Dottie number
In mathematics, the Dottie number is a constant that is the unique real root of the equation
- ,
where the argument of is in radians.
The decimal expansion of the Dottie number is given by:
Since is decreasing and its derivative is non-zero at , it only crosses zero at one point. This implies that the equation has only one real solution. It is the single real-valued fixed point of the cosine function and is a nontrivial example of a universal attracting fixed point. It is also a transcendental number because of the Lindemann–Weierstrass theorem.[1] The generalised case for a complex variable has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.
The name of the constant originates from a professor of French named Dottie who observed the number by repeatedly pressing the cosine button on her calculator.[2][nb 1]
The Dottie number, for which an exact series expansion can be obtained using the Faà di Bruno formula, has interesting connections with the Kepler and Bertrand's circle problems.[4]
Identities
[edit]The Dottie number appears in the closed form expression of some integrals:[5][6]
Using the Taylor series of the inverse of at (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series:
where each is a rational number defined for odd n as[2][7][8][nb 2]
The Dottie number can also be expressed as:
where is the inverse of the regularized beta function. This value can be obtained using Kepler's equation, along with other equivalent closed forms. [4]
In Microsoft Excel and LibreOffice Calc spreadsheets, the Dottie number can be expressed in closed form as SQRT(1-(2*BETA.INV(1/2,1/2,3/2)-1)^2)
. In the Mathematica computer algebra system, the Dottie number is Sqrt[1 - (2 InverseBetaRegularized[1/2, 1/2, 3/2] - 1)^2]
.
Another closed form representation:
where is the inverse survival function of Student's t-distribution. In Microsoft Excel and LibreOffice Calc, due to the specifics of the realization of `TINV` function, this can be expressed as formulas 2 *SQRT(3)* TINV(1/2, 3)/(TINV(1/2, 3)^2+3)
and TANH(2*ATANH(1/SQRT(3) * TINV(1/2,3)))
.
Notes
[edit]- ^ If a calculator is set to take angles in degrees, the sequence of numbers will instead converge to ,[3] the root of .
- ^ Kaplan does not give an explicit formula for the terms of the series, which follows trivially from the Lagrange inversion theorem.
References
[edit]- ^ Eric W. Weisstein. "Dottie Number".
- ^ a b Kaplan, Samuel R (February 2007). "The Dottie Number" (PDF). Mathematics Magazine. 80: 73. doi:10.1080/0025570X.2007.11953455. S2CID 125871044. Retrieved 29 November 2017.
- ^ Sloane, N. J. A. (ed.). "Sequence A330119". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b Pain, Jean-Christophe (2023). "An exact series expansion for the Dottie number". arXiv:2303.17962.
- ^ Michos, Alexander (2023-03-03), A Brief Investigation of an Integral Representation of Dottie's Number, doi:10.31219/osf.io/3rzj5, retrieved 2024-09-24
- ^ "Integral Representation of the Dottie Number". Mathematics Stack Exchange.
- ^ "OEIS A302977 Numerators of the rational factor of Kaplan's series for the Dottie number". oeis.org. Retrieved 2019-05-26.
- ^ "A306254 - OEIS". oeis.org. Retrieved 2019-07-22.
External links
[edit]- Miller, T. H. (Feb 1890). "On the numerical values of the roots of the equation cosx = x". Proceedings of the Edinburgh Mathematical Society. 9: 80–83. doi:10.1017/S0013091500030868.
- Salov, Valerii (2012). "Inevitable Dottie Number. Iterals of cosine and sine". arXiv:1212.1027.
- Azarian, Mohammad K. (2008). "ON THE FIXED POINTS OF A FUNCTION AND THE FIXED POINTS OF ITS COMPOSITE FUNCTIONS" (PDF). International Journal of Pure and Applied Mathematics.