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+\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out! +\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes +\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}} +\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}} + \begin{document} \begin{multicols}{2} @@ -71,5 +81,62 @@ & \qquad \text{(rationalise denominator)} \end{align*} + \subsection*{Argand planes} + + \begin{tikzpicture}\begin{scope}[thick,font=\scriptsize] + \draw [->] (-1.5,0) -- (1.5,0) node [above left] {$\operatorname{Re}(z)$}; + \draw [->] (0,-1.5) -- (0,1.5) node [below right] {$\operatorname{Im}(z)$}; + + % If you only want a single label per axis side: + \draw (1,-3pt) -- (1,0pt) node [below] {$1$}; + \draw (-1,-3pt) -- (-1,0pt) node [below] {$-1$}; + \draw (-3pt,1) -- (0pt,1) node [left] {$i$}; + \draw (-3pt,-1) -- (0pt,-1) node [left] {$-i$}; + \end{scope}\end{tikzpicture} + + Multiplication by \(i \implies\) anticlockwise rotation of \(\frac{\pi}{2}\) + + \subsection*{de Moivres' theorem} + + \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\] + + \subsection*{Complex polynomials} + + Include \(\pm\) for all solutions, incl. imaginary + +\newcolumntype{R}{>{\raggedleft\arraybackslash}X} +\newcolumntype{L}{>{\raggedright\arraybackslash}X} + \begin{tabularx}{\columnwidth}{rX} + \hline + Sum of squares & \(\begin{aligned} + z^2 + a^2 &= z^2-(ai)^2\\ + &= (z+ai)(z-ai) \end{aligned}\) \\ + \hline + Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\ + \hline + Division & \(P(z)=D(z)Q(z)+R(z)\) \\ + \hline + \parbox[t]{2cm}{Remainder} & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\ + \hline +\end{tabularx} + +\subsection*{Roots} + +\(n\)th roots of \(z=r\operatorname{cis}\theta\) are: + +\[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\] + +\begin{itemize} + + \item{Same modulus for all solutions} + \item{Arguments are separated by \(\frac{2\pi}{n}\)} + +\item{Solutions of \(z^n=a\) where \(a \in \mathbb{C}\) lie on the circle \(x^2+y^2=\left(|a|^{\frac{1}{n}}\right)^2\)} +\end{itemize} + +\subsubsection*{Conjugate root theorem} + +If \(a+bi\) is a solution to \(P(z)=0\), then the conjugate \(\overline{z}=a-bi\) is also a solution. + \end{multicols} \end{document} -- 2.47.1