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Hi to all 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.

flag	asked Oct 22 at 13:58 RH 105●1●9

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NO, a=0, u(z)=1. – Marc Palm Oct 22 at 14:10

Thank you very much. – RH Oct 22 at 14:39

A more profound example is $u(z)=1,$ but $a=2\pi.$ In that case, both $\sin$ and $\cos$ satisfy the equation. – Igor Rivin Oct 22 at 15:41

You must say, WHERE is your function analytic, to obtain a non-trivial answer. – Alexandre Eremenko Oct 22 at 16:21

The function is analytic in the whole complex plane. – RH Oct 22 at 17:03

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Alexandre Eremenko · Accepted Answer · 2012-10-22 21:32:30Z UTC

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 $p(z)=p(a-z)$.

Robert Israel · Answer 2 · 2012-10-22 19:55:00Z UTC

A necessary condition for a nonzero solution is $u(z) u(a-z) = 1$. If that is true and $u$ is entire, it has an entire square root. Now $\sqrt{u(z)} \sqrt{u(a-z)} = u(a/2) = \pm 1$: if it is $1$, then $f(z) = \sqrt{u(z)}$ is a solution. If it is $-1$, then $f(z) = (z - a/2) \sqrt{u(z)}$ is a solution.

functional equation

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