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$$f(z)=\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)-1$$

How many zeros does the above function have in $\Bbb{C}$?

Laura
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    That product is famous. Do you know an elementary function representation of it? Where is the problem from? – Jonas Meyer May 22 '13 at 03:45
  • @JonasMeyer. someone ask me the number of zeros of $f(z)=\sin(z)-z$ in $\Bbb{C}$. and I use the fact $\frac{\sin \pi z}{\pi}=z\prod\left(1-\frac{z^2}{n^2}\right)$ to get this problem. – Laura May 22 '13 at 03:48
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    Rush: Why not tell us that before? Anyway, thank you very much for clarifying. – Jonas Meyer May 22 '13 at 03:50
  • @JonasMeyer. thanks for you advice. I will do it next time – Laura May 22 '13 at 03:52
  • Since it does not have a pole, I guess that argument principle will be useful. – Guillermo May 22 '13 at 03:53

1 Answers1

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As noted in a comment, this is essentially the problem of determining the number of solutions to $\sin(z)=z$. There are infinitely many. It turns out more generally, as I learned from Aryabhata's answer at this link, with reference to an example in Markuševič's book at this link, that for every complex number $A$, the equation $\sin(z)=Az$ has infinitely many solutions in the complex plane.

Community
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Jonas Meyer
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