1.5 The Exponential Representation

We can write any complex number \(z = x + iy\) in the form \(\displaystyle {z = e^{i\theta }}\) where \(\quad \displaystyle {e^{i\theta } = \cos \theta + i \sin \theta }\)

One justification for using this definition is usually the series expansion for exp, cos and sin. We have \begin {align*} e^{i\theta } & = 1 + i\theta + \frac {\big (i\theta \big )^2}{2!} + \frac {\big (i\theta \big )^3}{3!}+ \cdots \\\\ & = 1 + i\theta - \frac {\theta ^2}{2!} - \frac {i \theta ^3}{3!} + \cdots \\\\ & = \Bigg (1 - \frac {\theta ^2}{2!} + \frac {\theta ^4}{4!} + \cdots \Bigg ) + i \Bigg ( \theta - \frac {\theta ^3}{3!}+ \cdots \Bigg )\\\\ & = \cos \theta + i \sin \theta \\\\ \end {align*}

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.