Analysis IV: Function Sequences, Power Series, Fourier Series

This course will cover the concepts of sequences and series of functions, as well as various types of convergence. Integral series and Fourier series will also be discussed.

Sequences of Functions: Simple Convergence and Uniform Convergence of a Sequence of Functions

- Definitions and the 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 (or differentiability, integrability) via uniform convergence

Function series

- Simple and uniform convergence

- Normal convergence

• Continuity, Differentiability, and Integrability of a Sequence of Functions

 Complete series.

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

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

 Functions that can be expanded as a series.

Applications to the Solution of Differential Equations: Solution Using Integer and Exponential Series of Matrices.

 Fourier series.

• Why Fourier series (problems and definitions)?
• Convergence (quadratic, simple, normal) of Fourier series
• Applications to the Calculation of Certain Series and Differential Equations

HAX201X – Analysis II: Sequences, Series, and Limited Expansions

HAX302X: Analysis III: Integration and Elementary Differential Equations

Recommended prerequisites: First-year math

Hourly volumes:

            CM: 39 hours

            TD: 39 hours

            Practical Work:

            Lot: