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THE GEOMETRY OF MATHEMATICAL METHODS

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 \(z=re^{i\phi}\text{.}\)