Analysis IV Function sequences, integer series, Fourier

  • ECTS

    8 credits

  • Component

    Faculty of Science

Description

This course covers the notions of function sequences and series, and the various convergences. Integer and Fourier series will also be developed.

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Objectives

Sequence of functions Simple convergence and uniform convergence of a sequence of functions

- Definitions and link between simple and uniform convergence of a sequence of functions

- Uniform Cauchy criterion

- Dini theorems

- Stone Weierstrass theorem using Bernstein polynomials

- Stability of continuity (resp. derivability, integration) by uniform convergence

Function series

- Simple and uniform convergence

- Normal convergence

  • Continuity, derivability, integrability of a series of functions

 Integral series.

Definitions, radius of convergence, Hadamard's formula, d'Alembert's rule.

Properties of the sum of the whole series: continuity, derivability, integrability.

 Functions that can be developed into an integer series.

Applications to solving differential equations: solving matrices by integer and exponential series.

Fourier series.

  • Why Fourier series (issues and definitions)?
  • Convergence (root mean square, simple, normal) of Fourier series
  • Applications to series calculations and differential equations
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Necessary prerequisites

HAX201X - Analysis II Sequences, series, limited developments

HAX302X: Analysis III integration and elementary differential equations

 

Recommended prerequisites: L1 maths

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Further information

Hourly volumes :

            CM: 39h

            TD: 39h

            TP :

            Terrain :

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