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Given an entire function $f(z)$, and $0\neq a\in \mathbb R$. We define the translation operator: $$T_{a}f(z)=f(z-a).$$ What properties the new function $f(z-a)$ could have? It is entire function! What about the zeros of $f(z-a)$?

I know it is an open question, but anything you know could help me.

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If $z$ is a zero of $f$ then $a+z$ is a zero of $T_a$, and the converse is also true. –  Davide Giraudo May 14 '12 at 16:30
    
I don't get the point of your edit. –  Gigili May 14 '12 at 16:33
    
You mean the ",......"! means "etc."; for more questions/properties comes to your mind. –  Clay May 14 '12 at 16:37

1 Answer 1

When $a$ is a complex, there is a one-to-one correspondence between the zeros of $f\colon z\mapsto f(z)$ and $g\colon z\mapsto f(z-a)$. This map is given by $Z(z)=z+a$ (which is obviously one-to-one). Indeed, if $f(z)=0$ then $f((z+a)-a)=0$ so $T(z)$ is a zero of $g$, and conversely if $z$ is a zero of $g$, we have $g(z)=f(z-a)=0$ hence $z-a=Z^{-1}(z)$ is a zero of $f$.

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