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\begin{center}
$\vspace*{-0.4cm}$

\hspace{-2.3cm}$%
\begin{tabular}
[c]{lll}%
\textbf{USTHB 2015-2016 Semestre 2} & \hspace{1.3cm}%
\raisebox{1cm}{\logousthb}\hspace{3.3cm} & \textbf{Analyse complexe\vspace
{-0.25cm}}\\
\textbf{Faculté de Mathématiques} &  & \textbf{2}$^{\mathbf{\grave{e}me}}%
$\textbf{\ année Lic Math}$\vspace*{0.1cm}$%
\end{tabular}
\ \ $

\textbf{Série d'exercices n}$%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$\textbf{\ 2 : Dérivation dans }$\mathbb{C}$\textbf{\ \ \ \ }$\vspace
*{-0.6cm}$
\end{center}

\begin{tikzpicture}[overlay, remember picture]
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\draw[line width=1.5pt] (-2.25,0.1)  -- (17.55,0.1);
\end{tikzpicture}$\vspace*{-0.2cm}$

\hspace*{-2cm}\underline{\textbf{Exercice 1}}\textbf{\ : }\label{key2}

\noindent Soit $f:\mathbb{C}\rightarrow\mathbb{C}$ la fonction définie par
$f\left(  z\right)  =\left\vert z^{2}\right\vert $ et soit $z_{0}\in
\mathbb{C}$.

\textbf{a)} Montrer que la limite $\lim\limits_{z\rightarrow0}\dfrac
{\overline{z}}{z}$ n'existe pas.

\textbf{b)} Montrer que pour tout $z\neq z_{0}$ on a $\dfrac{f\left(
z\right)  -f\left(  z_{0}\right)  }{z-z_{0}}=\overline{z}+z_{0}\left(
\dfrac{\overline{z}-\overline{z_{0}}}{z-z_{0}}\right)  .$

\textbf{c)} En déduire que $f$ n'est pas dérivable en aucun point sauf en
$z=0$.

\hspace*{-2cm}\underline{\textbf{Exercice 2}}\textbf{\ :}

\noindent Montrer que les fonctions complexes suivantes ne sont pas dérivables
aux points indiqués.

$\hspace{0.75cm}$\textbf{a)} $f\left(  z\right)  =\overline{z},$ pour
$z\in\mathbb{C}$ \textbf{b)} $f\left(  z\right)  =\operatorname{Re}z,$ pour
$z\in\mathbb{C},$ \textbf{c)} $f\left(  z\right)  =\operatorname{Im}z,$ pour
$z\in\mathbb{C}.$

\hspace*{-2cm}\underline{\textbf{Exercice 3}}\textbf{\ :}

\noindent Examiner si les fonctions suivantes sont holomorphes sur le domaine indiqué.

$\hspace{0.5cm}$\textbf{a)} $f\left(  z\right)  =\left(  \overline
{z}+i\right)  ^{2},$ sur $\mathbb{C}$, \textbf{b)} $f\left(  z\right)
=\dfrac{x}{x^{2}+y^{2}}+i\dfrac{y}{x^{2}+y^{2}},$ sur $\mathbb{C}%
\backslash\left\{  0\right\}  $,\medskip

$\hspace{0.5cm}$\textbf{c)} $f\left(  z\right)  =\operatorname{Re}\left(
\dfrac{z}{z-1}\right)  ,$ sur $\left.  \mathbb{C}\right\backslash \left\{
1\right\}  $, \textbf{d)} $f\left(  z\right)  =\left(  x^{2}-y^{2}-2xy\right)
+i\left(  x^{2}-y^{2}+2xy\right)  ,$ sur $\mathbb{C}$.

\hspace*{-2cm}\underline{\textbf{Exercice 4}}\textbf{\ : }

\noindent Montrer qu'il n'existe pas de fonction holomorphe sur $\mathbb{C}$
de partie réelle $\left\vert z\right\vert ^{2}$.

\hspace*{-2cm}\underline{\textbf{Exercice 5}}\textbf{\ :}

\noindent Soit $f:\mathbb{C}\rightarrow\mathbb{C}$ la fonction définie par%
\[
f\left(  z\right)  =\left\{
\begin{array}
[c]{lll}%
\dfrac{z^{5}}{\left\vert z^{4}\right\vert } &  & \text{si \ }z\neq0\\
0 &  & \text{si \ }z=0
\end{array}
\right.  .
\]


\textbf{a) }Montrer que la fonction $f$ vérifie les conditions de
Cauchy-Riemann au point $\left(  0,0\right)  $.

\textbf{b) }La fonction $f$ est-elle dérivable au point $\left(  0,0\right)  $?

\textbf{c) }Discuter le résultat.

\hspace*{-2cm}\underline{\textbf{Exercice 6}}\textbf{\ :}

\noindent Supposons que $f$ est holomorphe dans un domaine $D\subset
\mathbb{C}$ et que à chaque point $z\in D$, soit $f\left(  z\right)  =0$ ou
bien $f^{\prime}\left(  z\right)  =0$. Montrer que $f$ est constante sur $D$.

\hspace*{-2cm}\underline{\textbf{Exercice 7}}\textbf{\ :}

\noindent Supposons que $f$ est une fonction entière de la forme $f\left(
z\right)  =u\left(  x\right)  +iv\left(  y\right)  $.\newline Montrer que $f$
est un polynôme linéaire.

\hspace*{-2cm}\underline{\textbf{Exercice 8}}\textbf{\ :}

\noindent Montrer que les équations de Cauchy-Riemann s'écrivent en
coordonnées polaires sous la forme%
\[
\frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial\theta
}\text{\thinspace\ et \ }\frac{\partial v}{\partial r}=-\frac{1}{r}%
\frac{\partial u}{\partial\theta}.
\]


\hspace*{-2cm}\underline{\textbf{Exercice 9}}\textbf{\ : }

\noindent Montrer que la fonction $u$ définie ci-dessous est harmonique.%
\[
u\left(  x,y\right)  =\dfrac{x}{x^{2}+y^{2}},\ \left(  x,y\right)  \in\left.
\mathbb{R}^{2}\right\backslash \left\{  \left(  0,0\right)  \right\}  .
\]
Trouver une fonction $v$ pour que la fonction $f=u+iv$ soit holomorphe sur
$\mathbb{C}^{\ast}$.

\hspace*{-2cm}\underline{\textbf{Exercice 10}}\textbf{\ :}

\noindent Quelle est la nature des singularités de chacune des fonctions suivantes?

\textbf{a)} $f\left(  z\right)  =\dfrac{z+3}{z^{2}-1}$, \ \ \ \textbf{b)}
$f\left(  z\right)  =\dfrac{1}{\sin\left(  \frac{1}{z^{2}}\right)  }%
$,\ \ \ \ \textbf{c)} $f\left(  z\right)  =\dfrac{\operatorname{Log}\left(
z-2\right)  }{\left(  z^{2}+2z+2\right)  ^{4}}$.
\end{document}