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\begin{multicols}{2}
\section{Complex numbers}
\[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
\begin{multicols}{2}
\section{Complex numbers}
\[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
\item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
\item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
\item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
\item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
\item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
\item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
- \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
-
- \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
+ For length of \(f(x)\) from \(x=a \rightarrow x=b\):
+ \begin{align*}
+ &\text{Cartesian} \> & L &= \int^b_a \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} \> dx \\
+ &\text{Parametric} \> & L & = \int^b_a \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} \> dt
+ \end{align*}