Analysis IV Function sequences, integer series, Fourier

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

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

HAX201X - Analysis II Sequences, series, limited developments

HAX302X: Analysis III integration and elementary differential equations

Recommended prerequisites: L1 maths

Hourly volumes :

            CM: 39h

            TD: 39h

            TP :

            Terrain :