Série (mathématiques)

Résumé

En mathématiques, la notion de série permet de généraliser la notion de somme finie. Étant donné une suite de terme général u, étudier la série de terme général u c'est étudier la suite obtenue en prenant la somme des premiers termes de la suite (u), autrement dit la suite de terme général S défini par : S_n=u_0+u_1+\cdots+u_n=\sum_{k=0}^nu_k. L'étude d'une série peut passer par la recherche d'une écriture simplifiée des sommes finies en jeu et par la recherche éventuelle d'une limite finie quand n tend vers l'infini. Quand cette limite existe, la série est dite convergente, et la limite de la suite (S) est alors appelée somme de la série, et notée \sum_{k=0}^{+\infty}u_k. Le calcul d'une somme finie ne pouvant pas toujours être simplifié, un certain nombre de méthodes permettent de déterminer la nature (convergence ou non) d'une série sans réaliser explicitement les calculs. Toutefois, certaines règles de calcul sur les sommes finies ne sont pas nécessai

Source officielle

https://fr.wikipedia.org/wiki/S%C3%A9rie_(math%C3%A9matiques)

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (4)

Publications associées

Chargement

Personnes associées

Aucun résultat

Unités associées

Chargement

Concepts associés (160)

Concepts associés

Chargement

Cours associés (67)

Cours associés

Chargement

Séances de cours associées (137)

Séances de cours associées

Chargement

Let d(n) denote Dirichlet's divisor function for positive integer numbers. This work is primarily concerned with the study of We are interested, in the error term where Ρ3 is a polynomial of degree 3 ; more precisely xΡ3(log x) is the residue of in s = 1. A. Ivić showed that E(x) = Ο(x1/2+ε) for all ε > 0 (cf. [9], p.394). We will prove that for all x > 0, we have With this intention, we apply Perron's formula to the generating function ζ4(s)/ ζ (2s) and Landau's finite difference method. It was conjectured that E(x) = Ο(x1/4+ε) for ε > 0. The existence of non-trivial zeros of the Riemann ζ function implies that we cannot do better, that is The study of the Riesz means for ρ sufficiently large shows that their error term, is an infinite series , on the zeros of the Riemann Zeta function added with a development , into a series of Hardy-Voronoï's type, both being convergent. To find the "meaning" of , one could consider the difference But the series (probably) doesn't converge.We will thus substract only a finite part of , weighted by a smooth function ω, the number of terms of the finite part depending on x. If we consider this new error term , we obtain, using a classical method due to Hardy, that for x ≥ 1.

EPFL2007

Chargement

Degree and valuation of the Schur elements of cyclotomic Hecke algebras

Following the generalization of the notion of families of characters, defined by Lusztig for Weyl groups, to the case of complex reflection groups, thanks to the definition given by Rouquier, we show that the degree and the valuation of the Schur elements (functions A and a) remain constant on the "families" of the cyclotomic Hecke algebras of the exceptional complex reflection groups. The same result has already been obtained for the groups of the infinite series and for some special cases of exceptional groups.

Elsevier2008

Chargement

Surface polaritons of small coated cylinders illuminated by normal incident TM and TE plane waves

Olivier Martin, Juan Ramon Mosig

The surface polariton properties of TM or TE plane wave scattered by a coated cylinder are investigated in this paper. The coated cylinder (whose outer radius is much smaller than the wavelength) is assumed to be electrically small and low dissipative. Analytical formulas of the plasmonic resonances are derived and found to agree well with those obtained from exact expressions in the classical scattering theory. The behaviors of the scattering coefficients at resonances are also discussed and compared for different cases. While a single cylinder has the resonance at the relative permittivity of epsilon(r) = -1 (or relative permeability of mu r = -1) for the TE (or TM) polarization, the resonances of the coated cylinders change with different n values (where n denotes the series term or mode of the field), and also the inner and outer radii. It is shown that the scattered field in the near zone can be enhanced significantly compared to the incident wave. For the TE incident case, we take a silver coated nano-cylinder as an example to illuminate the near-field optical effect. Also, we have studied the peak values of the nth order scattered field for different n values and electrical parameter k(0)b (where k(0) is the wavenumber of the free space and b denotes the outer radius of the cylinder) around the cylinder. The derived new formulas for total cross sections are given and they may provide us with some potential photonic applications such as surface cleaning and etching. (c) 2008 Optical Society of America.

Optical Society of America2008

Chargement