1.5 參考解答

(2) If $z =re^{i\theta}$, $z \neq 0$, then $z^n=r^ne^{in\theta}$, $\overline{z}^n=r^ne^{i(-n\theta)}$

And $z^n+(\overline{z})^n=r^ne^{in\theta}+r^ne^{i(-n\theta)}$

$=r^n[(\cos{(n\theta)}+i\sin{(n\theta)})+(\cos{(-n\theta)}+i\sin{(-n\theta}))]$

$=r^n[\cos{(n\theta)}+i\sin{(n\theta)}+\cos{(n\theta)}-i\sin{(n\theta})]$

$=2r^n \cos{(n\theta)} \in \mathbb{R}$