By Roger Godement

ISBN-10: 3540634142

ISBN-13: 9783540634140

Les deux premiers volumes sont consacrés aux fonctions dans R ou C, y compris los angeles 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, aspect 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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Additional resources for Analyse Mathématique II: Calculus différentiel et intégral, séries de Fourier, fonctions holomorphes

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XΓ(x)](n) = P x + 1; xΓ(x), xΓ(1) (x) + Γ(x), . . , xΓn (x) + nΓ(n−1) (x) so we can define a second polynomial Q, defined by the transformation Q(x; y0 , y1 , . . , yn ) = P x + 1; xy0 , xy1 + y0 , . . , xyn + ny(n−1) and Q x; Γ(x), Γ (x), . . , Γ(n) (x) = 0 is also an algebraic differential equation for Γ(x). Q and P both have the same degree and P must divide Q otherwise there would be a remainder and that would mean P was not of 29 minimal degree. ,hn ) (x + 1) · (y0 )h0 · (y1 )h1 · .

Yn )an . Also assume that P is of lowest possible degree. ,an ) have no common factor of the form (x − γ) and so P is not divisible by any factor of (x − γ). It also means that P is not the product of any two polynomials of lower degree. Consider the relations P (x + 1; Γ(x + 1), Γ(1) (x + 1), . . , Γ(n) (x + 1)) = = P x + 1; xΓ(x), [xΓ(x)](1) , [xΓ(x)](2) , . . , [xΓ(x)](n) = P x + 1; xΓ(x), xΓ(1) (x) + Γ(x), . . , xΓn (x) + nΓ(n−1) (x) so we can define a second polynomial Q, defined by the transformation Q(x; y0 , y1 , .

An ) (x) are polynomials in x acting as coefficients of polynomial P . ,an ) (x) may be constants or zero. 1. If P (x; y0 , y1 , y2 ) = x2 y2 +xy1 +(x2 −α2 )y0 then A(0,0,1) (x) = x2 , A(0,1,0) (x) = x and A(1,0,0) (x) = (x2 − α2 ) where α is a constant. All the other coefficients in the summation are zero. Then P (z; f, f , f ) = x2 f + xf + (x2 − α2 )f = 0 is an algebraic differential equation which, in this example, has solutions f = Jα (x) and f = Yα (x), the Bessel functions of either the first or second kind.

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


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