

A222362


Decimal expansion of the ratio of the area of the latus rectum segment of any equilateral hyperbola to the square of its semiaxis: 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 semiaxes are equal (and are called the radius of the circle), equilateral (or rectangular) hyperbolas are hyperbolas whose semiaxes 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 semiaxis 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)).  JeanFrançois 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.


FORMULA

Sqrt(2)  arcsinh(1), also equals Integral_{1..infinity} 1/(x^2*(1+x)^(1/2)) dx.  JeanFrançois 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



MATHEMATICA

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


PROG



CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



