Analytical and Numerical Approaches for Finding Functional Iterates and Roots
We investigate solutions to the functional equation $f(f(x)) = e^x$, which can be interpreted as the problem of finding a half iterate of the exponential map. While no elementary solution exists, we construct and analyze non-elementary solutions using methods based on the Lambert W function, tetration, and Abel's functional equation. We examine structural properties of possible solutions, including monotonicity, injectivity, and behavior across different intervals, and provide a piecewise-defined framework that extends to the entire real domain. Building on this, we introduce the super-logarithm and its inverse, the super-root, as analytic tools for defining fractional iterates of $e^x$. Using a power-series expansion near $x = 1$, we numerically approximate the super-logarithm and demonstrate a procedure for computing fractional iterates, including the half-iterate of the exponential function. Our approach is validated by comparisons to known constructions such as Kneser's tetration, with an emphasis on computational feasibility and numerical stability. Finally, we explore the broader landscape of fractional iteration, showing that similar techniques can be applied to other functions beyond $e^x$. Through numerical approximations and series-based methods using genetic algorithms and other optimization techniques, we confirm that fractional iterates not only exist for many analytic functions but can be computed with practical accuracy, opening pathways to further applications in dynamical systems and functional analysis.