Inevitable Dottie Number. Iterals of cosine and sine
  The unique real root of cos(x) = x, recently referred to as the Dottie number, is expressed as an iteral of cosine. Using the derivatives of iterals, it is shown why this number is achieved starting from any real number, when the iterates of cosine successfully approach infinity, and how this affects the Maclaurin series of the iterals. Properties of the iterals of cosine and sine and their derivatives are considered. A C++ template for iteral is applied for computation of Julia sets. 
Valerii Salov
math.HO math.DS