Functional Square Root

[Leo.W]

What about that (x) = f(sqrt(x))? Is f(x) = x^4 the only answer to it?

Hi Catullus

The answer can be different seen from varied viewpoints.

If you're talking about the functional equation \[\sqrt{f}(z)=f(\sqrt{z})\] where f can be piecewise analytic, then f(x)=x^4 is not the only answer, or even not an answer. This is because every function have at least 2 different functional square root, for this f, it can be x^2 and -x^2, for a simple identity function, it has infinitely many functional square root, like f(x)=c-x, f(x)=c/x, f(x)=x, for any constant c, for all these f(x), f(f(x))=x.
And for most cases, some of the functional square roots can even be not analutic in the whole complex plane, which means your equation could be a little bit badly-defined if only considering singlevalued functions.

For multivalued functions, we can say f(x)=x^4 is a nice solution.
Or if we consider a properly defined or regularized functional square root, then f(x)=x^4 can be a nice singlevalued analytic solution, and the only one.
In this way, you're actually asking for an analytic function g which satisfies,
\[g(x)=\sqrt{f}(x)\Leftrightarrow g(g(x))=f(x),g(x)=g(g(\sqrt{x}))\]
Since g is now analytic, it has its maybe-multivalued inverse function g^-1
Let \[g^{-1}(g(x))=x\text{ takes true for some x, but for all x: }g^{-1}(g(x))=\sigma(x)\]
Then we take g^-1 for both side and substitute sqrt(x)=t, this gives \[\sigma(t^2)=\sigma(g(t))\]
This functional equation has infinitely many solutions, each combines with such a symmetry, thus we can't get any information about g(z).

But there's ways to find some other solutions which may give rise to multivalued or piecewise analytic solutions.
For example, we derive a new sigma(z) from our old solution f(x)=x^4, we use its functional square root g(x)=x^2 and its multivalued inverse, gives \[\sigma(z)=e^{\frac{ln(z^2)}{2}},f(z)=\sqrt{z^8}\]
which is also a solution, only piecewise analytic, can be proved easily using singlevalued complex analysis.

regards
leo