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How can I simplify $|\sin(\sqrt{iz})/\sin(\sqrt{i\bar{z}})|$, where $z$ is a complex number?

What I did so far: let $\sqrt{iz}=u$, then $iz=u^2$, and $i\bar{z}=-\bar{u}^2$, so $\sqrt{i\bar{z}}=iu$ (I'm not sure about the last one!)

so: $\sin(\sqrt{iz})/\sin(\sqrt{i\bar{z}})=\sin(u)/\sin(iu)$

and now I can use the representation of $\sin(u)$ in terms of $e^{iu}$.

Anything wrong with the method above!

// I've been asked to simplify this expression, no mention for the branch!//

So, no help!

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2  
Have you tried using the identity $i\sin x = \dfrac{e^x-e^{-x}}{2}$? –  Asaf Karagila Nov 22 '11 at 5:49
    
What do you mean by $\sqrt{iz}$? Which branch of the square root are you concerned with? –  Willie Wong Nov 22 '11 at 15:33
1  
@Willie: But, but ... when you take the absolute value, the choice of branch doesn't matter ... of course you should prove that! –  GEdgar Nov 22 '11 at 16:20

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