From 8ca72786957b19e5866fa2a1e86155ab80d9686d Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Wed, 20 Feb 2019 11:05:32 +1100 Subject: [PATCH] solving complex quadratics --- spec/complex.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/spec/complex.md b/spec/complex.md index 54c8082..b1688eb 100755 --- a/spec/complex.md +++ b/spec/complex.md @@ -106,13 +106,15 @@ 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* + ## 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 -- 2.43.2