has infinitely many roots, but unlike the Dottie number, they are not attracting fixed points.
[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] Using the Taylor series of the inverse of
(or equivalently, the Lagrange inversion theorem), the Dottie number can be expressed as the infinite series: where each
This value can be obtained using Kepler's equation, along with other equivalent closed forms.
[5] 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].