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What are the values of $z\in\mathbb{C}$, such that there is a non-constant sequence $z_n\in\mathbb{C}$ and $\sin z_n\to z$ ? How to find such a sequence if it exists ?

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Do you know for which $z \in \mathbb{C}$ there exists $w$ in $\mathbb{C}$ such that $z=\sin w$? –  Chris Eagle Sep 24 '12 at 13:10

2 Answers 2

up vote 4 down vote accepted

The following is what you want: Little Picard's Theorem

Please do observe that in fact

$$\forall \,z\in\Bbb C\,\,\exists\,w\in\Bbb C\,\,\,s.t.\,\,\,\sin w =z$$

except, perhaps, for one single element $\,z\in\Bbb C\,$

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However, the possibly omitted element would have to be real (for reasons of symmetry), and it is fairly easy to see that all of the reals are in fact hit by the complex sine. [This doesn't matter for the actual question, of course, since even $\mathbb C$ with one point missing is still dense in $\mathbb C$. –  Henning Makholm Sep 24 '12 at 13:46

What is sine in the complex plain?

It is defined by $\sin z = \displaystyle\frac{e^{zi}-e^{-zi}}{2i}$. And, this is continuous, so for any sequence $z_n \to z$, we have $\sin z_n \to \sin z$.

So, any number in the range of sine will arise as you wish, and the contant sequence is a good choice.

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Actually I was interested in non-constant sequences, edited. –  pritam Sep 24 '12 at 13:14
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@pritan It doesn't matter, every $z$ has a non-constant sequence, $z_n\to z$. –  Thomas Andrews Sep 24 '12 at 13:15
    
Ok, I got it. I think it can be $\sin^{-1} (z+\frac{1}{n}) $ right ? –  pritam Sep 24 '12 at 13:19

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