• cons
• easy
• nonn

Comments

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.

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)).

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."

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)

Examples

			2.29558714939263807403429804918949038759783220363858348392997534664...
		

References

• H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
• P. Lockhart, Measurement, Harvard University Press, 2012, p. 369.
• 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, Pohle Colloquium Video Lecture: The universal parabolic constant, Feb 02 2005
• S. Reese, Jonathan Sondow, Eric W. Weisstein, MathWorld: Universal Parabolic Constant
• Manas Shetty, Prajwal DSouza, Sparsha Kumari, Vinton Adrian Rebello, Where is the Parabola? The Parabolic Constant Mystery
• Jonathan 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.
• Wikipedia, Universal parabolic constant
• Index entries for transcendental numbers

Crossrefs

A002193 + A091648.

Cf. A103711, A103712, A222362, A232716, A232717.

Programs

Mathematica

RealDigits[ Sqrt[2] + Log[1 + Sqrt[2]], 10, 111][[1]] (* Robert G. Wilson v Feb 14 2005 *)

Maxima

fpprec: 100$ ev(bfloat(sqrt(2) + log(1 + sqrt(2)))); /* Martin Ettl, Oct 17 2012 */

PARI

sqrt(2)+log(1+sqrt(2)) \\ Charles R Greathouse IV, Mar 08 2013

Formula

Equals 2*Integral_{x = 0..1} sqrt(1 + x^2) dx. - Peter Bala, Feb 28 2019