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Dottie number

Mathematical constant related to the cosine function

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

${\displaystyle \cos x=x}$,

where the argument of ${\displaystyle \cos }$ is in radians.

The decimal expansion of the Dottie number is ${\displaystyle 0.739085133215160641655312087673873404...}$.^[1]

Since ${\displaystyle \cos($x)-x} isdecreasing and its derivative is non-zero at ${\displaystyle \cos($x)-x=0} , it only crosses zero at one point. This implies that the equation ${\displaystyle \cos($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 ${\displaystyle \cos z=z}$ for a complex variable ${\displaystyle z}$ has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.

Using the Taylor series of the inverse of ${\displaystyle f($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 ${\displaystyle a_{n}}$ is a rational number defined for odd nas^{[3][4][5][nb 1]}

${\displaystyle {\begin{aligned}a_{n}&={\frac {1}{n!2^{n}}}\lim _{m\to {\frac {\pi }{2}}}{\frac {\partial ^{n-1}}{\partial m^{n-1}}}{\left({\frac {\cos m}{m-\pi /2}}-1\right)^{-n}}\\&=-{\frac {1}{4}},-{\frac {1}{768}},-{\frac {1}{61440}},-{\frac {43}{165150720}},\ldots \end{aligned}}}$

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 ${\displaystyle 0.999847...}$,^[6] the root of ${\displaystyle \cos \left({\frac {\pi }{180}}x\right)=x}$.

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.^[7]

Closed forms[edit]

The Dottie number can be expressed as

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

where ${\displaystyle I^{-1}}$ is the inverse of the regularized beta function. This value can be obtained using Kepler's equation, along with other equivalent closed forms.^[8][7]

InMicrosoft 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:

${\displaystyle D=-\tanh \left(2\tanh ^{-1}\left({\frac {1}{\sqrt {3}}}\operatorname {InvT} \left({\frac {1}{4}},3\right)\right)\right)=-{\frac {2{\sqrt {3}}{\operatorname {InvT} \left({\frac {1}{4}},3\right)}}{\operatorname {InvT} ^{2}\left({\frac {1}{4}},3\right)+3}},}$

where ${\displaystyle \operatorname {InvT} }$ 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[edit]

Dottie number can be represented as

${\displaystyle D={\sqrt {1-\left(1-\left(\int _{0}^{\infty }{\frac {32(z-\sinh($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}}}} .^[9]

Notes[edit]

References[edit]

External links[edit]

• 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.