Dottie number

In mathematics, the Dottie number or the cosine constant 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 given by:

D = 0.739085133215160641655312087673... (sequence A003957 in the OEIS).

Since ${\displaystyle \cos(x)-x}$ is decreasing 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.^[1] 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.

The constant appeared in publications as early as 1860s.^[2] Norair Arakelian used lowercase ayb (ա) from the Armenian alphabet to denote the constant.^[2]

The constant name was coined by Samuel R. Kaplan in 2007. It originates from a professor of French named Dottie who observed the number by repeatedly pressing the cosine button on her calculator.^{[3][nb 1]}

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.^[5]

The Dottie number appears in the closed form expression of some integrals:^[6][7]

${\displaystyle \int _{0}^{\infty }\ln \left({\frac {4\left(x+\sinh x\right)^{2}+\pi ^{2}}{4(x-\sinh x)^{2}+\pi ^{2}}}\right)\mathrm {d} x=\pi ^{2}-2\pi D}$

${\displaystyle \int _{0}^{\infty }{\frac {3\pi ^{2}+4(x-\sinh x)^{2}}{(3\pi ^{2}+4(x-\sinh x)^{2})^{2}+16\pi ^{2}(x-\sinh x)^{2}}}\,\mathrm {d} x={\frac {1}{8+8{\sqrt {1-D^{2}}}}}}$

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:

${\displaystyle D={\frac {\pi }{2}}+\sum _{n\,\mathrm {odd} }a_{n}\pi ^{n}}$

where each ${\displaystyle a_{n}}$ is a rational number defined for odd n as^{[3][8][9][nb 2]}

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

The Dottie number can also be expressed as:

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

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. ${\displaystyle I_{\frac {1}{2}}^{-1}\left({\tfrac {1}{2}},{\tfrac {3}{2}}\right)\approx 0.16319}$ is the median of a beta distribution with parameters 1/2 and 3/2. ^[5]

In Microsoft Excel and LibreOffice Calc spreadsheets, the Dottie number can be expressed in closed form as SQRT(1-(1-2*BETA.INV(1/2,1/2,3/2))^2). In the Mathematica computer algebra system, the Dottie number is Sqrt[1 - (1-2 InverseBetaRegularized[1/2, 1/2, 3/2])^2].

Another closed form representation:

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

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))).

• 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.