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.
Backlinks
Metadata
date: 2021-02-22 10:31
tags:
- '@?further-maths'
- '@?complex-numbers'
- '@?public'
title: Further Maths - Exponential Form of Complex Numbers