\subsection*{Sketching complex graphs}
- \subsubsection*{Linear}
+ \subsubsection*{Rays/lines \qquad \(\operatorname{Arg}( z\pm b)=\theta\)}
+
+ \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
+ \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
+ \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
+ \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
+ \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
+ \begin{scope}
+ \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
+ \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
+ \end{scope}
+ \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
+ \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
+ \node [brown, mydot] at (-0.25,0) {};
+ \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
+ \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
+ \node [below, font=\footnotesize] at (1,0) {\(1\)};
+ \end{tikzpicture}\end{center}
\begin{itemize}
+ \item \(\operatorname{Arg}(z \pm b) = \theta\) (ray)
\item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
\item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
- \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
+ \item \(|z - (a+bi)|=|z - (c+di)| \\ \implies \frac{2(c-a)x + a^2 + b^2 - c^2 - d^2}{2(b-d)}\)
+ \item \(\operatorname{Re}(z) \pm \operatorname{Im}(z) = c\)
\end{itemize}
\subsubsection*{Circles}
\begin{itemize}
\item \(|z-z_1|^2=c^2|z_2+2|^2\)
\item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
+ \item \(z \overline{z} = r^2\)
\end{itemize}
- \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
+ \subsubsection*{Regions \qquad \(\operatorname{Arg}(z) \lessgtr \theta\)}
\begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
\draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
\node [blue, mydot] {};
\end{tikzpicture}\end{center}
- \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
-
- \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
- \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
- \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
- \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
- \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
- \begin{scope}
- \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
- \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
- \end{scope}
- \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
- \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
- \node [brown, mydot] at (-0.25,0) {};
- \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
- \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
- \node [below, font=\footnotesize] at (1,0) {\(1\)};
- \end{tikzpicture}\end{center}
\section{Vectors}
\begin{center}\begin{tikzpicture}
\begin{align*}
V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
- &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
+ &= 2\pi \int^{x=b}_{x=a} x|f(x)| \> dx
\end{align*}
- \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
+ \subsubsection*{Rotating the area between two graphs}
- \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
+ \[V = \pi \int^b_a \left( f(x)^2 - g(x)^2 \right) \> dx\]
\hfill where \(f(x) > g(x)\)
\subsection*{Length of a curve}