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Find all the complex roots of the equation $\cos z=3$.
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Since $\cos z=\left(e^{i z}+e^{-i z}\right) / 2$, it comes down to solve the equation $e^{i z}+$ $e^{-i z}=6$, i.e.,
$$w+w^{-1}=6 \Leftrightarrow w^{2}-6 w+1=0$$
if we let $w=e^{i z}$. The roots of $w^{2}-6 w+1=0$ are $w=3 \pm 2 \sqrt{2}$. Therefore, the solutions for $\cos z=3$ are
$$i z=\log (3 \pm 2 \sqrt{2}) \Leftrightarrow z=-i(\ln (3 \pm 2 \sqrt{2})+2 n \pi i)=2 n \pi-i \ln (3 \pm 2 \sqrt{2})$$
for $n$ integers.
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