Dottie number



In mathematics, the Dottie number 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 $$0.739085133215160641655312087673873404...$$.

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. 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.



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 $\frac{\pi}{2}+\sum_{n\,\mathrm{odd}} a_{n} \pi^{n}$  where each $$a_n$$ is a rational number defined for odd n as


 * $$\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 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.

If a calculator is set to take angles in degrees, the sequence of numbers will instead converge to $$0.999847...$$, 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.

Closed forms
The Dottie number can be expressed as
 * $$D=\sqrt{1-\left(2I^{-1}_\frac12\left(\frac 12,\frac 32\right)-1\right)^2},$$

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 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].

Another closed form representation:
 * $$D=- \tanh\left(2\tanh^{-1}\left(\frac1{\sqrt3} \operatorname{InvT} \left(\frac14,3\right)\right)\right)=-\frac{2\sqrt3 {\operatorname{InvT}\left(\frac14,3\right)}}{\operatorname{InvT}^2\left(\frac14,3\right)+3},$$

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 2 *SQRT(3)* TINV(1/2, 3)/(TINV(1/2, 3)^2+3) and TANH(2*ATANH(1/SQRT(3) * TINV(1/2,3))).

Integral representations
Dottie number can be represented as


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