Exploring the mathematics of functional square roots, the Composita method, and the deep structure of equations like \(f(f(x)) = F(x)\).
An introduction to functional iteration, fixed points, conjugacy, and the solvable vs. unsolvable dichotomy.
ReadExplore \(f(f(x)) = F(x)\) in two ways: a paper-backed composita mode for iterative roots and a local analytic fixed-point mode with rendered formulas and verification plots.
Interactive SolverCompute and explore the composita \(F^\Delta(n,k)\) of generating functions. Based on Kruchinin & Kruchinin (2013).
ToolReferences, academic papers, external resources, and credits for the tools and mathematics behind this site.
ReferencesFinished notes on fixed points, composita half-iterates, and numerical/analytic solution strategies, each linked to the upgraded solver and bibliography.
ArticlesView the full source code, report issues, or contribute to the project on GitHub.
GitHubGiven a function \(F(x)\), a functional square root (or half-iterate) is a function \(f(x)\) satisfying \[ f(f(x)) = F(x). \] Equivalently, \(f^{\circ 2} = F\), where \(f^{\circ n}\) denotes the \(n\)-th iterate of \(f\).
Some equations, like \(f(f(x)) = x^2 - 2\), have elegant closed-form solutions connected to Chebyshev polynomials. Others, like \(f(f(x)) = x^2 + 1\), have no known closed form — their dynamics lack real fixed points and diverge. The Composita method, developed by V.V. and D.V. Kruchinin, provides a systematic framework for analyzing and numerically solving such equations via generating functions.