Dottie number

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

\cos x=x

,

The Dottie number is the unique real fixed point of the cosine function.

where the argument of $\cos$ is in radians. The decimal expansion of the Dottie number is $0.739085...$ .[1]

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.[2] 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.

The solution of quadrisection of circle into four parts of the same area with chords coming from the same point can be expressed via Dottie number.

Using the Taylor series of the inverse of $f(x)=\cos(x)-x$ at ${\textstyle {\frac {\pi }{2}}}$ (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series ${\textstyle {\frac {\pi }{2}}+\sum _{n\,\mathrm {odd} }a_{n}\pi ^{n}}$ where each $a_{n}$ is a rational number defined for odd n as[3][4][5][nb 1]

{\begin{aligned}a_{n}&={\frac {1}{n!2^{n}}}\lim _{m\to {\frac {\pi }{2}}}{\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{aligned}}

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.[3]

If a calculator is set to take angles in degrees, the sequence of numbers will instead converge to $0.999847...$ ,[6] the root of $\cos \left({\frac {\pi }{180}}x\right)=x$ .

Closed form

The Dottie number can be expressed as

D={\sqrt {1-\left(2I_{\frac {1}{2}}^{-1}\left({\frac {1}{2}},{\frac {3}{2}}\right)-1\right)^{2}}},

where $I^{-1}$ is the inverse regularized Beta function. This value can be obtained using Kepler's equation.[7]

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].

Integral representations

Dottie number can be represented as

D={\sqrt {1-\left(1-\left(\int _{0}^{\infty }{\frac {32(z-\sinh(z))^{2}+24\pi ^{2}}{\left(4(z-\sinh(z))^{2}+3\pi ^{2}\right)^{2}+16\pi ^{2}(z-\sinh(z))^{2}}}\,dz\right)^{-1}\right)^{2}}}

.

Another integral representation:

D={\frac {\pi }{2}}-{\frac {1}{2\pi }}\int _{0}^{\infty }\ln \left({\frac {2\pi \cosh(x)+\pi ^{2}}{x^{2}+\cosh ^{2}(x)}}+1\right)\,dx

Notes

Kaplan does not give an explicit formula for the terms of the series, which follows trivially from the Lagrange inversion theorem.

References

Sloane, N. J. A. (ed.). "Sequence A003957". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Eric W. Weisstein. "Dottie Number".
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.
"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.
Sloane, N. J. A. (ed.). "Sequence A330119". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Gaidash, Tyma (2022-02-23). "Why Dottie$=2\sqrt{I^{-1}_\frac12(\frac 12,\frac 32)-I^{-1}_\frac12(\frac 12,\frac 32)^2} = \sin^{-1}\big(1-2I^{-1}_\frac12(\frac 12,\frac 32)\big)$?". Math Stack Exchange. Retrieved 2023-08-11.

External links

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.

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[6] Kaplan does not give an explicit formula for the terms of the series, which follows trivially from the Lagrange inversion theorem.

[1] Sloane, N. J. A. (ed.). "Sequence A003957". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[2] Eric W. Weisstein. "Dottie Number".

[Kaplan-3] 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.

[4] "OEIS A302977 Numerators of the rational factor of Kaplan's series for the Dottie number". oeis.org. Retrieved 2019-05-26.

[5] "A306254 - OEIS". oeis.org. Retrieved 2019-07-22.

[7] Sloane, N. J. A. (ed.). "Sequence A330119". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[8] Gaidash, Tyma (2022-02-23). "Why Dottie$=2\sqrt{I^{-1}_\frac12(\frac 12,\frac 32)-I^{-1}_\frac12(\frac 12,\frac 32)^2} = \sin^{-1}\big(1-2I^{-1}_\frac12(\frac 12,\frac 32)\big)$?". Math Stack Exchange. Retrieved 2023-08-11.