The decimal expansion of the Dottie number is .[1]
Since is decreasing and its derivative is non-zero at , it only crosses zero at one point. This implies that the equation has only one real solution. It is the single real-valuedfixed 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 for a complex variable has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.
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 ,[6] the root of .
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 formedit
The Dottie number can be expressed as
where 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]
^"Integral Representation of the Dottie Number". Mathematics Stack Exchange.
External linksedit
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.