I have the following question:
Let $f$ be an analytic function satisfying the functional equation: $f(z)=u(z)f(a-z)$ where $a$ is a real constant. Let $g$ be another function satisfying the same functional equation. In this case I asking if $f=g$. 
Thank you in advance.



functional equation
In the solution by Robert Israel, the uniqueness question (which was asked) is not addressed.


Of course one cannot conclude that $f=g$, because if $f$ satisfies the equation,
and $p$ is any function such that $p(z)=p(a-z)$ then $fp$ satisfies the equation.
And this is the general description of all solutions because if $f$ and $g$ satisfy the
equation, then their ratio has the property that $p(z)=p(a-z)$.


