조건을 만족시키는 복소수 구하기

$\displaystyle \frac{z - \overline{z}}{i} < 0$이므로 $\displaystyle z - \overline{z} \ne 0$ $\displaystyle \frac{z}{1 + z ^{2}}$이 실수이므로 $\displaystyle \frac{z}{1 + z ^{2}}$$\displaystyle = \overline{\frac{z}{1 + z ^{2}}}$ 즉, $\displaystyle \frac{z}{1 + z ^{2}}$$\displaystyle = \frac{\overline{z}}{1 + {\overline{z}} ^{2}}$ $\displaystyle z + z {\overline{z}} ^{2} - z - z ^{2} \overline{z} = 0$ 즉, $\displaystyle ( z - \overline{z} ) ( z \overline{z} - 1 ) = 0$ $\displaystyle \therefore z \overline{z} = 1$ (∵ $\displaystyle z - \overline{z} \ne 0$) 또한, $\displaystyle \frac{z ^{2}}{1 + z}$이 실수이므로 $\displaystyle \frac{z ^{2}}{1 + z}$$\displaystyle = \overline{\frac{z ^{2}}{1 + z}}$ 즉, $\displaystyle \frac{z ^{2}}{1 + z}$$\displaystyle = \frac{{\overline{z}} ^{2}}{1 + {\overline{z}}}$ $\displaystyle z ^{2} + z ^{2} \overline{z} - {\overline{z}} ^{2} - z {\overline{z}} ^{2} = 0$ 즉, $\displaystyle ( z - \overline{z} ) ( z + \overline{z} + z \overline{z} ) = 0$ ∴ $\displaystyle z + \overline{z} = - z \overline{z} = - 1$ (∵ $\displaystyle z - \overline{z} \ne 0$, $\displaystyle z \overline{z} = 1$) $z$, $\displaystyle \overline{z}$를 두 근으로 하는 이차방정식은 $x ^{2} + x + 1 = 0$ $\displaystyle \therefore x = \frac{- 1 \pm \sqrt{3} i}{2}$ $\displaystyle \frac{z - \overline{z}}{i} < 0$이어야 하므로 $\displaystyle z = \frac{- 1 - \sqrt{3} i}{2}$