Further Maths - Exponential Form of Complex Numbers

Sergeant

What is Euler’s relation?

\[e^{i\theta} = \cos \theta + i \sin \theta\]

Why can you rewrite $e^{i\theta}$ as $\cos\theta + i\sin\theta$?

Because the Maclaurin series of $\sin x$, $\cos x$ and $e^x$ match up.

How can you write a complex number with argument $\theta$ and modulus $r$ in exponential form?

\[re^{i\theta}\]

\[e^{\pi i} = -1\]

What is this identity a special case of?

Euler’s relation.

\[z _ 1 = r _ 1 e^{\theta _ 1 i} \\ z _ 2 = r _ 2 e^{\theta _ 2 i}\]

What is $z _ 1 z _ 2$?

\[r _ 1 r _ 2 e^{(\theta _ 1 + \theta _ 2)i}\]

\[z _ 1 = r _ 1 e^{\theta _ 1 i} \\ z _ 2 = r _ 2 e^{\theta _ 2 i}\]

What is $\frac{z _ 1}{z _ 2}$?

\[\frac{r _ 1}{r _ 2} e^{(\theta _ 1 - \theta _ 2)i}\]

\[z = r e^{\theta i}\]

What is $z^n$?

\[r^n e^{n\theta i}\]

What is De Moivre’s Theorem?

\[z = r(\cos\theta + i \sin\theta)\]

Then

\[z^n = r^n (\cos n\theta + i \sin n\theta)\]

What’s the process (but not the actual steps) for proving De Moivre’s Theorem using Euler’s relation?

Rewrite the modulus-argument form using $e$ and apply the laws of indices.