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Decimal expansion of arccosh(sqrt(2)), the inflection point of sech(x).
+10
26
8, 8, 1, 3, 7, 3, 5, 8, 7, 0, 1, 9, 5, 4, 3, 0, 2, 5, 2, 3, 2, 6, 0, 9, 3, 2, 4, 9, 7, 9, 7, 9, 2, 3, 0, 9, 0, 2, 8, 1, 6, 0, 3, 2, 8, 2, 6, 1, 6, 3, 5, 4, 1, 0, 7, 5, 3, 2, 9, 5, 6, 0, 8, 6, 5, 3, 3, 7, 7, 1, 8, 4, 2, 2, 2, 0, 2, 6, 0, 8, 7, 8, 3, 3, 7, 0, 6, 8, 9, 1, 9, 1, 0, 2, 5, 6, 0, 4, 2, 8, 5, 6
COMMENTS
Asymptotic growth constant in the exponent for the number of spanning trees on the 2 X infinity strip on the square lattice. - R. J. Mathar, May 14 2006 Arccosh(sqrt(2)) = (1/2)*log((sqrt(2)+1)/(sqrt(2)-1)) = log(tan(3*Pi/8)) = int(1/cos(x),x=0..Pi/4). Therefore, in Gerardus Mercator's (conformal) map this is the value of the ordinate y/R (R radius of the spherical earth) for latitude phi = 45 degrees north, or Pi/4. See, e.g., the Eli Maor reference, eqs. (5) and (6). This is the latitude of, e.g., the Mission Point Lighthouse, Michigan, U.S.A. - Wolfdieter Lang, Mar 05 2013 Decimal expansion of the arclength on the hyperbola y^2 - x^2 = 1 from (0,0) to (1,sqrt(2)). - Clark Kimberling, Jul 04 2020
REFERENCES
L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (85) page 16-17.
E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, chapter 13, A Mapmaker's Paradise, pp. 163-180.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:4 at page 283.
FORMULA
Equals Sum_{n>=1, n odd} binomial(2*n,n)/(n*4^n) [see Lehmer link]. - R. J. Mathar, Mar 04 2009 Equals arcsinh(1), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013 Equals lim_{n->oo} Sum_{k=1..n} 1/sqrt(n^2+k^2). - Amiram Eldar, May 19 2022 Equals Sum_{n >= 1} 1/(n*P(n, sqrt(2))*P(n-1, sqrt(2))), where P(n, x) denotes the n-th Legendre polynomial. The first twenty terms of the series gives the approximate value 0.88137358701954(24...), correct to 14 decimal places. - Peter Bala, Mar 16 2024 Equals 2F1(1/2,1/2;3/2;-1) [Krupnikov]. - R. J. Mathar, May 13 2024
EXAMPLE
0.8813735870195430252326093249797923090281603282616...
MATHEMATICA
RealDigits[Log[1 + Sqrt[2]], 10, 100][[1]] (* Alonso del Arte, Aug 11 2011 *)
PROG
(Maxima) fpprec : 100$ ev(bfloat(log(1 + sqrt(2)))); /* Martin Ettl, Oct 17 2012 */
Decimal expansion of the ratio of the length of the latus rectum arc of any parabola to its semi latus rectum: sqrt(2) + log(1 + sqrt(2)).
+10
11
2, 2, 9, 5, 5, 8, 7, 1, 4, 9, 3, 9, 2, 6, 3, 8, 0, 7, 4, 0, 3, 4, 2, 9, 8, 0, 4, 9, 1, 8, 9, 4, 9, 0, 3, 8, 7, 5, 9, 7, 8, 3, 2, 2, 0, 3, 6, 3, 8, 5, 8, 3, 4, 8, 3, 9, 2, 9, 9, 7, 5, 3, 4, 6, 6, 4, 4, 1, 0, 9, 6, 6, 2, 6, 8, 4, 1, 3, 3, 1, 2, 6, 6, 8, 4, 0, 9, 4, 4, 2, 6, 2, 3, 7, 8, 9, 7, 6, 1, 5, 5, 9, 1, 7, 5
COMMENTS
The universal parabolic constant, equal to the ratio of the latus rectum arc of any parabola to its focal parameter. Like Pi, it is transcendental.
