From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Mon, 18 Feb 2019 03:56:38 +0000 (+1100)
Subject: complex conjugate
X-Git-Tag: yr12~251
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/47385e8e6477197a86b27dee64a4cd305b871d47

complex conjugate
---

diff --git a/spec/complex.md b/spec/complex.md
index 10eba3f..e7760be 100755
--- a/spec/complex.md
+++ b/spec/complex.md
@@ -111,9 +111,11 @@ $z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$
 
 General form:
 $z=r \operatorname{cis} \theta$
-$= r\operatorname{cos}\theta+r\operatorname{sin}\theta i$
+$= r(\operatorname{cos}\theta+i \operatorname{sin}\theta)$
+
+$z=a+bi$  
+$z=r\operatorname{cis}\theta$  
 
-where
 
 - $z=a+bi$
 - $r$ is the distance from origin, given by Pythagoras ($r=\sqrt{x^2+y^2}$)
@@ -122,12 +124,16 @@ where
 Note each complex number has multiple polar representations:
 $z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) where $n$ is integer number of revolutions
 
+### Conjugate in polar form
+
+$$(r \operatorname{cis} \theta)^{-1} = r\operatorname{cis} (- \theta)$$
+
 ### Multiplication and division in polar form
 
 $z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)
 
 ${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divide moduli, subtract angles)
 
-## de Moivres' Theorum
+## de Moivres' Theorem
 
-$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$
+$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$ where $n \in \mathbb{Z}$