Analyse mathématique III: Fonctions analytiques, by Roger Godement

By Roger Godement

Ce vol. III disclose l. a. théorie classique de Cauchy dans un esprit orienté bien davantage vers ses innombrables utilisations que vers une théorie plus ou moins complète des fonctions analytiques. On montre ensuite remark les intégrales curvilignes à los angeles Cauchy se généralisent à un nombre quelconque de variables réelles (formes différentielles, formules de style Stokes). Les bases de los angeles théorie des variétés sont ensuite exposées, principalement pour fournir au lecteur le langage "canonique" et quelques théorèmes importants (changement de variables dans les intégrales, équations différentielles). Un dernier chapitre montre touch upon peut utiliser ces théories pour construire los angeles floor de Riemann compacte d'une fonction algébrique, sujet rarement traité dans los angeles littérature non spécialisée bien que n'éxigeant que des innovations élémentaires. Un quantity IV exposera, outre, l'intégrale de Lebesgue, un bloc de mathématiques spécialisées vers lequel convergera tout le contenu des volumes précédents: séries et produits infinis de Jacobi, Riemann, Dedekind, fonctions elliptiques, théorie classique des fonctions modulaires et l. a. model moderne utilisant los angeles constitution de groupe de Lie de SL (2, R).

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5–8) provides elements of distributions, Sobolev and Besov spaces, pseudo-differential operators and maximum principles for second-order elliptic Waldenfels operators which play a crucial role throughout the book. Each chapter has its own focus. Chapter 5 is a summary of the basic definitions and results about the theory of distributions or generalized functions which will be used in subsequent chapters. Distribution theory has become a convenient tool in the study of partial differential equations.

9 and 10. In Sect. 5 we describe the classical surface and volume potentials arising in boundary value problems for elliptic differential operators in terms of pseudo-differential operators. One of the important questions in the theory of elliptic boundary value problems is that of the smoothness of a solution near the boundary. In Sect. 6, following Boutet de Monvel [Bo], we introduce a condition concerning symbols in the normal direction at the boundary (the transmission property) in order to ensure the boundary regularity property.

10. Chapter 4 is devoted to the general theory of semigroups. In Sects. 3 we study Banach space valued functions, operator valued functions and exponential functions, generalizing the numerical case. 4 is devoted to the theory of contraction semigroups. 11). We consider when a linear operator is the infinitesimal generator of some contraction semigroup. 10). In Sect. 28), generalizing the theory of contraction semigroups developed in Sect. 4. 30). 10 in Chap. 13. Part II (Chaps. 5–8) provides elements of distributions, Sobolev and Besov spaces, pseudo-differential operators and maximum principles for second-order elliptic Waldenfels operators which play a crucial role throughout the book.

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