From 524425c84045eff029812a22c41216e612dd6a21 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Wed, 6 Mar 2019 16:36:24 +1100 Subject: [PATCH] [spec] condense complex & vector reference sheets --- spec/complex-ref.pdf | Bin 36486 -> 36864 bytes spec/complex.md | 50 ++++++++++++++++++++++++++++--------------- spec/vectors-ref.pdf | Bin 60473 -> 47615 bytes spec/vectors.md | 2 +- 4 files changed, 34 insertions(+), 18 deletions(-) diff --git a/spec/complex-ref.pdf b/spec/complex-ref.pdf index 96ab8bf9abca3b5f93134bda60b0fb18de3c0a94..6119f104d99772ff1228f88b7b19a18189f294c4 100644 GIT binary patch delta 35397 zcmY(KQ+Opxv~D}Ltra`z*tTukwzE36ZQFLzvDL9{+sWPcJe~7cud}NDZ;m;tLUTa> z82R~OU0j{bjO<`N*ROP>9Oy<7dQUW_2-BStsxgpV84U@YS?=wH5EI2+ z8j<*qZo;Z-s=BJuGRne!MHpRJEvmppT}WVl+@C=F&io0AlK3_-;Qwy_{JgqfsB;7< z8sqHt?J}rEM#t_3FA4a)WB=njkGs>{n&0Ae0^A7&Le;>uJRd(vZ{I_=FDaBp2O$9l z`w-$-+CJMBct=|j?hh%745TK15ABRWX=#UY`+lvo^%?lTEsud?)Z`?1i7E023!j2z 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Remainder of $P(z) \div (z - \alpha)$ is $P(\alpha)$ #### Factor theorem + If $a+bi$ is a solution to $P(z)=0$, then: - $P(a+bi)=0$ - $z-(a+bi)$ is a factor of $P(z)$ +#### Sum of two cubes + +$$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$$ + ## Conjugate root theorem -If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, then the conjugate $\overline{z}=a-bi$ is also a solution. +If $a+bi$ is a solution to $P(z)=0$, then the conjugate $\overline{z}=a-bi$ is also a solution. ## Polar form @@ -146,8 +151,8 @@ If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, then the conjug - $\theta=\operatorname{arg}(z)$ (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 +Each complex number has multiple polar representations: +$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) with $n \in \mathbb{Z}$ revolutions ### Conjugate in polar form @@ -157,9 +162,9 @@ 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) +$$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$$ -${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divide moduli, subtract angles) +$${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$$ ## de Moivres' Theorem @@ -179,6 +184,17 @@ $$x^2 + y^2 = (|a|^{1 \over n})^2$$ ## 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:** $\operatorname{Arg}(z) < \theta$ +### Straight line + +- $\operatorname{Re}(z) = c$ or $\operatorname{Im}(z) = c$ (perpendicular bisector) +- $\operatorname{Arg}(z) = \theta$ +- $|z+a|=|z+bi|$ where $m={a \over b}$ +- $|z+a|=|z+b| \longrightarrow 2(a-b)x=b^2-a^2$ + +### Circle + +$|z-z_1|^2 = c^2 |z_2+2|^2$ or $|z-(a + bi)| = c$ + +### Locus + +$\operatorname{Arg}(z) < \theta$ diff --git a/spec/vectors-ref.pdf b/spec/vectors-ref.pdf index bd9863571d3f5e6dd032ca6daa920f79408745f3..c1c46b6a2850bc89cd18ee4dcda6a3935f956917 100644 GIT binary patch delta 28404 zcmV()K;OT)*aQFb0+1V%L{dkyGcpDs0W-5YQmG7+kZUG?F)%ScJ_>Vma%Ev{3V7PJ zTg$E^xeeXtEBpi2NTi-X<1T_EK(ZU8cR?1{cHh-Bf~@lUAw@l_RAt}pnR&F!Wr?CF z^6;g;{P$umn*KGH0Q8@}+`nJ`{Ppt7-^|5Wh1SF6>&NA4oI)Kg^ry9#uaAFy(b|8! 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