# Real Analysis: Course Contents (MAT 409)

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable.

## Course Objective

At the end of this course the students will be able to review elementary mathematics from an advanced standpoint, insight into the role of rigor in mathematics. Build understanding of the foundations and concepts of analysis and strengthen powers of logic (reasoning). Define Riemann integral and Riemann sums, prove various theorems about Riemann sums and Riemann integrals and emphasize the proofs’ development. Define uniform convergence and prove their theorems. Study and apply Fourier series theorems (e.g convergence criteria etc.)

## Course Contents

The Riemann Integral; Upper and lower sums, Definition of a Riemann integral, Integrability criterion, Classes of integrable functions, Properties of the Riemann integral,

Infinite Series; Review of sequences, Geometric series, Tests for convergence, Conditional and absolute convergence. Regrouping and rearrangement of series. Power series, radius of convergence,

Uniform Convergence; Uniform convergence of a sequence and a series, the M-test, properties of uniformly convergent series. Weierstrass approximation theorem,

Improper Integrals; Classification, tests for convergence, Absolute and conditional convergence, Convergence of cos(x) sinx dx, the Gamma function. Uniform convergence of integrals, M-text, properties of uniformly convergent integrals,

Fourier series;  Orthogonal functions, Legendre, Hermite and Laguerre polynomials, Convergence in the mean. Fourier-Legendre and  Fourier-Bessel series, Bessel inequality, Parseval equality. Convergence of the trigonometric Fourier series.