There exist a function such that $f\circ f(x)=e^x$?

• How to calculate $f(x)$ in $f(f(x)) = e^x$?

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In fact, you can find such an $f$ that is analytic. From this answer on MathOverflow:

A real-analytic solution in this case was constructed by H. Kneser, “Reelle analytische Lösungen der Gleichung $φ(φ(x))=e^x$ und verwandter Funktional-gleichungen”, J. Reine Angew. Math. 187 (1949), 56-67.

I’m afraid I don’t know the details of the construction, but hopefully this reference is helpful.

• Characterising functions $f$ that can be written as $f = g \circ g$?
• The graph of $f(x)$ where $f(f(x))=e^x$