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A003957 The Dottie number: decimal expansion of root of cos(x) = x. 43
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
From Ben Branman, Dec 28 2011: (Start)
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.
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)
REFERENCES
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.
A. Baker, Theorem 1.4 in Transcendental Number Theory. Cambridge, England: Cambridge University Press, 1975.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..499 from Ben Branman)
Hrant Arakelian, New Fundamental Mathematical Constant: History, Present State and Prospects, Nonlinear Science Letters B, Vol. 1, No. 4, pp. 183-193.
Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, 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.
J. Bertrand, Exercise III, in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie Hachette et Cie, p. 285, 1865.
Samuel R. Kaplan, The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 73-74.
T. H. Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc. 21, 160-162, 1902.
V. Salov, Inevitable Dottie Number. Iterals of cosine and sine, arXiv preprint arXiv:1212.1027 [math.HO], 2012.
Eric Weisstein's World of Mathematics, Dottie Number
FORMULA
Equals twice A197002. - Hugo Pfoertner, Feb 20 2024
EXAMPLE
0.73908513321516064165531208767387340401341175890075746496568063577328...
MAPLE
evalf(solve(cos(x)=x, x), 140); # Alois P. Heinz, Feb 20 2024
MATHEMATICA
RealDigits[ FindRoot[ Cos[x] == x, {x, {.7, 1} }, WorkingPrecision -> 120] [[1, 2] ]] [[1]]
FindRoot[Cos[x] == x, {x, {.7, 1}}, WorkingPrecision -> 500][[1, 2]]][[1]] (* Ben Branman, Apr 12 2008 *)
N[NestList[Cos, 1, 100], 20] (* Clark Kimberling, Jul 01 2019 *)
RealDigits[Root[{# - Cos[#] &, 0.739085}], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)
RealDigits[Sqrt[1 - (2 InverseBetaRegularized[1/2, 1/2, 3/2] - 1)^2], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)
PROG
(PARI) solve(x=0, 1, cos(x)-x) \\ Charles R Greathouse IV, Dec 31 2011
(Python)
from sympy import Symbol, nsolve, cos
x = Symbol("x")
a = list(map(int, str(nsolve(cos(x)-x, 1, prec=110))[2:-2]))
print(a) # Michael S. Branicky, Jul 15 2022
CROSSREFS
Cf. A330119 (degrees-based analog).
Sequence in context: A368654 A358184 A361604 * A021579 A139788 A093525
KEYWORD
nonn,cons
AUTHOR
EXTENSIONS
More terms from David W. Wilson
Additional references from Ben Branman, Dec 28 2011
STATUS
approved

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