Download e-book for kindle: Analyse mathematique III: Fonctions analytiques, by Roger Godement

By Roger Godement

ISBN-10: 3540661425

ISBN-13: 9783540661429

Ce vol. III divulge los angeles th?orie classique de Cauchy dans un esprit orient? bien davantage vers ses innombrables utilisations que vers une th?orie plus ou moins compl?te des fonctions analytiques. On montre ensuite remark les int?grales curvilignes ? l. a. Cauchy se g?n?ralisent ? un nombre quelconque de variables r?elles (formes diff?rentielles, formules de kind Stokes). Les bases de los angeles th?orie des vari?t?s sont ensuite expos?es, principalement pour fournir au lecteur le langage "canonique" et quelques th?or?mes importants (changement de variables dans les int?grales, ?quations diff?rentielles). Un dernier chapitre montre touch upon peut utiliser ces th?ories pour construire l. a. floor de Riemann compacte d'une fonction alg?brique, sujet rarement trait? dans l. a. litt?rature non sp?cialis?e bien que n'?xigeant que des thoughts ?l?mentaires. Un quantity IV exposera, outre,l'int?grale de Lebesgue, un bloc de math?matiques sp?cialis?es vers lequel convergera tout le contenu des volumes pr?c?dents: s?ries et produits infinis de Jacobi, Riemann, Dedekind, fonctions elliptiques, th?orie classique des fonctions modulaires et l. a. model moderne utilisant l. a. constitution de groupe de Lie de SL(2,R).

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Additional info for Analyse mathematique III: Fonctions analytiques, differentielles et varietes, surfaces de Riemann

Example text

Let f be a continuous real-valued function on R. Show that the inverse image with respect to f of an open set is open, of a closed set is closed, and of a Borel set is Borel. 59. A sequence f f,,) of real-valued functions defined on a set E is said to converge uniformly on E to a function f if given c > 0, there is an N such that for all x E E and all n >_ N, we have I f (x) - f (x) I < E. Let { f } be a sequence of continuous functions defined on a set E. Prove that if { f on on E. f 60. Prove Proposition 21.

In fact, it is not even finitely additive (see Theorem 18): there are disjoint sets A and B for which m*(AU B)

With this extended concept of convergence we may assert that any monotone sequence (an } of real numbers, bounded or unbounded, converges to an extended real number and therefore limn an is properly defined. The extended concept of supremum and infimum of a set and of convergence for any monotone sequence of real numbers allows us to make the following definition. Definition Let {an} be a sequence of real numbers. The limit superior of {an}, denoted by lim sup{an }, is defined by lim sup{an} = lim [sup {ak I n +00 k > nj The limit inferior of {an}, denoted by lini ml (a,,), is defined by liminf{an} = lim [inf {ak I k > n}].

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Analyse mathematique III: Fonctions analytiques, differentielles et varietes, surfaces de Riemann by Roger Godement

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