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Could anyone help me find this limit $$\limsup_{|z|\to\infty}\frac{\log|e^{-iz}|}{|z|}$$

where $z=x+iy, x,y\in \mathbb R$.

I guess we need to use that $e^{-iz}=\cos z- i\sin z$, then $$\limsup_{|z|\to\infty}\frac{\log|e^{-iz}|}{|z|}=\limsup_{|z|\to\infty}\frac{\log|\cos z- i\sin z|}{|z|}$$

but what next?

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    Better use $|e^z| = e^{\text{Re}(z)}$.2012-04-18
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    @Dirk I nicer template of the `Re` symbol is obtained by $\Re$ (`\Re`). I guess you might also use $\mathfrak{Re}$ (`\mathfrak{Re}`)2012-04-18
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    Seems to be a matter of taste and habit. I feel like $\text{Re}$ is more popular that $\Re$ in Germany...2012-04-19

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