39. What is the greatest value that the modulus of a complex number $Z$ can take, satisfying the equation $\left|Z+\frac{1}{z}\right|=1$?
39. Let $z=r(\cos \varphi+i \sin \varphi), z^{2}=r^{2}(\cos 2 \varphi+i \sin 2 \varphi)$,

$$
\begin{gathered}
\left|z^{2}+1\right|=\sqrt{\left(r^{2} \cos 2 \varphi+1\right)^{2}+\left(r^{2} \sin 2 \varphi\right)^{2}}=\sqrt{r^{4}+2 r^{2} \cos 2 \varphi+1} \\
\left|z+\frac{1}{z}\right|=\frac{\left|z^{2}+1\right|}{|z|}=\frac{\sqrt{r^{4}+2 r^{2} \cos 2 \varphi+1}}{r}=1
\end{gathered}
$$

From here, $r^{4}+r^{2}(2 \cos 2 \varphi-1)+1=0, \quad r^{2}=\frac{1-2 \cos 2 \varphi \pm \sqrt{(1-2 \cos 2 \varphi)^{2}-4}}{2}$.

$r^{2}$ takes the maximum value when $\cos 2 \varphi=-1 ; r^{2}=\frac{3+\sqrt{5}}{2}$;

$r=\frac{1+\sqrt{5}}{2}$.