How one can find the positive integer $k$

Question

Let us consider the non constant complex analytic function $f$ on all $ℂ$ verifying

$$f(s)=(-1)^{k}f(2-s)$$

where $k$ is a positive integer.

My question is: How one can find the positive integer $k$.

Answer 1

Define $g(z) = f(1+z)$ to simplify the problem. Then the given identity is equivalent to $$ g(z) = (-1)^kg(-z) $$ and that holds exactly if

• $g$ is an even function and $k$ is an even integer,
• or $g$ is an odd function and $k$ is an odd integer.