A222362 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)).

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, 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, 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

Offset: 0

Author

Sylvester Reese and Jonathan Sondow, Mar 01 2013

Keywords

• cons
• easy
• nonn

Comments

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.

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)).

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".

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

Examples

			0.532839975353552023569079399229905769541511547115312662423384129337355...
		

References

• H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
• P. Lockhart, Measurement, Harvard University Press, 2012, p. 369.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• J.-F. Alcover, Asymptote of the logarithmic curve involute.
• I.N. Bronshtein, Handbook of Mathematics, 5th ed., Springer, 2007, p. 202, eq. (3.338a).
• Steven R. Finch, Mathematical Constants, Errata and Addenda, 2012, section 8.1.
• J. Pahikkala, Arc Length Of Parabola, PlanetMath.
• Sylvester Reese and Jonathan Sondow, Universal Parabolic Constant, MathWorld.
• Eric Weisstein's World of Mathematics, Rectangular hyperbola.
• Wikipedia, Equilateral hyperbola.
• Wikipedia, Universal parabolic constant.
• Index entries for transcendental numbers.

Crossrefs

Cf. A002193, A091648, A103710, A103711, A180434, A278386.

Programs

Magma

Sqrt(2) - Log(Sqrt(2)+1); // G. C. Greubel, Feb 02 2018

Maple

Digits:=100: evalf(sqrt(2)-arcsinh(1)); # Wesley Ivan Hurt, Nov 27 2016

Mathematica

RealDigits[Sqrt[2] - Log[1 + Sqrt[2]], 10, 111][[1]]

PARI

sqrt(2)-log(sqrt(2)+1) \\ Charles R Greathouse IV, Apr 18 2013

PARI

sqrt(2)-asinh(1) \\ Charles R Greathouse IV, Dec 04 2020

Formula

Sqrt(2) - arcsinh(1), also equals Integral_{1..oo} 1/(x^2*(1+x)^(1/2)) dx. - Jean-François Alcover, Apr 16 2015

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