By Roger Godement

ISBN-10: 3540632123

ISBN-13: 9783540632122

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 options 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 courtroom 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 type inimitable de l'auteur, et pas seulement par son refus de l'écriture condensée en utilization dans de nombreux manuels.

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**Additional info for Analyse mathématique I: Convergence, fonctions élémentaires**

**Sample text**

B) Elliptic transformation. Note (i and Qs are images with respect to all circles shown here as enclosing them. (c) Parabolic transformations. REVIEW OF COMPLEX ANALYSIS 42 LEMMA 2-9 Two points are images with respect to a Mobius circle 91Z if and only if all Mobius circles joining these points are orthogonal to M. 5 oo, we write instead of (2-36a) W= (2-42a) w1 . Z Then substituting the variables in (2-32), w= + (2-42b) z=E+ We have a new transformation (with double-fixed point at - ), necessarily of the form (2-34b) W=Z+B (2-43) B50 This is called a "parabolic" transformation.

C) If it is a circle, the images are pairs consisting of an interior and an exterior point on a ray through the center, situated so that the radial distances have as their product the square of the radius of the circle. ) We note the formula for images a, 0 with regard to the circle Izi = R, namely (2-29) a5 = R2 or a$ = R2 so that images are, for example, (2-30) where arg li = arg a = X. in optics. a = lal exp ix z 6 = Ial exp A Thus images agree with the cognate concept EXERCISES 1 Usually euclidean geometry is defined as the study of affine transformations together with symmetry.

This gives the classical representation of the rotation group of the sphere in terms of quaternions ± (p + qI + rJ + sK). Re w11B -E (a) Deduce Lemma 2-10 from (2-46a) by using the fact that if f(a) = 0, then f(1/a) = -o (the image of 0). This determines numerator and denominator to within a constant factor. This factor is determined by If(z)l = 1 when jzj = 1. Note that (2-45) embraces "in the limit" f (z) = exp iO/z (with a = oo) but a more useful form would be f(z) _ (2-49) flz - a -az+$ -aa (b) Show (2-49) covers all cases of (2-45) except that 0 and a can be replaced by -0 and -a in (2-49) producing the same transformation.

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