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\title{Nombres complexes}

\section*{Ensemble des nombres complexes}
\[
\C = \{ a + \i b\quad,\quad a\in\R,\ b\in\R,\ \i^2=-1\}
\]


\section*{Les différentes formes}

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\begin{tabular}{*3{|>{\centering\arraybackslash}m{0.3\textwidth}}|}
\hline
\rowcolor{bleu}
\color{white}Forme algébrique & \color{white}Forme trigonométrique & \color{white}Forme exponentielle\\
\hline
$z = a + \i b$ & $a = r(\cos\theta + \i\sin\theta)$ & $z=r\e^{\i\theta}$\\
\hline
\end{tabular}
\[
\begin{cases}
r = \sqrt{a^2+b^2}\\[6pt]
\cos\theta=\dfrac{a}{r}\text{ et }\sin\theta=\dfrac{b}{r}
\end{cases}
\]

\paragraph*{Exemple.} 

\[
z = 1+\i \Longrightarrow r = \sqrt{1^2+1^2}=\sqrt2 \Longrightarrow \begin{cases}
\cos\theta=\dfrac{1}{\sqrt2}=\dfrac{\sqrt2}{2}\\
\sin\theta=\dfrac{1}{\sqrt2}=\dfrac{\sqrt2}{2}
\end{cases} \Longrightarrow\theta=\dfrac{\pi}{4}\quad[2\pi] \Longrightarrow z = \sqrt2\left(\cos\dfrac{\pi}{4}+\i\sin\dfrac{\pi}{4}\right)=\sqrt2\e^{\i\frac{\pi}{4}}.
\]


\section*{Définitions}

\begin{minipage}[t]{0.3\textwidth}
\subsection*{Parties réelles et\\imaginaires}
$z=a+\i b \iff \begin{cases}
a = \Re\text{e}(z)\\
b = \Im\text{m}(z)
\end{cases}$

\subsection*{Conjugué}
$z = a+\i b \iff \overline{z} = a - \i b$
\end{minipage}
\begin{minipage}[t]{0.3\textwidth}
\subsection*{Module}
$z = a + \i b \Longrightarrow |z| = \sqrt{a^2+b^2}$
\end{minipage}
\begin{minipage}[t]{0.3\textwidth}
\subsection*{Arguments}
$\begin{array}{l}
z = a + \i b \Longrightarrow \arg(z)=\theta,\\
\text{ où }\begin{cases}
\cos\theta=\dfrac{a}{|z|}\\
\sin\theta=\dfrac{b}{|z|}
\end{cases}
\end{array}$
\end{minipage}

\section*{Propriétés}

\begin{minipage}{0.3\textwidth}
\subsection*{Conjugué}
\begin{itemize}
\item[$\bullet$] $\overline{z+z'}=\overline{z}+\overline{z'}$\\[-2mm]
\item[$\bullet$] $\overline{z\times z'}=\overline{z}\times\overline{z'}$\\[-2mm]
\item[$\bullet$] $\overline{\dfrac{z}{z'}}=\dfrac{\overline{z}}{\overline{z'}}$\\[-1mm]
\item[$\bullet$] $\overline{z^n}=\overline{z}^n$
\end{itemize}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\subsection*{Module}
\begin{itemize}
\item[$\bullet$] $|z|=\left|\overline{z}\right|$\\[-2mm]
\item[$\bullet$] $|z \times z'|=|z| \times |z'|$\\[-2mm]
\item[$\bullet$] $\left|\dfrac{z}{z'}\right|=\dfrac{|z|}{|z'|}$\\[-1mm]
\item[$\bullet$] $|z^n| = |z|^n$
\end{itemize}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\subsection*{Arguments}
\begin{itemize}
\item[$\bullet$] $\arg(\overline{z}) = -\arg(z)$\\[-2mm]
\item[$\bullet$] $\arg(z\times z')=\arg(z)+\arg(z')$\\[-2mm]
\item[$\bullet$] $\arg\left(\dfrac{z}{z'}\right)=\arg(z)-\arg(z')$\\[-1mm]
\item[$\bullet$] $\arg(z^n)=n\times\arg(z)$
\end{itemize}
\end{minipage}

\newpage

\section*{Polynômes}

\begin{minipage}[t]{0.45\textwidth}
\subsection*{\'Equations du second degré}
$
\begin{cases}
az^2+bz+c=0\\
\Delta=b^2-4ac<0
\end{cases} \Longrightarrow \begin{cases}
z_1=\dfrac{-b-\i\sqrt{|\Delta|}}{2a}\\
z_2=\dfrac{-b+\i\sqrt{|\Delta|}}{2a}=\overline{z_1}
\end{cases}
$
\end{minipage}
\begin{minipage}[t]{0.45\textwidth}
\subsection*{Factorisation de $\text{z}^n-\text{a}^n$}
$z^n - a^n = (z-a)\displaystyle\sum_{k=0}^{n-1} a^kz^{n-k-1}$
\end{minipage}


\section*{Trigonométrie}

\begin{minipage}[t]{0.45\textwidth}
\subsection*{Formules d'Euler}
$\cos\theta=\dfrac{\e^{\i\theta}+\e^{-\i\theta}}{2}
\qquad
\text{et}
\qquad
\sin\theta=\dfrac{\e^{\i\theta}-\e^{-\i\theta}}{2\i}$
\end{minipage}
\begin{minipage}[t]{0.45\textwidth}
\subsection*{Formule de Moivre}
$
(\cos\theta+\i\sin\theta)^n=\cos(n\theta)+\i\sin(n\theta)
$
\end{minipage}

\bigskip

\begin{minipage}[t]{0.45\textwidth}
\subsection*{Formules d'addition}
\begin{itemize}
\item[$\bullet$] $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$
\item[$\bullet$] $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$
\end{itemize}
\end{minipage}
\begin{minipage}[t]{0.45\textwidth}
\subsection*{Formules de duplication}
\begin{itemize}
\item[$\bullet$] $\sin(2a)=2\sin(a)\cos(a)$
\item[$\bullet$] $\begin{aligned}[t]\cos(2a)&=\cos^2(a)-\sin^2(a)\\
& = 2\cos^2(a)-1\\
& = 1-2\sin^2(a)
\end{aligned}$
\end{itemize}
\end{minipage}

\subsection*{Distances et angles}

\begin{minipage}[t]{0.45\linewidth}
Soient $A$, $B$ et $C$ d'affixes respectives $a$, $b$ et $c$.

\bigskip

\begin{itemize}
\item[$\bullet$] $|b-a| = AB$\\
\item[$\bullet$] $\arg(b-a)=\angleor{u}{AB}$\\
\item[$\bullet$] $\left| \dfrac{c-a}{b-a} \right| = \dfrac{AC}{AB}$\\[2ex]
\item[$\bullet$] $\arg\left(  \dfrac{c-a}{b-a} \right) = \angleor{AB}{AC}$
\end{itemize}
\end{minipage}
\hfill
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\end{minipage}

\subsection*{Racines n-ièmes de l'unités}

Pour $n$ entier naturel, on appelle \textit{racines n-ième} de l'unité tous les nombres complexes $z$ tels que $z^n=1$.

\medskip

On note $\mathbb{U}_n$ l'ensemble de toutes ces valeurs:
\[
\mathbb{U}_n = \{ z \in \C | z^n = 1 \} = \left\{ \e^{\frac{2\i k\pi}{n}}\,,\, k\in\llbracket 0\;;\; n-1\rrbracket \right\}
\]

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