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tidy up Z notes
author
Andrew Lorimer
<andrew@lorimer.id.au>
Tue, 19 Feb 2019 11:09:15 +0000
(22:09 +1100)
committer
Andrew Lorimer
<andrew@lorimer.id.au>
Tue, 19 Feb 2019 11:09:15 +0000
(22:09 +1100)
spec/complex.md
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diff --git
a/spec/complex.md
b/spec/complex.md
index e7760be2d467eeee835e0783238c7cb8d0d4a891..54c8082b9a4cbee945288ab43da60c99d946f195 100755
(executable)
--- a/
spec/complex.md
+++ b/
spec/complex.md
@@
-45,7
+45,7
@@
Therefore..
- $i^{4n+2} = -1$
- $i^{4n+3} = -i$
- $i^{4n+2} = -1$
- $i^{4n+3} = -i$
-
Divide by 4 and take remainder
.
+
For $i^n$, divide $n$ by 4 and let remainder $= r$. Then $i^n = i^r$
.
### Multiplying complex expressions
### Multiplying complex expressions
@@
-68,7
+68,6
@@
Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2 = |z|^2$
- $z \overline{z} = |z|^2$
- $z + \overline{z} = 2 \operatorname{Re}(z)$
- $z \overline{z} = |z|^2$
- $z + \overline{z} = 2 \operatorname{Re}(z)$
-
### Modulus
Distance from origin.
### Modulus
Distance from origin.
@@
-109,17
+108,11
@@
$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$
## Polar form
## Polar form
-General form:
-$z=r \operatorname{cis} \theta$
-$= r(\operatorname{cos}\theta+i \operatorname{sin}\theta)$
-
-$z=a+bi$
-$z=r\operatorname{cis}\theta$
+$$\begin{equation}\begin{split}z & =r \operatorname{cis} \theta \\ & = r(\operatorname{cos}\theta+i \operatorname{sin}\theta) \\ & = a + bi \end{split}\end{equation}$$
-
-- $z=a+bi$
-- $r$ is the distance from origin, given by Pythagoras ($r=\sqrt{x^2+y^2}$)
-- $\theta$ is the argument of $z$, CCW from origin
+- $r=|z|$, given by Pythagoras ($r=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}$)
+- $\theta=\operatorname{Arg}(z)$ (on CAS: `arg(a+bi)`)
+- **principal argument** is $\operatorname{Arg}(z) \in (-\pi, \pi]$
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
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
@@
-128,6
+121,8
@@
$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) where $n$
$$(r \operatorname{cis} \theta)^{-1} = r\operatorname{cis} (- \theta)$$
$$(r \operatorname{cis} \theta)^{-1} = r\operatorname{cis} (- \theta)$$
+Reflection of $z$ across horizontal axis.
+
### Multiplication and division in polar form
$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)
### Multiplication and division in polar form
$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)