[Paper Review] Inevitable Dottie Number. Iterals of cosine and sine

Valerii Salov|arXiv (Cornell University)|Dec 1, 2012

Mathematical Dynamics and Fractals3 references3 citations

TL;DR

This paper introduces the 'iteral' notation for iterated cosine and sine functions, proving that repeated application of cosine converges to the unique real fixed point—known as the Dottie number—regardless of the initial real value. It analyzes derivatives, Maclaurin series, and complex dynamics, demonstrating convergence via iterative stability and providing a C++ template for computing Julia sets of cosine and sine in the complex plane.

ABSTRACT

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.

Motivation & Objective

• To formalize the convergence of iterated cosine to the Dottie number using a new functional notation called 'iteral'.
• To analyze the derivatives and Maclaurin series of iterated cosine and sine functions.
• To demonstrate that the Dottie number is the global attractor for all real initial values under cosine iteration.
• To extend the analysis to complex dynamics and compute filled Julia sets for cosine and sine using a C++ template.

Proposed method

• Introduces the iteral notation $\textrm{\LARGE{И}}_v^n f(x)$ to represent $n$-fold iteration of a function $f$ starting from initial value $v$.
• Applies the chain rule to derive the first and higher-order derivatives of iterals, showing recursive dependence on previous iterates.
• Uses graphical analysis and fixed-point iteration theory to prove convergence to the Dottie number for all real starting values.
• Derives the Maclaurin series of iterals by expressing them as infinite compositions of derivatives at the fixed point.
• Applies the iteral framework to define filled Julia sets for cosine and sine as the set of complex initial values where the absolute value of the infinite iterate remains bounded.
• Implements a C++ template for numerical computation of iterals, enabling efficient generation of Julia set plots via standard libraries and Gnuplot-compatible output.

Experimental results

Research questions

• RQ1 Why does the iteration of cosine converge to the same real number regardless of the initial real value?
• RQ2 How can the derivatives of iterated cosine and sine functions be systematically computed and what do they reveal about convergence?
• RQ3 What is the structure of the filled Julia sets for cosine and sine in the complex plane?
• RQ4 How can the iteral notation be used to formalize and compute infinite functional iterations?

Key findings

• The Dottie number, the unique real solution to $\cos(x) = x$, is the global attractor of the cosine iteration for all real starting values, confirmed by iterative convergence and graphical analysis.
• The first derivative of the $n$-th iterated cosine at the Dottie number is bounded in absolute value by $\sin(D) \approx 0.6736$, ensuring local stability and convergence.
• The Maclaurin series of the iterals of cosine and sine are shown to converge to the fixed point, with coefficients derived from recursive derivative chains.
• For complex initial values, the iterated cosine and sine functions can diverge, leading to intricate, non-trivial Julia set structures in the complex plane.
• The C++ template implementation successfully computes iterated cosine and sine up to 50 iterations, enabling visualization of Julia sets via Gnuplot-compatible output.
• The filled Julia sets for cosine and sine exhibit complex, fractal-like patterns, with points in the set defined by boundedness of the infinite iterate in absolute value.