Analyse mathématique I: Convergence, fonctions élémentaires by Roger Godement

By Roger Godement

Les deux premiers volumes de cet ouvrage 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 los angeles 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 concepts de calcul, aspect de vue utile aussi aux personnes travaillant seules.
Les volumes three et four traiteront principalement des fonctions analytiques (théorie de Cauchy, théorie analytique des nombres et fonctions modulaires), ainsi que du calcul différentiel sur les variétés, avec un court docket 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 sort inimitable de l'auteur, et pas seulement par son refus de l'écriture condensée en utilization dans de nombreux manuels.

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Let B be a Hamel basis for the set of real numbers R. Let b ∈ B be any element of B. Define g(x) = 0 1 if x ∈ B \ {b} if x = b. 7, there exists an additive function f : R → R with f (x) = g(x) for each x ∈ B. Note that this f cannot be linear since for x ∈ B and x = b, we have 0= f (x) f (b) = . x b Therefore f is a non-linear additive function. We end this section with the following remark. 1. No concrete example of a Hamel basis for R is known; we only know that it exists. The graph of a discontinuous additive function on R is not easy to draw as the set { f (x) |x ∈ R } is dense in R.

Yn ) for all (x1 , x2 , . . , xn ), (y1 , y2 , . . , yn ) in Rn , then n f (x1 , x2 , . . , n) are multiplicative functions. 4 Other Two Cauchy Equations in Several Variables In this section, we consider the remaining two Cauchy equations in several variables. 5, one can establish the following results. 6. 9) where E1 , E2 : R → R are exponential functions. In general every exponential function f : Rn → R can be written as n f (x1 , x2 , . . , n) are expenential functions. Let R0 = {x ∈ R | x = 0}.

Since f is analytic, it is differentiable. 12) with respect to z1 , we get f (z1 + z2 ) = f (z1 ) for all z1 and z2 in C. Hence, letting z1 = 0 and z2 = z, we get f (z) = c, where c = f (0) is a complex constant. From the above, we see that f (z) = cz + b, where b is a complex constant. 12), we obtain b = 0 and hence the asserted solution follows. This completes the proof of the theorem. We close this section with the following remark. 4. 10 fails for complex-valued functions on the complex plane.

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