Exponential Form

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Section 2.7 Exponential Form

Definition 2.9. The Exponential Form of a Complex Number.

If we use polar coordinates

\begin{align} x\amp = r\cos\phi\tag{2.7.1}\\ y\amp = r\sin\phi\tag{2.7.2} \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 $\boldsymbol{z = re^{i\phi}}$:

\begin{align} z\amp = x+iy\notag\\ \amp = r\cos\phi + i r\sin\phi\notag\\ \amp = r(\cos\phi +i\sin\phi)\notag\\ \amp = re^{i\phi}\tag{2.7.3} \end{align}

Geometric Interpretation.

The parameters $r$ and $\phi$ inherit their geometric interpretation from polar coordinates, i.e. $r$ represents the distance of $z$ from the origin in the complex plane and $\phi$ represents the polar angle, measured in radians, counterclockwise from the real axis, as shown in Figure 2.10

Figure 2.10. The point $z=re^{i\phi}\text{.}$

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