Dottie number

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 $\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{180\}x\backslash right)\; =\; x$.}}

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}}

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}}

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

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

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

:$D\; =\; \backslash frac\{\backslash pi\}\{2\}+\backslash sum\_\{n\backslash ,\backslash mathrm\{odd\}\}\; a\_\{n\}\; \backslash pi^\{n\}$

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.}}

:$\backslash begin\{align\}$

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:

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

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.

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

Another closed form representation:

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

where $\backslash operatorname\{InvT\}$ 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}}.

{{reflist|group=nb}}

{{Reflist}}

• {{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)

{{Irrational number}}