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%I #129 Feb 20 2024 08:34:36
%S 7,3,9,0,8,5,1,3,3,2,1,5,1,6,0,6,4,1,6,5,5,3,1,2,0,8,7,6,7,3,8,7,3,4,
%T 0,4,0,1,3,4,1,1,7,5,8,9,0,0,7,5,7,4,6,4,9,6,5,6,8,0,6,3,5,7,7,3,2,8,
%U 4,6,5,4,8,8,3,5,4,7,5,9,4,5,9,9,3,7,6,1,0,6,9,3,1,7,6,6,5,3,1,8,4,9,8,0,1,2,4,6
%N The Dottie number: decimal expansion of root of cos(x) = x.
%C Let P be the point in quadrant I where the curve y=sin(x) meets the circle x^2+y^2=1. Let d be the Dottie number. Then P=(d,sin(d)), and d is the slope at P of the sine curve. - _Clark Kimberling_, Oct 07 2011
%C From _Ben Branman_, Dec 28 2011: (Start)
%C The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator users before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.
%C The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian small letter ayb (ա, the first letter in the Armenian alphabet) to denote this constant. (End)
%D H. Arakelian, The Fundamental Dimensionless Values (Their Role and Importance for the Methodology of Science). [In Russian.] Yerevan, Armenia: Armenian National Academy of Sciences, 1981.
%D A. Baker, Theorem 1.4 in Transcendental Number Theory. Cambridge, England: Cambridge University Press, 1975.
%H G. C. Greubel, <a href="/A003957/b003957.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..499 from Ben Branman)
%H Hrant Arakelian, <a href="http://www.nonlinearscience.com/paper.php?pid=0000000113">New Fundamental Mathematical Constant: History, Present State and Prospects</a>, Nonlinear Science Letters B, Vol. 1, No. 4, pp. 183-193.
%H Mohammad K. Azarian, <a href="http://www.ijpam.eu/contents/2008-46-1/3/3.pdf">On the Fixed Points of a Function and the Fixed Points of its Composite Functions</a>, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
%H Trefor Bazett, <a href="https://www.youtube.com/watch?v=qHnXE_h5c2M">What is cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos(...?? // Banach Fixed Point Theorem</a>, YouTube video, 2022.
%H J. Bertrand, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k6570951r/f297.item.texteImage">Exercise III</a>, in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie Hachette et Cie, p. 285, 1865.
%H Samuel R. Kaplan, <a href="https://www.maa.org/sites/default/files/Kaplan2007-131105.pdf">The Dottie Number</a>, Math. Magazine, 80 (No. 1, 2007), 73-74.
%H T. H. Miller, <a href="http://dx.doi.org/10.1017/S0013091500030868">On the imaginary roots of cos x = x</a>, Proc. Edinburgh Math. Soc. 21, 160-162, 1902.
%H V. Salov, <a href="http://arxiv.org/abs/1212.1027">Inevitable Dottie Number. Iterals of cosine and sine</a>, arXiv preprint arXiv:1212.1027 [math.HO], 2012.
%H David R. Stoutemyer, <a href="https://arxiv.org/abs/2207.00707">Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses</a>, arXiv:2207.00707 [math.GM], 2022.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DottieNumber.html">Dottie Number</a>
%F Equals twice A197002. - _Hugo Pfoertner_, Feb 20 2024
%e 0.73908513321516064165531208767387340401341175890075746496568063577328...
%p evalf(solve(cos(x)=x,x), 140); # _Alois P. Heinz_, Feb 20 2024
%t RealDigits[ FindRoot[ Cos[x] == x, {x, {.7, 1} }, WorkingPrecision -> 120] [[1, 2] ]] [[1]]
%t FindRoot[Cos[x] == x, {x, {.7, 1}}, WorkingPrecision -> 500][[1, 2]]][[1]] (* _Ben Branman_, Apr 12 2008 *)
%t N[NestList[Cos, 1, 100], 20] (* _Clark Kimberling_, Jul 01 2019 *)
%t RealDigits[Root[{# - Cos[#] &, 0.739085}], 10, 100][[1]] (* _Eric W. Weisstein_, Jul 15 2022 *)
%t RealDigits[Sqrt[1 - (2 InverseBetaRegularized[1/2, 1/2, 3/2] - 1)^2], 10, 100][[1]] (* _Eric W. Weisstein_, Jul 15 2022 *)
%o (PARI) solve(x=0,1,cos(x)-x) \\ _Charles R Greathouse IV_, Dec 31 2011
%o (Python)
%o from sympy import Symbol, nsolve, cos
%o x = Symbol("x")
%o a = list(map(int, str(nsolve(cos(x)-x, 1, prec=110))[2:-2]))
%o print(a) # _Michael S. Branicky_, Jul 15 2022
%Y Cf. A009442, A177413, A182503, A197002, A200309, A212112, A212113, A217066.
%Y Cf. A330119 (degrees-based analog).
%K nonn,cons
%O 0,1
%A _Leonid Broukhis_
%E More terms from _David W. Wilson_
%E Additional references from _Ben Branman_, Dec 28 2011
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