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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)). 4
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

Table of n, a(n) for n=0..109.

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

Wikipedia, Equilateral hyperbola

Wikipedia, Universal parabolic constant

FORMULA

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

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

CROSSREFS

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

Sequence in context: A271523 A125844 A171025 * A176524 A268690 A065627

Adjacent sequences:  A222359 A222360 A222361 * A222363 A222364 A222365

KEYWORD

cons,easy,nonn

AUTHOR

Sylvester Reese and Jonathan Sondow, Mar 01 2013

STATUS

approved

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Last modified January 21 08:26 EST 2017. Contains 281102 sequences.