Topics in Analysis

Math 4516: Topics in Analysis

Topics covered: Taylor series and approximation, Measure theory, Lebesque integral, Metric spaces; additional topics as appropriate for the particular class.

This class is a continuation of Math 3515, Analysis 1. Hence, I will frequently refer to it as "Analysis 2".

Prerequisite: Math 3515 (Analysis 1)

Textbook: We will be using the online resource Interactive Real Analysis, as well as the cheap and useful Dover^(*) text book Introductory Real Analysis by Kolmogorov and Fomin. Additional resources will be made available as needed.

(*) Dover text books for Math are classic text books in mathematics, reprinted in paperback format, available for an incredibly low price - get as many Dover books as you can get, they are (almost) all good to have and for the price you can't go wrong..

Software: We will make use of the DyKnow client software. Please download and install that software.

General Information

Syllabus

Exams

Final Exam

Resources

Notes

05: Metric Spaces

Part 1: Intro and Examples

Part 2: Continuity

04: Lebesgue Integration

Part 1: Outer measure

Part 1(A): Open Covers and Compact Sets

Part 2: (Lebesgue) Measure

Part 3: Lebesgue Integral for bdd functions (+ worksheet)

Part 4: General Lebesgue Integral and thms

Summary of Leb measure & integration

03: Sounds and Fourier Transform

Part 1: The Basics of Sound

Part 2: The Fourier Transform
(chord-D-major, chord1, note-guitar, note-piano)

Part 3: Fast Fourier Transform
(fft.nb Mathematica notebook)

Part 4: Saving Digital Data: Saving Digital Data

02: Fourier Series

Part 1: Introduction and Basic Formulas

Part 2: Convergence Theorems

Part 3: Changing the Scale

01: Power and Taylor Series

Part 1: Power series

Part 2: Basic Algebra

Part 3: Multiplying and Dividing

Part 4: Taylor series

Part 5: Taylor's Theorem, Lagrange Reminder, & "most beautiful formula in mathematics"

Assignments

Archive: none
Bert G. Wachsmuth