Analysis IV Function sequences, entire series, Fourier

This course will cover the concepts of sequences and series of functions and various types of convergence. Entire series and Fourier series will also be discussed.

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

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

- Uniform Cauchy criterion

- Dini's theorems

- Stone-Weierstrass theorem using Bernstein polynomials

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

Series of functions

- Simple and uniform convergence

- Normal convergence

• Continuity, differentiability, and integrability of a series of functions

 Complete series.

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

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

 Functions that can be developed as entire series.

Applications to solving differential equations: solving using entire and exponential series of matrices.

 Fourier series.

• Why Fourier series (issues and definitions)?
• Convergence (quadratic mean, simple, normal) of Fourier series
• Applications to calculations of certain series and differential equations

HAX201X – Analysis II Sequences, series, limited developments

HAX302X: Analysis III: Integration and Elementary Differential Equations

Recommended prerequisites: L1 math

Hourly volumes:

            CM: 39 hours

            Tutorial: 39 hours

            TP:

            Land: