A003957 The Dottie number: decimal expansion of root of cos(x) = x.

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

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.

Trefor Bazett, What is cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos(...?? // Banach Fixed Point Theorem, YouTube video, 2022.

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.

David R. Stoutemyer, Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses, arXiv:2207.00707 [math.GM], 2022.

Eric Weisstein's World of Mathematics, Dottie Number

Example

0.73908513321516064165531208767387340401341175890075746496568063577328...

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. A009442, A177413, A182503, A200309, A212112, A212113, A217066.

Cf. A330119 (degrees-based analog).

Sequence in context: A072334 A358184 A361604 * A021579 A139788 A093525

Adjacent sequences: A003954 A003955 A003956 * A003958 A003959 A003960

Keyword

nonn,cons

Author

Leonid Broukhis

Extensions

More terms from David W. Wilson

Additional references from Ben Branman, Dec 28 2011

Status

approved