Math 5308—Mathematical Analysis

1. Course Description

Infinite series, sequences and series of real-valued functions, concept and basic properties of Lebesgue integral, Fourier series and Fourier integrals, analysis in Rⁿ, including continuity, differentiation, and Taylor’s formula.

Prerequisite: Math 5307

2. Material Covered

A. Infinite Series and Infinite Products

1) Limit superior and limit inferior of a real-valued sequence

2) Cauchy condition for series

3) Alternating series

4) Absolute and conditional convergence

5) Various tests for series

6) Dirichlet’s test and Abel’s test

7) Infinite products

B. Sequences and Series of Functions

1) Uniform convergence with continuity, differentiation, and integration

2) The Cauchy condition for uniform convergence

3) Weierstrass test

4) Dirichlet’s test for uniform convergence

5) Mean convergence

6) Uniform convergence of power series

C. The Lebesgue Integral

1) Integral of a step function

2) Basic properties of the Lebesgue integral

3) The Levi monotone convergence theorem

4) The Lebesgue dominated convergence theorem

5) Continuity and differentiation under the integral sign

D. Fourier Series and Fourier Integrals

1) Orthogonal systems of functions

2) Properties of the Fourier coefficients

3) The Riesz-Fisher theorem

4) The Riemann-Lebesgue lemma

5) Parseval’s formula

6) The Weierstrass approximation theorem

7) Convolution theorem for Fourier transforms

E. Multivariable Differential Calculus

1) Directional derivatives and continuity

2) Total derivative

3) The Jacobian matrix

4) The chain rule

5) The Mean-value theorem for differentiable functions

6) Taylor’s formula

3. Reference Books

1) T.M. Apostol, Mathematical Analysis, Second Edition

2) W. Rudin, Principles of Mathematical Analysis