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Section 2.6 Exponential Form

If we use polar coordinates

\begin{align} x\amp = r\cos\theta\tag{2.6.1}\\ y\amp = r\sin\theta\tag{2.6.2} \end{align}

to describe the complex number \(z=x+iy\text{,}\) we can factor out the \(r\) and use Euler's formula to obtain the exponential form of the complex number \(z\text{:}\)

\begin{align} z\amp = x+iy\notag\\ \amp = r\cos\theta + i r\sin\theta\notag\\ \amp = r(\cos\theta +i\sin\theta)\notag\\ \amp = re^{i\theta}\tag{2.6.3} \end{align}

Just as \(r\) represents the distance of \(z\) from the origin in the complex plane, \(\theta\) represents the polar angle, measured in radians, counterclockwise from the real axis, as shown in Figure 2.6.1

Figure 2.6.1. The point \(z=re^{i\theta}\text{.}\)

It is easiest to do multiplication and division of two complex numbers \(z_1=r_1e^{i\theta_1}\) and \(z_2 = r_2 e^{i\theta_2}\) in exponential form:

\begin{align} z_1 z_2 \amp = r_1 e^{i\theta_1}\, r_2 e^{i\theta_2}\notag\\ \amp = r_1 r_2 e^{i(\theta_1 + \theta_2)}\tag{2.6.4}\\ \frac{z_1}{ z_2} \amp = r_1 e^{i\theta_1}/ r_2 e^{i\theta_2}\notag\\ \amp = \frac{r_1}{ r_2} e^{i(\theta_1 - \theta_2)}\tag{2.6.5} \end{align}

Notice that the magnitudes of the two complex numbers multiply (or divide) whereas the angles add (or substract).