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%I #70 Sep 08 2022 08:46:04
%S 5,3,2,8,3,9,9,7,5,3,5,3,5,5,2,0,2,3,5,6,9,0,7,9,3,9,9,2,2,9,9,0,5,7,
%T 6,9,5,4,1,5,1,1,5,4,7,1,1,5,3,1,2,6,6,2,4,2,3,3,8,4,1,2,9,3,3,7,3,5,
%U 5,2,9,4,2,4,0,0,8,0,9,5,1,0,1,6,6,8,0,6,4,2,4,1,7,3,8,5,5,2,9,8,7,8,2,7,4,0,3,0,0,3
%N Decimal expansion of the ratio of the area of the latus rectum segment of any equilateral hyperbola to the square of its semi-axis: sqrt(2) - log(1 + sqrt(2)).
%C Just as circles are ellipses whose semi-axes are equal (and are called the radius of the circle), equilateral (or rectangular) hyperbolas are hyperbolas whose semi-axes are equal.
%C Just as the ratio of the area of a circle to the square of its radius is always Pi, the ratio of the area of the latus rectum segment of any equilateral hyperbola to the square of its semi-axis is the universal equilateral hyperbolic constant sqrt(2) - log(1 + sqrt(2)).
%C Note the remarkable similarity to sqrt(2) + log(1 + sqrt(2)), the universal parabolic constant A103710, which is a ratio of arc lengths rather than of areas. 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 This constant is also the abscissa of the vertical asymptote of the involute of the logarithmic curve (starting point (1,0)). - _Jean-François Alcover_, Nov 25 2016
%D H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
%D P. Lockhart, Measurement, Harvard University Press, 2012, p. 369.
%H G. C. Greubel, <a href="/A222362/b222362.txt">Table of n, a(n) for n = 0..10000</a>
%H J.-F. Alcover, <a href="/A222362/a222362.pdf">Asymptote</a> of the logarithmic curve involute.
%H I.N. Bronshtein, <a href="http://books.google.com/books?id=gCgOoMpluh8C&lpg=PA202&vq=%22areas%20in%20the%20hyperbola%22&pg=PA202#v=onepage&q&f=false">Handbook of Mathematics</a>, 5th ed., Springer, 2007, p. 202, eq. (3.338a).
%H S. R. Finch, <a href="http://arxiv.org/abs/2001.00578">Mathematical Constants, Errata and Addenda</a>, 2012, section 8.1.
%H J. Pahikkala, <a href="http://planetmath.org/arclengthofparabola">Arc Length Of Parabola</a>, PlanetMath.
%H S. Reese, J. Sondow, <a href="http://mathworld.wolfram.com/UniversalParabolicConstant.html">Universal Parabolic Constant</a>, MathWorld
%H E.W. Weisstein, <a href="http://mathworld.wolfram.com/RectangularHyperbola.html">Rectangular hyperbola</a>, MathWorld
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hyperbola#Rectangular_hyperbola_with_horizontal.2Fvertical_asymptotes_.28Cartesian_coordinates.29">Equilateral hyperbola</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 Sqrt(2) - arcsinh(1), also equals Integral_{1..infinity} 1/(x^2*(1+x)^(1/2)) dx. - _Jean-François Alcover_, Apr 16 2015
%F Equals Integral_{x = 0..1} x^2/sqrt(1 + x^2) dx. - _Peter Bala_, Feb 28 2019
%e 0.532839975353552023569079399229905769541511547115312662423384129337355...
%p Digits:=100: evalf(sqrt(2)-arcsinh(1)); # _Wesley Ivan Hurt_, Nov 27 2016
%t RealDigits[Sqrt[2] - Log[1 + Sqrt[2]], 10, 111][[1]]
%o (PARI) sqrt(2)-log(sqrt(2)+1) \\ _Charles R Greathouse IV_, Apr 18 2013
%o (PARI) sqrt(2)-asinh(1) \\ _Charles R Greathouse IV_, Dec 04 2020
%o (Magma) Sqrt(2) - Log(Sqrt(2)+1) // _G. C. Greubel_, Feb 02 2018
%Y Cf. A002193, A091648, A103710, A103711, A180434, A278386.
%K cons,easy,nonn
%O 0,1
%A Sylvester Reese and _Jonathan Sondow_, Mar 01 2013
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