sketching complex graphs

diff --git a/spec/complex.md b/spec/complex.md
index 54c8082b9a4cbee945288ab43da60c99d946f195..c5ad6be5bee90287f816804de6db4ecb759626df 100755 (executable)
--- a/spec/complex.md
+++ b/spec/complex.md
@@ -106,13 +106,33 @@ To solve $z^2+a^2=0$ (sum of two squares):
 
 $z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$
 
+*Must include $\pm$ in solutions*
+
+## Solving complex polynomials
+
+Include $\pm$ for all solutions, including imaginary.
+
+#### Dividing complex polynomials
+
+Dividing $P(z)$ by $D(z)$ gives quotient $Q(z)$ and remainder $R(z)$ such that:
+
+$$P(z) = D(z)Q(z) + R(z)$$
+
+#### Remainder theorem
+
+Let $\alpha \in \mathbb{C}$. Remainder of $P(z) \div (z - \alpha)$ is $P(\alpha)$
+
+## Conjugate root theorem
+
+If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, the the conjugate $a-bi$ is also a solution.
+
 ## Polar form
 
 $$\begin{equation}\begin{split}z & =r \operatorname{cis} \theta \\ & = r(\operatorname{cos}\theta+i \operatorname{sin}\theta) \\ & = a + bi \end{split}\end{equation}$$
 
 - $r=|z|$, 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 $\operatorname{Arg}(z) \in (-\pi, \pi]$
+- $\theta=\operatorname{arg}(z)$ (on CAS: `arg(a+bi)`)
+- **principal argument** is $\operatorname{Arg}(z) \in (-\pi, \pi]$ (note capital $\operatorname{Arg}$)
 
 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
@@ -132,3 +152,16 @@ ${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divid
 ## de Moivres' Theorem
 
 $(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$ where $n \in \mathbb{Z}$
+
+## Roots of complex numbers
+
+$n$th roots of $r \operatorname{cis} \theta$ are:  
+$z={r^{1 \over n}} \cdot (\cos ({{\theta + 2k \pi} \over n}) + i \sin ({{\theta + 2 k \pi} \over n}))$
+
+Same modulus for all solutions. Arguments are separated by ${2 \pi} \over n$
+
+## Sketching complex graphs
+
+- **Straight line:** $\operatorname{Re}(z) = c$ or $\operatorname{Im}(z) = c$ (perpendicular bisector) or $\operatorname{Arg}(z) = \theta$
+- **Circle:** $|z-z_1|^2 = c^2 |z_2+2|^2$ or $|z-(a + bi)| = c$
+- **Locus:** $\operatorname{Arg}(z) \lt \theta$