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\begin{center}
$%
\begin{tabular}
[c]{lll}%
\textbf{U.S.T.H.B. 2014-2015 Semestre 2} & \hspace{3.75cm} & \textbf{Math 4} :
\textbf{Analyse complexe\vspace{-0.25cm}}\\
\textbf{Faculté de Mathématiques} &  & \textbf{2}$^{\mathbf{\grave{e}me}}%
$\textbf{\ Lic, ST, Section G}%
\end{tabular}
\ \ \ \ $\textbf{\vspace{0.1cm}}

\underline{\textbf{Série d'exercices n}$%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
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$\textbf{\ 0 : Nombres complexes}}\textbf{\vspace{-0.3cm}}
\end{center}

\hspace{-2cm}=============================================================

\hspace{-2cm}\underline{\textbf{Exercice 1}}\textbf{\ :\label{key1} }Soient
$z=2-i,w=1+3i.$

$\hspace{2.1cm}$Écrire les nombres complexes suivants sous forme $x+iy.$

$\hspace{3cm}$\textbf{a)} $\dfrac{z}{w}$, b) $\dfrac{zw}{z+w}$.\vspace*{0.7cm}



\hspace{-2cm}\underline{\textbf{Exercice 2}}\textbf{\ : }Trouver le module et
l'argument principal des nombres complexes suivants :

$\hspace{1.8cm}$\textbf{a) }$z=4+3i$, \textbf{b)} $z=-\cos\frac{\pi}{5}%
+i\sin\frac{\pi}{5}$, c) $z=\cos\theta-i\sin\theta$ $\left(  \pi<\theta
<\frac{3\pi}{2}\right)  $.\vspace*{0.7cm}

\hspace{-2cm}\underline{\textbf{Exercice 3}}\textbf{\ : }Représenter les
ensembles des points suivants dans le plan complexe.\newline a) $\left\{
z\in\mathbb{C\ }/\ \left\vert z-3i\right\vert \leq\left\vert z-3\right\vert
\right\}  ,$ \textbf{b)} $\left\{  z\in\mathbb{C}\ /\ \left\vert
z-i\right\vert <3\right\}  ,$ \textbf{c)} $\left\{  z\in\mathbb{C}%
\ /\ \left\vert z-i\right\vert >3\right\}  ,$

$\hspace{1.8cm}$\textbf{d)} $\left\{  z\in\mathbb{C}\ /\ \operatorname{Re}%
\left(  z\right)  -\operatorname{Im}z<1\right\}  .$\vspace*{0.7cm}

\hspace{-2cm}\underline{\textbf{Exercice 4}}\textbf{\ : }Résoudre les
équations : a) $z^{3}+3z^{2}+3z+3=0,$ \textbf{b) }$\left(  z-1\right)
^{4}=1.$

\hspace{-2cm}\underline{\textbf{Exercice 5}}\textbf{\ : }Donner les nombres
complexes suivants sous forme $x+iy.$

$\hspace{2.1cm}$\textbf{a) }$\left(  1+i\right)  ^{1000},$ b)\textbf{
}$\left(  \sqrt{3}-i\right)  ^{3}\left(  -1+i\sqrt{3}\right)  ^{-5}.$%
\vspace*{0.7cm}



\hspace{-2cm}\underline{Exercice\textbf{ }6}\textbf{\ : }Calculer $i^{\frac
{1}{6}}$ et représenter les résultats dans le plan complexe.\vspace*{0.7cm}

\hspace{-2cm}\underline{Exercice\textbf{ }7}\textbf{\ : }Calculer les sommes
suivantes :

$\hspace{2cm}$a)\textbf{ }$\sin x+\sin2x+......+\sin nx,$ b)\textbf{ }$\cos
x+\cos2x+......+\cos nx.$

\hspace{-2cm}=============================================================


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