# Questions tagged [fractional-iteration]

The study of fractional self-iterations of a map. A basic example is the analysis of functional square roots of a map $g$, i.e. solutions $f$ to the functional equation $f \circ f = g$. The continuous version of fractional iteration concerns maps which have flows. This case is also known as continuous iteration. A classic example is the problem of extending tetration to the real and complex numbers.

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### Does the formal power series solution to $f(f(x))= \sin( x) $ converge?

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### How to solve $f(f(x)) = \cos(x)$?

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### solving $f(f(x))=g(x)$

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### f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential

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### Does the exponential function have a (compositional) square root?

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### An explicit series representation for the analytic tetration with complex height

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### “Closed-form” functions with half-exponential growth

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### Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

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### What's a natural candidate for an analytic function that interpolates the tower function?

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### $f\circ f=g$ revisited

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### Rational functions with a common iterate

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### Do complex iterates of functions have any meaning?

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### The non-convergence of f(f(x))=exp(x)-1 and labeled rooted trees

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### Smoothness in Ecalle's method for fractional iterates

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### Analytic Tetration and Natural Rates of Growth

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### solution of functional equation $f^{\circ k}(x) = x$

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