Analysis IV Sequences of functions, integer series, Fourier

  • ECTS

    8 credits

  • Component

    Faculty of Science

Description

This course will cover the concepts of sequences and series of functions and the various convergences. The 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's theorems

- Stone Weierstrass theorem by Bernstein polynomials

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

Series of functions

- Simple and uniform convergence

- Normal convergence

  • Continuity, differentiability, integrability of a series of functions

 Whole series.

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

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

 Integer developable functions.

Applications to the solution of differential equations: resolution by integer and exponential series of matrices.

Fourier series.

  • Why Fourier series (problematic and definitions)?
  • Convergence (root mean square, simple, normal) of Fourier series
  • Applications to the calculation of certain series and differential equations
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Necessary pre-requisites

HAX201X - Analysis II Sequences, series, limited developments

HAX302X: Analysis III integration and elementary differential equations

 

Recommended prerequisites: L1 math

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Additional information

Hourly volumes:

            CM : 39h

            TD : 39h

            TP:

            Terrain:

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