I'm looking to study a function $φ$ that verifies $(φ \circ φ)(x) = \sqrt{2^x}$.
My approach is as follows: try to express $φ$ in the form of a limited expansion.
Posing $φ(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$.
I then replace $x$ by $φ(x)$ to obtain the development of $φ \circ φ$.
I thus obtain a system of equations because the coefficients must correspond to those of $\sqrt{2^x}$.
Here are the first equations with a 6th-order approximation.
Unfortunately , the task seems difficult , which makes me I wonder if this approach is right ...
Would you have alternative approaches & any references on this subject that I might find pertinent ?
Thanks in advance!
[[ Difficult to get through with this !
I am Posting this longish Comment to aid OP. ]]
We should have a way to get $y_1(y_1(x))=e^x$
We will then have to tweak that to get $y_2(y_2(x))=e^{ax}$
We can then let $a=\log(2)/2$
That will give this : $y_3(y_3(x))=e^{x\log(2)/2}=\sqrt{e^{\log(2)}}^x=\sqrt{2}^x$
Eventually , the closed-form function might not [ will not ?? ] exist & other ways to express the function might be very cumbersome & tough to make.
Check out :
thoughts about $f(f(x))=e^x$
https://en.wikipedia.org/wiki/Half-exponential_function
https://mathoverflow.net/questions/12081/does-the-exponential-function-have-a-compositional-square-root
https://www.quora.com/Is-there-a-function-f-such-that-f-f-x-exp-x
https://mathoverflow.net/questions/45477/closed-form-functions-with-half-exponential-growth
https://koreascience.kr/article/JAKO201611639306363.page