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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, GP, Section G}%
\end{tabular}
\ \ \ $\textbf{\vspace{0.25cm}}

\underline{\textbf{Série d'exercices n}$%
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%BeginExpansion
{{}^\circ}%
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$\textbf{\ 1 : Fonctions élémentaires}}
\end{center}

\noindent\underline{\textbf{Exercice 1}}\textbf{\ : \label{key2}}Séparer les
parties réelles et imaginaires des fonctions suivantes :

$\hspace{0.75cm}$\textbf{a)} $f\left(  z\right)  =e^{-z},$ \textbf{b)}
$f\left(  z\right)  =\sin z,$ c) $f\left(  z\right)  =\ $Ch$\,z,$ d) $f\left(
z\right)  =2^{z^{2}},$ e) $f\left(  z\right)  =z^{2-i}.$\vfill


\noindent\underline{\textbf{Exercice 2}}\textbf{\ : }Démontrer les relations
suivantes :

$\hspace{1.5cm}$\textbf{a)} $\left\vert \sin z\right\vert =\sqrt{\text{Ch}%
^{2}y-\cos^{2}x},$ b) $\left\vert \cos z\right\vert =\sqrt{\text{Ch}^{2}%
y-\sin^{2}x},$

$\hspace{1.5cm}$c) $\left\vert \text{Sh\ }z\right\vert =\sqrt{\text{Ch}%
^{2}x-\cos^{2}y},$ d) $\left\vert \text{Ch\ }z\right\vert =\sqrt{\text{Ch}%
^{2}x-\sin^{2}y}.$\vfill


\noindent\underline{\textbf{Exercice 3}}\textbf{\ : }Résoudre dans
$\mathbb{C}$ les équations suivantes :

$\hspace{0.75cm}$\textbf{a)} $\operatorname{Im}\left(  \sin z\right)  =0$, b)
$\operatorname{Re}\left(  \hspace{0.05cm}\text{Sh}\hspace{0.05cm}z\right)
=0$, c) $\sin z=\frac{4}{3}i$, d) Sh$\hspace{0.05cm}z=\frac{i}{2}$, e)
$e^{z}=-2$.\vfill


\noindent\underline{Exercice 4}\textbf{\ : }Démontrer que $e^{\left(
\text{Log\ }z\right)  }=z$ et montrer que

$\hspace{2.05cm}$l'égalité Log$\left(  e^{z}\right)  =z$ n'est pas toujours
vérifiée.\vfill


\noindent\underline{\textbf{Exercice 5}}\textbf{\ : }Calculer

$\hspace{0.75cm}$a) Log$\left(  1+i\right)  ,$ \textbf{b)} $i^{i},$ c)
$\left(  1-i\right)  ^{3-3i}.$\vfill


\noindent\underline{\textbf{Exercice 6}}\textbf{\ : }Calculer les limites
suivantes :

$\hspace{0.75cm}$a) $\lim\limits_{z\rightarrow\frac{\pi}{4}}\dfrac{\cos\left(
2z\right)  }{\text{ch}\left(  iz\right)  +i\ \text{sh}\left(  iz\right)  }$,
\textbf{b)} $\lim\limits_{z\rightarrow-i\frac{\pi}{2}}\dfrac{e^{2z}+1}%
{e^{z}+i}$.\vfill


\noindent\underline{\textbf{Exercice 7}}\textbf{\ : }Montrer que la limite
$\lim\limits_{z\rightarrow0}\dfrac{\overline{z}}{z}$ n'existe pas.


\end{document}