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Titre: Cours d’Analyse IV Suites et Séries de fonctions

Auteurs: L. Pujo-Menjouet

Ecole: Université Claude Bernard, Lyon I

Résumé: Le but de ce cours est de généraliser la notion de somme finie de termes en étudiant comment cette dernière se comporte lorsque l’on considère une succession infinie de termes. La clé sera de considérer ces sommes infinies, aussi appelées séries, comme la limite de suites. Autrement dit, quand on se souvient du cours sur les suites, il sera plus facile d’assimiler le cours sur les séries C’est pour cela que les deux premiers chapitres concernant des rappels ne doit pas être négligé. Un des points clés de ce cours sera l’étude des séries de Fourier dont les applications sont assez nombreuses dans d’autres domaines des mathématiques (notamment les équations différentielles et les équations aux dérivées partielles).

Pour arriver au chapitre concernant les séries de Fourier, il faudra cependant faire un petit chemin qui nous y amènera de façon moins abrupte. Comme nous l’avons écrit plus haut, nous rappellerons la structure de R, puis la notion de suites dans R ou C. Nous considèrerons ensuite les séries dans leur généralité, puis les suites et séries de fonction, pour ensuite passer aux séries entières, aux fonctions développables en séries entière et enfin les séries de Fourier. Nous pourrons alors résoudre quelques équations différentielles à l’aide de cette théorie.

L’objectif de la deuxième partie du cours sera de résoudre des équations différentielles à l’aide des transformées de Laplace. Cet outil mathématique ne pourra s’appliquer rigoureusement sans un petit travail préliminaire sur les intégrales dépendant d’un paramètre. Une fois ces concepts assimilés, vous serez en possession d’outils solides pour résoudre plusieurs types d’équations différentielles et équations aux dérivées partielles mais également des problèmes un peu plus théoriques.

Extrait du sommaire:
1 Structure de R, suites dans R ou C : 5
1.1 La crise des nombres chez les grecs 5
1.2 Suites et voisinages : 6
1.3 Limites de suites 7
1.4 Borne sup ou inf, max ou min 9
1.5 Suites adjacentes 10
2 Rappels suites complexes, limsup de suites réelles 11
2.1 Suites complexes 11
2.2 Limite sup et inf 14
3 Séries dans R ou C : 17
3.1 Premiers critères de convergence 18
3.2 Séries réelles à termes positifs 19
3.3 Comparaison d’une série et d’une intégrale impropre 22
3.4 Séries à termes quelconques 23
3.5 Sommation par paquets, produit 24
4 Suites de fonctions 27
4.1 Propriétés des limites uniformes 30
5 Série de fonctions 33
5.1 DEFINITION 33
6 Séries entières 37
6.1 Opérations sur les séries entières 39
6.2 Propriétés fonctionnelles d’une série entière 40
7 Fonctions développables en séries entières 43
7.1 L’exemple de l’exponentielle complexe 43
7.2 Développement en série entière 44
7.3 Développement des fonctions usuelles 46
8 Séries de Fourier 49
8.1 Interprétation géométrique des séries de Fourier 54
9 INTEGRALES DEPENDANT D’UN PARAMETRE 57
9.1 Intervalle d’intégration J compact 58
9.1.1 Bornes d’intégration constantes 58
9.1.2 Bornes d’intégration variables 60
9.2 Intervalle d’intégration J non borné 61
9.2.1 Rappel 61
9.2.2 Convergence 62
10 Fonctions Eulériennes 65
11 Transformées de Laplace 67
11.1 Rappel 67
11.2 Définition 68
11.3 Quelques fonctions élémentaires 68
11.4 Existence de L 69
11.5 Transformée inverse et transformée de dérivées 70
11.5.1 Transformée inverse 70
11.5.2 Transformer une dérivée 71
11.6 Résolution d’équations différentielles 72
11.7 Théorème de translation 73
11.7.1 Translation sur l’axe des s 73
11.7.2 Translation sur l’axe des t 73
11.8 Propriétés additionnelles 73
11.8.1 Multiplier une fonction par tn 73
11.8.2 Convolution 73
11.8.3 Transforme d’une intégrale 73
11.8.4 Equation intégrale de Volterra 73
11.8.5 Transforme de fonction périodique 74
11.8.6 Dirac 74

Mathématique appliquée cours 5

Télécharger le fichier PDF: Cours d’Analyse IV Suites et Séries de fonctions

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