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Suppose I have two special functions $f_1$ and $f_2$. Is there an algorithm which can tell me whether there exists elementary $g$ such that $f_1 = g\circ f_2$? Furthermore, is there any possibility to construct $g$ when it exists?

Elementary to me means something like, a function constructed from a finite number of function compositions of rational functions, exponentials and logarithms, trig functions and their inverses. Special means non-elementary. If it's necessary to give different definitions to these ones to answer my question, then please go ahead and redefine.

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  • $\begingroup$ Since all you know about $f_1$ and $f_2$ is that they're not constructed in the manner you describe, how are they fed to the algorithm? If you assume that your computing platform can test equality of functions, then it's easy to get a semi-positive algorithm by trying all possible $g$ and seeing if any works. $\endgroup$
    – LSpice
    Commented Feb 26, 2022 at 21:06
  • $\begingroup$ @LSpice, an expression for $f_1$ and $f_2$ would be given. Eg. Suppose I have some elementary function which I know has no elementary antiderivative. So, I define its antiderivative to be $f_1$. How can I know whether $f_1$ is expressible in terms of already known special functions or not? Maybe it's equal to $g(\Gamma(x))$, in which case I don't want to define a new special function for it $\endgroup$
    – Jojo
    Commented Feb 26, 2022 at 21:52
  • $\begingroup$ Related. $\endgroup$
    – Dan Romik
    Commented Feb 28, 2022 at 2:56

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