Dottie number
thumb|The Dottie number is the unique real [[fixed point (mathematics)|fixed point of the cosine function.]]
In mathematics, the Dottie number is a constant that is the unique real root of the equation
- [math]\displaystyle{ \cos x = x }[/math],
where the argument of [math]\displaystyle{ \cos }[/math] is in radians.
The decimal expansion of the Dottie number is [math]\displaystyle{ 0.739085133215160641655312087673873404... }[/math].[1]
Since [math]\displaystyle{ \cos(x) - x }[/math] is decreasing and its derivative is non-zero at [math]\displaystyle{ \cos(x) - x = 0 }[/math], it only crosses zero at one point. This implies that the equation [math]\displaystyle{ \cos(x) = x }[/math] 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 [math]\displaystyle{ \cos z = z }[/math] for a complex variable [math]\displaystyle{ z }[/math] has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.
thumb|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 [math]\displaystyle{ f(x) = \cos(x) - x }[/math] at [math]\displaystyle{ \frac{\pi}{2} }[/math] (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series [math]\displaystyle{ \frac{\pi}{2}+\sum_{n\,\mathrm{odd}} a_{n} \pi^{n} }[/math] where each [math]\displaystyle{ a_n }[/math] is a rational number defined for odd n as[3][4][5][nb 1]
- [math]\displaystyle{ \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} }[/math]
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 [math]\displaystyle{ 0.999847... }[/math],[6] the root of [math]\displaystyle{ \cos\left(\frac{\pi}{180}x\right) = x }[/math].
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.[7]
Closed form
The Dottie number can be expressed as
- [math]\displaystyle{ D=\sqrt{1-\left(2I^{-1}_\frac12\left(\frac 12,\frac 32\right)-1\right)^2}, }[/math]
where [math]\displaystyle{ I^{-1} }[/math] is the inverse regularized Beta function.[1] This value can be obtained using Kepler's equation, along with other equivalent closed forms.[8]
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
- [math]\displaystyle{ 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} }[/math].[2]
Or as
- [math]\displaystyle{ D=\frac{\pi }{2}-\frac1{2 \pi } \int_0^{\infty } \ln \left(\frac{2 \pi \cosh (x)+\pi ^2}{x^2+\cosh ^2(x)}+1\right) \, dx }[/math]
Two more representations are given as follows
- [math]\displaystyle{ D=\frac1\pi \int_0^{\pi } \arctan\left(\tan \left(\frac{t-\sin t+\frac{\pi }{2}}2\right)\right) \text{ d}t+\frac{1}{\pi } }[/math].[9]
- [math]\displaystyle{ D=\frac{\pi}{2}-\int_{0}^{\pi}\left\lfloor\frac{2\left(t-\sin t\right)}{\pi}\right\rfloor\,\mathrm{d}t }[/math].[10]
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". OEIS Foundation. https://oeis.org/A003957.
- ↑ Eric W. Weisstein. "Dottie Number". http://mathworld.wolfram.com/DottieNumber.html.
- ↑ 3.0 3.1 Kaplan, Samuel R (February 2007). "The Dottie Number". Mathematics Magazine 80: 73. doi:10.1080/0025570X.2007.11953455. https://www.maa.org/sites/default/files/Kaplan2007-131105.pdf. Retrieved 29 November 2017.
- ↑ "OEIS A302977 Numerators of the rational factor of Kaplan's series for the Dottie number.". https://oeis.org/A302977.
- ↑ "A306254 - OEIS". https://oeis.org/A306254.
- ↑ Sloane, N. J. A., ed. "Sequence A330119". OEIS Foundation. https://oeis.org/A330119.
- ↑ Pain, Jean-Christophe (2023). "An exact series expansion for the Dottie number". arXiv:2303.17962.
- ↑ Gaidash, Tyma (2022-02-23). "Why Dottie$=2\sqrt{I^{-1}_\frac12(\frac 12,\frac 32)-I^{-1}_\frac12(\frac 12,\frac 32)^2} = \sin^{-1}\big(1-2I^{-1}_\frac12(\frac 12,\frac 32)\big)$?". https://math.stackexchange.com/questions/4389528/why-dottie-2-sqrti-1-frac12-frac-12-frac-32-i-1-frac12-frac-12-fr.
- ↑ N/A, Anixx (2015-03-04). "Explaining $\cos^\infty$". https://math.stackexchange.com/a/1175016/595084.
- ↑ N/A, Jam (2020-01-24). "Explaining $\cos^\infty$". https://math.stackexchange.com/questions/227317/explaining-cos-infty?noredirect=1&lq=1#comment7241451_1175016.
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".
- Azarian, Mohammad K. (2008). "ON THE FIXED POINTS OF A FUNCTION AND THE FIXED POINTS OF ITS COMPOSITE FUNCTIONS". International Journal of Pure and Applied Mathematics. https://ijpam.eu/contents/2008-46-1/3/3.pdf.
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