Get Analyse Mathématique II: Calculus différentiel et intégral, PDF

By Roger Godement

ISBN-10: 3540634142

ISBN-13: 9783540634140

Les deux premiers volumes sont consacrés aux fonctions dans R ou C, y compris l. a. théorie élémentaire des séries et intégrales de Fourier et une partie de celle des fonctions holomorphes. L'exposé non strictement linéaire, mix symptoms historiques et raisonnements rigoureux. Il montre l. a. diversité des voies d'accès aux principaux résultats afin de familiariser le lecteur avec les méthodes de raisonnement et idées fondamentales plutôt qu'avec les suggestions de calcul, element de vue utile aussi aux personnes travaillant seules.
Les volumes three et four traitent principalement des fonctions analytiques (théorie de Cauchy, théorie analytique des nombres et fonctions modulaires), ainsi que du calcul différentiel sur les variétés, avec un exposé de l'intégrale de Lebesgue, en suivant d'assez près le célèbre cours donné longtemps par l'auteur à l'Université Paris 7.
On reconnaîtra dans ce nouvel ouvrage le kind inimitable de l'auteur, et pas seulement par son refus de l'écriture condensée en utilization dans ce nombreux manuels.

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Extra info for Analyse Mathématique II: Calculus différentiel et intégral, séries de Fourier, fonctions holomorphes

Example text

T > Suppose Then there such that to + (2k-l)r ~ tk ~ to + 2kr, k = 1,2, ••• , and By the assumption on t ~ to. x(t)! < L for all Therefore, on the intervals ! x(t)! > 5/2, and hence By taking a large ! xt ! > 5/2. Therefore, L, if necessary, we can assume that these intervals do not overlap, and hence Let K(50 ,L) be the smallest integer ~ v(eo)/«5/L)w(5/2». then 52 If k> 1 + K(50 ,L), which is a contradiction. we have 2rK( BO,L) Therefore, at some e. \x t \ < 1 t l , such that to ~ tl ~ to + This proves the theorem.

3. For nonlinear equations, the equivalence classes Va may involve the consideration of trajectories which have a point in common after any preassigned times. r = 1, The following example was given by A. Hausrath. consider the scalar equation x(t) = ~[I xtl - x(t)]. For ~ > 0, For a given ~ in C = C([-l,O],R), this equation through If ~(O) In fact, since implies o x(t) which is continuous in (O,~) 2: 0, F 0, ~ 2: 0, => x(O) > 0 then 2: ~(O) x(t) > 0 and for for t 2: of (~,~,t). I xtl = x(t) for > 1.

1) at tl > to. 1) is stable at ~ for which ~ 3rrj2). x(t - ~ = exp(-3rr~j2), Since there is a ~O tl > is unstable for any >0 the function x(t) = satisfying this equation, 3rr. It seems to be very difficult to determine in general when stability at t O implies stability at t >t 1 - but the following result is very easy and indi- 0' cates the simplicity of autonomous and periodic systems. 1. If f(t,~) is either independent of bility (or asymptotic stability) of (or asymptotic stability) of Proof.

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Analyse Mathématique II: Calculus différentiel et intégral, séries de Fourier, fonctions holomorphes by Roger Godement


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