<s>[INST] Powers of complex numbers : Which is the general formula? Let $z=\frac{1}{2}\left (\sqrt{3}+i\right )\in \mathbb{C}$.
I want to calcualte the powers $z^n$ for $n\in \mathbb{Z}$.
I have done the following: \begin{align*}&z^1=\frac{1}{2}\left (\sqrt{3}+i\right ) \\ &z^2=\left (\frac{1}{2}\left (\sqrt{3}+i\right )\right )^2=\frac{1}{4}\left (3+2\sqrt{3}i-1\right )=\frac{1}{4}\left (2+2\sqrt{3}i\right )=\frac{1}{2}\left (1+\sqrt{3}i\right ) \\ &z^3=z^2\cdot z=\frac{1}{2}\left (1+\sqrt{3}i\right )\cdot \frac{1}{2}\left (\sqrt{3}+i\right )=\frac{1}{4}\left (\sqrt{3}+i+3i-\sqrt{3}\right )=\frac{1}{4}\cdot 4i=i \\ & z^4=z^3\cdot z=i\cdot \frac{1}{2}\left (1+\sqrt{3}i\right )=\frac{1}{2}\left (-\sqrt{3}+i\right ) \end{align*} Is therea specific formula that we get for an arbitrary $n$ ?
 [/INST] It is easiest to express the general formula using the polar representation of $z = r e^{i\theta}$
$$
z^n = r^n e^{in\theta} = r^n\left(\cos(n\theta) + i \sin(n\theta)\right).
$$
See also Euler formula and De Moivre's formula.
</s>