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{\vspace*{0cm} }

\begin{center}
\textbf{{\large U}NIVERSITÉ DES {\large S}CIENCES ET DE LA {\large T}%
ECHNOLOGIE}\newline\textbf{{\large H}OUARI {\large B}OUMEDIENNE}%
\footnote{USTHB : Bab Ezzouar Alger, Algérie.}

\textbf{{\large F}ACULTÉ DES {\large M}ATHÉMATIQUES}

\textbf{{\large D}ÉPARTEMENT D'{\large A}NALYSE\vspace{0.8cm}}

\logousthb


\textbf{\vspace{0.6cm}}

\textbf{{\LARGE Notes de Cours du module}}\textbf{\vspace{0.4cm}}

\textbf{{\Huge Analyse Complexe (Math 4)}}\textbf{\vspace{0.4cm}}

\textbf{Par\vspace{0.3cm}}

\textbf{{\large LAADJ Toufik}}\footnote{Page Web :
\ \ \href{http://perso.usthb.dz/~tlaadj/}{http://perso.usthb.dz/\symbol{126}%
tlaadj/}}\vspace{0.4cm}

\textbf{Pour}\vspace{0.4cm}

\textbf{{\LARGE Deuxième année Licence}}

\textbf{{\LARGE Domaine : Sciences et Technologies}}
\end{center}

\vspace{0.4cm}

\begin{center}
Février 2014
\end{center}

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\newpage

\chapter*{Description du Cours}%

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\section*{\textbf{Objectif du Cours}\label{Objectif}}

L'objectif du module Analyse Complexe (Math 4) est de maîtriser les concepts
et les résultats fondamentaux de la théorie des fonctions complexes de
variables complexes de manière à pouvoir les utiliser dans d'autre
cours.\newline Ces notes de cours donnent les principales définitions et les
résultats fondamentaux, illustrés par des exemples.

\section*{\textbf{Contenu du Cours}}

\begin{itemize}
\item Les nombres complexes

\item Fonctions complexes

\item Dérivation complexe, équations de Cauchy-Riemann

\item Intégration complexe, Théorème de Cauchy

\item Séries infinies, séries de Taylor, séries de Laurent

\item Théorème des résidus
\end{itemize}

\section*{\textbf{Résultats d'apprentissage}}

À la fin du cours, l'étudiant doit avoir une bonne compréhension de la théorie
des fonctions complexes à variable complexe et devrait être en mesure
d'appliquer ces connaissances pour résoudre les exercices dans une variété de
contextes.\newline En particulier, l'étudiant doit être capable de :

\begin{itemize}
\item Comprendre ce qu'une dérivation complexe est.

\item Citer, tirer et appliquer les équations de Cauchy-Riemann.

\item Effectuer l'intégration curviligne de fonctions complexes.

\item Comprendre et appliquer le théorème de Cauchy et la formule intégrale de Cauchy

\item Étudier les propriétés de convergence d'une série de puissance complexe.

\item Appliquer les théorèmes de Taylor et de Laurent pour obtenir des
développements en série de puissance.

\item Identifier et classifier les singularités de fonctions complexes et
trouver des résidus.

\item Tirer et appliquer le théorème des résidus pour calculer des intégrales
réelles en utilisant des résidus.
\end{itemize}

\mainmatter


\setcounter{chapter}{-1}

\chapter{Les nombres complexes}%

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\section{L'ensemble des nombres complexes}

\subsection{Opérations sur les nombres complexes}

\subsection{Valeur absolue (ou module)}

\section{Représentation graphique des nombres complexes}

\section{Forme polaire des nombres complexes}

\subsection{Formule de De Moivre}

\subsection{Racines d'un nombre complexe}

\chapter{Fonctions élémentaires}%

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\section{Fonctions complexes}

\subsection{Fonctions uniformes et multiformes}

\subsection{Fonctions inverses}

\subsection{Transformations}

\subsection{Limites}

\subsection{Continuité}

\section{Fonctions élémentaires}

\subsection{Les fonctions polynômiales}

\subsection{Les fractions rationnelles}

\subsection{Les fonctions exponentielles}

\subsection{Fonctions trigonométriques}

\subsection{Les fonctions hyperboliques}

\subsection{Fonctions logarithmiques}

\subsection{La fonction $z^{\alpha}$}

\subsection{Fonctions trigonométriques inverses}

\subsection{Fonctions hyperboliques inverses}

\chapter{Dérivation dans le domaine complexe}%

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\section{Domaines dans le plan complexe}

\section{Fonctions holomorphes}

\subsection{Dérivées}

\subsection{Conditions de Cauchy-Riemann}

\subsubsection{Dérivées d'ordre supérieur}

\subsection{Fonctions harmoniques}

\subsection{Règles de dérivation}

\subsection{Règle de l'Hôpital}

\subsection{Points singuliers}

\subsubsection{Singularités apparentes}

\subsubsection{Singularités essentielles}

\chapter{Intégration dans le domaine complexe}

\vspace*{-1.2cm}%
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\section{Chemins et courbes dans le plan complexe}

\section{Intégration le long d'une courbe}

\subsection{Propriétés}

\subsubsection{Longueur d'une courbe}

\subsubsection{Théorème d'estimation}

\section{Théorèmes de Cauchy}

\subsection{Domaines simplement ou multiplement connexes}

\subsection{Théorème de Cauchy}

\subsection{Primitives et intégration}

\subsubsection{Théorème fondamental de l'intégration}

\subsection{Formule intégrale de Cauchy}

\subsubsection{Inégalité de Cauchy}

\subsubsection{Théorème de Liouville}

\subsubsection{Théorème fondamental de l'algèbre}

\chapter{Séries infinies, séries de Taylor, séries de Laurent}%

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\section{Séries de fonctions}

\subsection{Convergence absolue}

\section{Séries entières}

\subsection{Rayon de convergence}

\section{Séries de Taylor}

\subsection{Quelques séries particulières}

\section{Séries de Laurent}

\subsection{Classification des singularités}

\subsubsection{Pôles}

\subsubsection{Singularités apparentes}

\subsubsection{Singularités essentielles}

\subsubsection{Singularités à l'infini}

\chapter{Théorème des résidus}%

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\section{Résidus}

\subsection{Calcul des résidus}

\subsubsection{Pôle simple}

\subsubsection{Pôle d'ordre $k\geq2$}

\subsubsection{Point singulier essentiel}

\section{Le théorème des résidus}

\section{Application du théorème des résidus}

\subsection{Théorèmes particuliers utilisés pour le calcul d'intégrales}

\subsection{Application aux transformées de Fourier}

\subsubsection{Cas d'une fonction rationnelle}

\subsection{Calcul d'intégrales définies diverses}

\subsubsection{Intégrale du type ${%
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{x}dx,\alpha>0$}

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\begin{thebibliography}{9}                                                                                                %


\bibitem {Amroun}\textbf{A%
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N.} Cours de Fonctions de variables complexes. Université Djillali Liabès de
Sidi Bel Abbes Algérie, 2009.

\bibitem {BeckM}\textbf{B%
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M., M%
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G., P%
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D. and S%
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L.} A First Course in Complex Analysis. San Francisco State University, San
Francisco, CA, USA and Binghamton University, Binghamton, NY, USA, 2012.

\bibitem {Giroux}\textbf{G%
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A.} Analyse complexe. Département de mathématiques et statistique, Université
de Montréal, 2004.%
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\bibitem {Murray}\textbf{M%
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R. S. }Variables complexes : cours et problèmes, volume 12 de Série Schaum,
New York 1973.
\end{thebibliography}


\end{document}