Just as all circles are similar, all parabolas are similar. Just as the ratio of a semicircle to its radius is always Pi, the ratio of the latus rectum arc of any parabola to its semi latus rectum is sqrt(2) + log(1 + sqrt(2)).
Note the remarkable similarity to sqrt(2) - log(1 + sqrt(2)), the universal equilateral hyperbolic constant A222362, which is a ratio of areas rather than of arc lengths. 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." Is it a coincidence that the universal parabolic constant is equal to 6 times the expected distance A103712 from a randomly selected point in the unit square to its center? (Reese, 2004; Finch, 2012)
REFERENCES
H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
P. Lockhart, Measurement, Harvard University Press, 2012, p. 369.
C. E. Love, Differential and Integral Calculus, 4th ed., Macmillan, 1950, pp. 286-288.
C. S. Ogilvy, Excursions in Geometry, Oxford Univ. Press, 1969, p. 84.
S. Reese, A universal parabolic constant, 2004, preprint.
FORMULA
Equals 2*Integral_{x = 0..1} sqrt(1 + x^2) dx. - Peter Bala, Feb 28 2019
EXAMPLE
2.29558714939263807403429804918949038759783220363858348392997534664...
MATHEMATICA
RealDigits[ Sqrt[2] + Log[1 + Sqrt[2]], 10, 111][[1]] (* Robert G. Wilson v Feb 14 2005 *)
PROG
(Maxima) fpprec: 100$ ev(bfloat(sqrt(2) + log(1 + sqrt(2)))); /* Martin Ettl, Oct 17 2012 */
Decimal expansion of the expected distance from a randomly selected point in the unit square to its center: (sqrt(2) + log(1 + sqrt(2)))/6.
+10
10
3, 8, 2, 5, 9, 7, 8, 5, 8, 2, 3, 2, 1, 0, 6, 3, 4, 5, 6, 7, 2, 3, 8, 3, 0, 0, 8, 1, 9, 8, 2, 4, 8, 3, 9, 7, 9, 3, 2, 9, 7, 2, 0, 3, 3, 9, 3, 9, 7, 6, 3, 9, 1, 3, 9, 8, 8, 3, 2, 9, 2, 2, 4, 4, 4, 0, 6, 8, 4, 9, 4, 3, 7, 8, 0, 6, 8, 8, 8, 5, 4, 4, 4, 7, 3, 4, 9, 0, 7, 1, 0, 3, 9, 6, 4, 9, 6, 0, 2, 5, 9, 8, 6, 2, 5
COMMENTS
Is it a coincidence that this constant is equal to 1/6 of the universal parabolic constant A103710? (Reese, 2004; Finch, 2012) exp(d(2)) - exp(d(2))/Pi = 0.9994179247351742... ~ 1 - 1/1718. - Gerald McGarvey, Feb 21 2005 Take a point on a line of irrational slope and a line segment of a given length centered at the point, integrate the distance of a point on the line to the set of lattice points along the line segment, and divide by the length. The limit as the length approaches infinity can be shown by a generalization of the Equidistribution Theorem to give the expected distance of a point in the unit square to its corners, this constant. - Thomas Anton, Jun 19 2021
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 8.1.
S. Reese, A universal parabolic constant, 2004, preprint.
FORMULA
Equals (1/3)*Integral_{x = 0..1} sqrt(1 + x^2) dx. - Peter Bala, Feb 28 2019 Equals Integral_{x>=1} arcsinh(x)/x^4 dx. - Amiram Eldar, Jun 26 2021
EXAMPLE
0.38259785823210634567238300819824839793297203393976391398832922444...
