Further Maths - Exponential Form of Complex Numbers
See Also
Sergeant
further-maths/textbooks/year-2/chapter-1-complex-numbers/ex1a
Flashcards
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 Macluarin series of $\sin x$, $\cos x$ and $e^x$ match up.
How can you write a complex number with argument $\theta$ and moudlus $r$ in exponential form?
\[re^{i\theta}\]
\[e^{\pi i} = -1\]
What is this identity a special case of?
Euler’s relation.
2021-02-23
\[z _ 1 = r _ 1 e^{\theta _ 1 i} \\ z _ 2 = r _ 2 e^{\theta _ 2 i}\]
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 Do Moivre’s Theorem?
If
\[z = r(\cos\theta + i \sin\theta)\]Then
\[z^n = r^n (\cos n\theta + i \sin n\theta)\]#
What’s the process (but not the actual steps) for provind De Moivre’s Theorem using Euler’s relation?
Rewrite the modulus-argument form using $e$ and apply the laws of indicies.