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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
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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
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
J.-F. Alcover, Asymptote of the logarithmic curve involute.
I.N. Bronshtein, Handbook of Mathematics, 5th ed., Springer, 2007, p. 202, eq. (3.338a).
S. R. Finch, Mathematical Constants, Errata and Addenda, 2012, section 8.1.
J. Pahikkala, Arc Length Of Parabola, PlanetMath.
S. Reese, J. Sondow, Universal Parabolic Constant, MathWorld
E.W. Weisstein, Rectangular hyperbola, MathWorld
FORMULA
Sqrt(2) - arcsinh(1), also equals Integral_{1..infinity} 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
EXAMPLE
0.532839975353552023569079399229905769541511547115312662423384129337355...
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]]
PROG
(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
(Magma) Sqrt(2) - Log(Sqrt(2)+1) // G. C. Greubel, Feb 02 2018
CROSSREFS
Sequence in context: A352903 A125844 A171025 * A352024 A176524 A309276
KEYWORD
cons,easy,nonn
AUTHOR
Sylvester Reese and Jonathan Sondow, Mar 01 2013
STATUS
approved

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Last modified May 29 10:24 EDT 2024. Contains 372938 sequences. (Running on oeis4.)