Korean J. Math. Vol. 24 No. 1 (2016) pp.81-106
DOI: https://doi.org/10.11568/kjm.2016.24.1.81

Finding the natural solution to $f(f(x)) = \exp(x)$

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William Paulsen


In this paper, we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are an infinite number of solutions, and there is no agreement as to which solution is "best.'' We will approach the problem by first solving Abel's functional equation $\alpha(e^x) = \alpha(x) + 1$ by perturbing the exponential function so as to produce a real fixed point, allowing a unique holomorphic solution. We then use this solution to find a solution to the unperturbed problem. However, this solution will depend on the way we first perturbed the exponential function. Thus, we then strive to remove the dependence of the perturbed function. Finally, we produce a solution that is in a sense more natural than other solutions.

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