This is an exercise from a book I tried:
One would like to find all holomorphic equations that satisfy:$$i) \ f(z)+f''(z) = 0 \text{ in } \mathbb{C} $$$$ii)\ f(z^{2})=f(z)^{2} \text{ in } \mathbb{C}$$
I attempted this:
For i) one can use the Ansatz: $A\sin(bx)+C\cos(bx)$ and this also turns out to be the solution. How to show that these are all equations that satisfy the equation?
For ii) all terms of the form $x,x^2, x^3,\ldots, x^n$ satisfy the functional equation. I don't see any other, so I assume that these are the only ones.
How to show that these are all possible equations that satisfy the functional equations?