MATHEMATICA
RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/6, 10, 111][[1]] (* Robert G. Wilson v, Feb 14 2005 *)
PROG
(Maxima) fpprec: 100$ ev(bfloat((sqrt(2) + log(1 + sqrt(2)))/6)); /* Martin Ettl, Oct 17 2012 */ (PARI) (sqrt(2) + log(1 + sqrt(2)))/6 \\ G. C. Greubel, Sep 22 2017
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)).
+10
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
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.
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
EXAMPLE
0.532839975353552023569079399229905769541511547115312662423384129337355...
MATHEMATICA
RealDigits[Sqrt[2] - Log[1 + Sqrt[2]], 10, 111][[1]]
Decimal expansion of (sqrt(2) + arcsinh(1))/4.
+10
2
5, 7, 3, 8, 9, 6, 7, 8, 7, 3, 4, 8, 1, 5, 9, 5, 1, 8, 5, 0, 8, 5, 7, 4, 5, 1, 2, 2, 9, 7, 3, 7, 2, 5, 9, 6, 8, 9, 9, 4, 5, 8, 0, 5, 0, 9, 0, 9, 6, 4, 5, 8, 7, 0, 9, 8, 2, 4, 9, 3, 8, 3, 6, 6, 6, 1, 0, 2, 7, 4, 1, 5, 6, 7, 1, 0, 3, 3, 2, 8, 1, 6, 7, 1, 0, 2, 3
COMMENTS
Maximal value of the average distance from the set of lattice points of any line in the plane, where average distance is the limit of the integral of the distance to the set over a segment of the line centered around a fixed point, divided by the length of the segment, as that length approaches infinity, achieved precisely by lines x = z + 1/2 and y = z + 1/2 for integers z.
The average distance between the center of a unit square to a randomly and uniformly chosen point on its perimeter. - Amiram Eldar, Jun 23 2022
FORMULA
Equals Integral_{x=0..1/2} 2*sqrt(x^2+1/4) dx.
EXAMPLE
0.57389678734815951850857451...
MATHEMATICA
First[RealDigits[N[(Sqrt[2]+ArcSinh[1])/4, 100]]] (* Stefano Spezia, Jun 21 2021 *)
Decimal expansion of the initial angle in radians above the horizon that maximizes the length of a projectile's trajectory.
+10
2
9, 8, 5, 5, 1, 4, 7, 3, 7, 8, 6, 2, 3, 1, 5, 4, 6, 2, 1, 1, 4, 9, 2, 8, 5, 3, 7, 2, 5, 7, 3, 0, 4, 6, 3, 8, 7, 7, 2, 4, 7, 2, 2, 0, 5, 9, 6, 7, 4, 2, 9, 6, 4, 8, 1, 2, 7, 8, 4, 5, 1, 1, 4, 0, 3, 2, 8, 2, 9, 5, 2, 7, 0, 5, 2, 0, 8, 0, 5, 3, 5, 7, 2, 5, 7, 1, 5
COMMENTS
A projectile is launched with an initial speed v at angle theta above the horizon. Assuming that the gravitational acceleration g is uniform and neglecting the air resistance, the trajectory is a part of a parabola whose length is maximized when the angle is the root of the equation csc(theta) = coth(csc(theta)). The maximal length is then u * v^2/g, where u = 1.1996... is the root of coth(x) = x ( A085984). The angle in degrees is 56.4658351274...
The initial angle that maximizes the horizontal distance is the well-known result theta = Pi/4 = 45 degrees. The corresponding length of trajectory in this case is u * v^2/g, where u = (sqrt(2) + arcsinh(1))/2 = 1.1477... ( A103711), which is 95.67...% of the maximum value.
REFERENCES
Thomas Szirtes, Applied Dimensional Analysis and Modeling, Butterworth-Heinemann, 2007, p. 578.
FORMULA
Equals arccsc(u) where u is the root of coth(x) = x ( A085984).
EXAMPLE
0.98551473786231546211492853725730463877247220596742...
MATHEMATICA
RealDigits[ArcCsc[x /. FindRoot[x == Coth[x], {x, 1}, WorkingPrecision -> 120]], 10, 100][[1]]
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