Mathematics for Electrical Engineering and Computing

Solving some other complex equations

If we have the equation zⁿ= c, where c is any complex number, then we write the right-hand side of the equation in exponential form and use the fact that we can add a multiply of 2π to the argument without changing the value of the number. Write

$\begin{array}{lll}c\hfill & =\hfill & r?{\text{e}}^{\text{j}\theta }?=?r?{\text{e}}^{\text{j}(\theta ?+?\text{2}\pi N)}?\text{where}?N?\in ?\mathbb{Z}\hfill \\ z^n\hfill & =\hfill & r?{\text{e}}^{\text{j}(\theta ?+?\text{2}\pi N)}\hfill \\ \iff \hfill & z?=?r^{(1/n)}?{\text{e}}^{(\text{j}(\theta ?+?\text{2}\pi N)/n)}\hfill \end{array}$

taking the nth root of both sides.

Example 10.23

Solve $z^3?=?-4?+?\text{j}4\sqrt{3.}$

Solution Write $-4?+?j4\sqrt{3}$ in ...