Como se expressa \(e^{a+ib}\) e como calcular \((e^{a+ib})^n\) para \(n\in\mathbb{N}\)?

\(e^{a+ib}=e^a(\cos b + i\sin b)\) e \((e^{a+ib})^n = e^{na}(\cos(nb)+i\sin(nb))\).

Exemplo:
Considere \(z=0.5+i\frac{\pi}{2}\) e \(n=3\).
\(e^z = e^{0.5}\big(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\big)=e^{0.5}(0+i)=i\,e^{0.5}.\)
\((e^z)^3=(i\,e^{0.5})^3=i^3 e^{1.5}=-i\,e^{1.5}.\)
Verificando por \(e^{nz}\): \(e^{3z}=e^{1.5}\big(\cos\frac{3\pi}{2}+i\sin\frac{3\pi}{2}\big)=e^{1.5}(0-i)=-i\,e^{1.5}.\)