Dottie number

{{Short description|Mathematical constant related to the cosine function}}

file:Cosine_fixed_point.svg of the cosine function.]]

In mathematics, the Dottie number or the cosine constant is a constant that is the unique real root of the equation

: $\cos x = x$ ,

where the argument of $\cos$ is in radians.

The decimal expansion of the Dottie number is given by:

: {{mvar|D}} = {{val|0.739085133215160641655312087673}}... {{OEIS|id=A003957}}.

Since $\cos(x) - x$ is decreasing and its derivative is non-zero at $\cos(x) - x = 0$ , it only crosses zero at one point. This implies that the equation $\cos(x) = x$ 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.{{cite web|url=http://mathworld.wolfram.com/DottieNumber.html|author=Eric W. Weisstein|authorlink=Eric W. Weisstein|title=Dottie Number}} The generalised case $\cos z = z$ for a complex variable $z$ has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.

file:quadrisection.svg

History

The constant appeared in publications as early as 1860s.{{Cite web |last=Weisstein |first=Eric W. |title=Dottie Number |url=https://mathworld.wolfram.com/DottieNumber.html |access-date=2025-01-29 |website=mathworld.wolfram.com |language=en}} Norair Arakelian used lowercase ayb (ա) from the Armenian alphabet to denote the constant.

The constant name was coined by Samuel R. Kaplan in 2007. It originates from a professor of French named Dottie who observed the number by repeatedly pressing the cosine button on her calculator.{{refn|group=nb|If a calculator is set to take angles in degrees, the sequence of numbers will instead converge to $0.999847...$ ,{{Cite OEIS|A330119}} the root of $\cos\left(\frac{\pi}{180}x\right) = x$ .}}

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.{{cite arXiv |last=Pain |first=Jean-Christophe |date=2023 |title=An exact series expansion for the Dottie number |class=math.NT |eprint=2303.17962}}

Identities

The Dottie number appears in the closed form expression of some integrals:{{Citation |last=Michos |first=Alexander |title=A Brief Investigation of an Integral Representation of Dottie's Number |date=2023-03-03 |url=https://osf.io/3rzj5 |access-date=2024-09-24 |doi=10.31219/osf.io/3rzj5}}{{cite web |title=Integral Representation of the Dottie Number |url=https://math.stackexchange.com/questions/2446725/integral-representation-of-the-dottie-number |website=Mathematics Stack Exchange |language=en}}

: $\int _0^{\infty }\ln \left(\frac{4\left(x+\sinh x\right)^2+\pi^2}{4(x-\sinh x)^2+\pi ^2}\right)\mathrm{d} x = \pi^2 - 2\pi D$

: $\int_{0 }^{\infty } \frac{3\pi^2+4(x-\sinh x)^2}{(3\pi^2+4(x-\sinh x)^2)^2 + 16\pi^2(x-\sinh x)^2} \, \mathrm dx = \frac1{8+8\sqrt{1-D^2}}$

Using the Taylor series of the inverse of $f(x) = \cos(x) - x$ at $\frac{\pi}{2}$ (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series:

: $D = \frac{\pi}{2}+\sum_{n\,\mathrm{odd}} a_{n} \pi^{n}$

where each $a_n$ is a rational number defined for odd n as{{cite journal |last1=Kaplan |first1=Samuel R |date=February 2007 |title=The Dottie Number |url=https://www.maa.org/sites/default/files/Kaplan2007-131105.pdf |journal=Mathematics Magazine |volume=80 |page=73 |doi=10.1080/0025570X.2007.11953455 |s2cid=125871044 |accessdate=29 November 2017}}{{Cite web |title=OEIS A302977 Numerators of the rational factor of Kaplan's series for the Dottie number. |url=https://oeis.org/A302977 |access-date=2019-05-26 |website=oeis.org}}{{Cite web |title=A306254 - OEIS |url=https://oeis.org/A306254 |access-date=2019-07-22 |website=oeis.org}}{{refn|group=nb|Kaplan does not give an explicit formula for the terms of the series, which follows trivially from the Lagrange inversion theorem.}}

: $\begin{align}
a_n&=\frac{1}{n!2^n}\lim_{m\to\frac\pi2}
\frac{\partial^{n-1}}{\partial m^{n-1}}{\left(\frac{\cos m}{m-\pi/2}-1\right)^{-n}}
\\&=-\frac{1}{4},-\frac{1}{768},-\frac{1}{61440},-\frac{43}{165150720},\ldots
\end{align}$

The Dottie number can also be expressed as:

: $D=\sqrt{1-\left(1-2I^{-1}_\frac12\left(\frac 12,\frac 32\right)\right)^2},$

where $I^{-1}$ is the inverse of the regularized beta function. This value can be obtained using Kepler's equation, along with other equivalent closed forms. $I^{-1}_\frac12\left(\tfrac 12,\tfrac 32\right) \approx 0.16319$ is the median of a beta distribution with parameters 1/2 and 3/2.

In Microsoft Excel and LibreOffice Calc spreadsheets, the Dottie number can be expressed in closed form as {{code|SQRT(1-(1-2*BETA.INV(1/2,1/2,3/2))^2)|dax|style=white-space:nowrap}}. In the Mathematica computer algebra system, the Dottie number is {{code|Sqrt[1 - (1-2 InverseBetaRegularized[1/2, 1/2, 3/2])^2]|mathematica|style=white-space:nowrap}}.

Another closed form representation:

: $D=- \tanh\left(2\text{ arctanh}\left(\frac1{\sqrt3} \operatorname{InvT} \left(\frac14,3\right)\right)\right)=-\frac{2\sqrt3 {\operatorname{InvT}\left(\frac14,3\right)}}{\operatorname{InvT}^2\left(\frac14,3\right)+3},$

where $\operatorname{InvT}$ 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 {{code|2 *SQRT(3)* TINV(1/2, 3)/(TINV(1/2, 3)^2+3)|dax|style=white-space:nowrap}} and {{code|TANH(2*ATANH(1/SQRT(3) * TINV(1/2,3)))|dax|style=white-space:nowrap}}.

Notes

References

External links

{{cite journal |journal=Proceedings of the Edinburgh Mathematical Society|title=On the numerical values of the roots of the equation cosx = x|first1=T. H. |last1=Miller|date=Feb 1890|volume=9|pages=80–83|doi=10.1017/S0013091500030868|doi-access=free}}
{{cite news |title=Inevitable Dottie Number. Iterals of cosine and sine |first1=Valerii |last1=Salov|date=2012|arxiv=1212.1027 }}
{{cite journal |url=https://ijpam.eu/contents/2008-46-1/3/3.pdf |journal=International Journal of Pure and Applied Mathematics |title=ON THE FIXED POINTS OF A FUNCTION AND THE FIXED POINTS OF ITS COMPOSITE FUNCTIONS|first1=Mohammad K. |last1=Azarian|year=2008}}

Category:Mathematical constants

Category:Real transcendental numbers

Category:Fixed points (mathematics)