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

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By Roger Godement

ISBN-10: 3540661425

ISBN-13: 9783540661429

Ce vol. III disclose l. a. 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 ? los angeles Cauchy se g?n?ralisent ? un nombre quelconque de variables r?elles (formes diff?rentielles, formules de sort 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 concepts ?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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Extra resources for Analyse mathematique III: Fonctions analytiques, differentielles et varietes, surfaces de Riemann

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1. The parallelogram of the double periodicity in the complex z-plane. 3) in the open disk lai< 8. 3) in the parameter a to the entire complex a-plane completes the proof. The following q-addition formulas for the basie trigonometrie functions have been established in [140]. 2. Cq (X,YiW) = Cq (XiW) Cq (YiW) - Bq (XiW) Bq (Yi W) , Bq (x, y;w) = Bq (x;w) Cq (y;w) + Cq (x;w) Bq (y;w) . PROOF. 1). 10) 3. 19) 2 eos2 WX, 2 sin2 WX, respectively. 12) if the Cq (Xj w) and Bq (Xj w) are real valued funetionsj in this ease Eq.

2) qa ,q with 1] = (q1/4 + q-1/4) /2. 3). 3. 18). 15) with x (z) = (cf + q-Z) /2 = coso, qZ = ei8 ; see Ex. 11. 11). 5, Eq. 1). 1). 18) (a2/ß2;q)kqk2/4 (ßei'P)k (q; q)k . xL00 qn(n-2k)/4 ßne-m'P (q; q)n n=O X ( _q(l- n+k)/2 ei8+ i'P a/ ß, _q(l-n+k)/2ei'P-i8 a/ ß; q) n . 1) onee again. The seeond sum ean be redueed to the sum of two 4CP3 series similar to those in 20 2. 17) is an analog of exp (ax + ßy). im Eq (x, Yj (1 - q) a/2, (1 - q) ß/2) q-tl- = f: (ß/~)n n=O n. e- inrp (1+e i8 +irp a/ß)n (1+e irp - i8a/ß)n n (-n)k ( x {; ~ =L (ß/2)n n!

The polynomial solutions Yn (x) and Ym (x) are orthogonal on a contour 0 in the complex z-plane: [Yn(x(z)) Ym(x(z)) p(z)Vxdz) dz=O, mi=n. [Note: In many cases a special choice of the contour 0 in this relation gives rise to the continuous orthogonality property for a realvalued system of polynomials. The most general case corresponds to the Askey-Wilson polynomials [15], [18]; see other examples in [9], [10], [16], [46], [86], [111], [113]. ] (23) Show that for the nonuniform lattices of the classical type v(n) 1 (z) = t (_l)n-k k=O 'Y (n)!

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

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