Can the function $(-1)^n$, $n=1,2,...$ be extended to an analytic function $f(z)$ defined on the right half complex plane satisfying the growth condition $$|f(x+iy)|\le C e^{Px+A|y|},$$ with $A<\pi$ and $C,P\in\mathbb{R}$?
I know that such an extension would be unique by Carlson's theorem and I showed that the obvious "power function" extension from complex analysis does not work for any choice of branch. I'm not sure where to go from here.
If $f$ is holomorphic in the right half-plane with $f(n) = (-1)^n$ for all positive integers $n$ and with the given growth restrictions, then $$ g(z) = f(z+1)+f(z) $$ satisfies $|g(x+iy)| \le C e^{Px+A|y|}$ for some constant $C$, and $g(n) = 0$ for all positive integers $n$. Carlson's theorem can be applied to $g$, and it follows that $g$ is identically zero.
It follows that $f(z+2) = f(z)$ for all $z$ in the right half-plane. This periodicity allows to extend $f$ to an entire function.
Finally we can apply Entire periodic function with bounded growth is constant to $F(z) = f(2z)$ and conclude that $f$ is constant.
Therefore such a function $f$ does not exist.