Английская Википедия:Dottie number
In mathematics, the Dottie number is a constant that is the unique real root of the equation
- <math> \cos x = x </math>,
where the argument of <math>\cos</math> is in radians.
The decimal expansion of the Dottie number is <math>0.739085133215160641655312087673873404...</math>.[1]
Since <math>\cos(x) - x</math> is decreasing and its derivative is non-zero at <math>\cos(x) - x = 0</math>, it only crosses zero at one point. This implies that the equation <math>\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> \cos z = z </math> for a complex variable <math> z </math> has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.
Using the Taylor series of the inverse of <math>f(x) = \cos(x) - x</math> at <math display="inline">\frac{\pi}{2}</math> (or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series <math display="inline">\frac{\pi}{2}+\sum_{n\,\mathrm{odd}} a_{n} \pi^{n}</math> where each <math>a_n</math> is a rational number defined for odd n as[3][4][5]Шаблон:Refn
- <math>\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>0.999847...</math>,[6] the root of <math>\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>D=\sqrt{1-\left(2I^{-1}_\frac12\left(\frac 12,\frac 32\right)-1\right)^2},</math>
where <math>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 Шаблон:Code. In the Mathematica computer algebra system, the Dottie number is Шаблон:Code.
Integral representations
Dottie number can be represented as
- <math>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>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>
Notes
References
External links
