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\begin{document}

\begin{center}
$\vspace*{-1cm}$

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

\underline{\textbf{Série d'exercices n}$%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$\textbf{\ 5 : Théorème des résidus}}
\end{center}

\noindent\underline{\textbf{Exercice 1}}\textbf{\ : \label{key7}}

\noindent Trouver les résidus de \textbf{(a)} $f\left(  z\right)
=\dfrac{z^{2}-2z}{\left(  z+1\right)  ^{2}\left(  z^{2}+4\right)  }$ et (b)
$f\left(  z\right)  =\dfrac{e^{z}}{\sin^{2}z}$\ en tous les pôles à distance
finie.$\bigskip$

\noindent\underline{\textbf{Exercice 2}}\textbf{\ :}

Calculer$\dfrac{1}{2\pi i}$ $%
%TCIMACRO{\doint \limits_{C}}%
%BeginExpansion
{\displaystyle\oint\limits_{C}}
%EndExpansion
\dfrac{e^{z}}{z^{2}\left(  z^{2}+2z+2\right)  }dz$\ le long du cercle $C$
d'équation \textbf{(a)} $\left\vert z\right\vert =3$ et \textbf{(b)}
$\left\vert z\right\vert =1.\bigskip$

\noindent\underline{\textbf{Exercice 3}}\textbf{\ :}

Évaluer \textbf{(a)} $%
%TCIMACRO{\dint \limits_{0}^{+\infty}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{+\infty}}
%EndExpansion
\dfrac{1}{x^{6}+1}dx$ et \textbf{(b)} $%
%TCIMACRO{\dint \limits_{-\infty}^{+\infty}}%
%BeginExpansion
{\displaystyle\int\limits_{-\infty}^{+\infty}}
%EndExpansion
\dfrac{x^{2}}{\left(  x^{2}+1\right)  ^{2}\left(  x^{2}+2x+2\right)  }%
dx.$\bigskip

\noindent\underline{\textbf{Exercice 4}}\textbf{\ :}

Évaluer \textbf{(a)} $%
%TCIMACRO{\dint \limits_{0}^{2\pi}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{2\pi}}
%EndExpansion
\dfrac{1}{3-2\cos\theta+\sin\theta}d\theta$ et (b) $%
%TCIMACRO{\dint \limits_{0}^{2\pi}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{2\pi}}
%EndExpansion
\dfrac{1}{2+\sin\theta}d\theta.$\bigskip

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

Montrer que $%
%TCIMACRO{\dint \limits_{0}^{\infty}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{\infty}}
%EndExpansion
\dfrac{\cos\left(  mx\right)  }{x^{2}+1}dx=\dfrac{\pi}{2}e^{-m},\ m>0.$%
\bigskip

\noindent\underline{\textbf{Exercice 6}}\textbf{\ :}

Montrer que $%
%TCIMACRO{\dint \limits_{0}^{+\infty}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{+\infty}}
%EndExpansion
\dfrac{\sin x}{x}dx=\dfrac{\pi}{2}.$\bigskip

\noindent\underline{\textbf{Exercice 7}}\textbf{\ :}

Montrer que $%
%TCIMACRO{\dint \limits_{0}^{+\infty}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{+\infty}}
%EndExpansion
\dfrac{x^{p-1}}{1+x}dx=\dfrac{\pi}{\sin\left(  p\pi\right)  },\ \ 0<p<1.$%
\bigskip

\noindent\underline{\textbf{Exercice 8}}\textbf{\ :}

Montrer que $%
%TCIMACRO{\dint \limits_{0}^{+\infty}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{+\infty}}
%EndExpansion
\dfrac{\operatorname{Ch}\left(  ax\right)  }{\operatorname{Ch}x}dx=\dfrac{\pi
}{2\cos\left(  \dfrac{\pi a}{2}\right)  },\ \ $\ où $\left\vert a\right\vert
<1.$\bigskip

\noindent\underline{\textbf{Exercice 9}}\textbf{\ :}

Démontrer que $%
%TCIMACRO{\dint \limits_{0}^{+\infty}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{+\infty}}
%EndExpansion
\dfrac{\operatorname{Log}\left(  x^{2}+1\right)  }{x^{2}+1}dx=\pi
\operatorname{Log}2.$


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