Math 522 HOMEPAGE

Instructor: Alexandru Ionescu, ionescu@math.wisc.edu
Lectures: MWF 11:00-11:50 in Room B-321
Textbook: Principles of Mathematical Analysis, third edition, by W. Rudin
Office hours: MW 3-4 or by appointment in office 619.

Information
Problem Set 1. Due in class on Monday, February 4. Solutions.
Problem Set 2. Due in class on Friday, February 15. Solutions.
Problem Set 3. Due in class on Wednesday, February 27. Solutions.
Midterm Exam 1. In class on Friday, February 29. Solutions.
Problem Set 4. Due in class on Friday, March 14. Solutions.
Problem Set 5. Due in class on Friday, April 4. Solutions.
Problem Set 6. Due in class on Wednesday, April 16. Solutions.
Midterm Exam 2. In class on Friday, April 18. Solutions.
Problem Set 7. Due in class on Monday, May 5. Solutions.

Course schedule:
Week 1 (Jan. 23 - Jan. 25): The Riemann-Stieltjes integral, basic definitions and properties.
Week 2 (Jan. 28 - Feb. 1): Integrability of functions: continuous and monotonic functions, algebraic properties.
Week 3 (Feb. 4 - Feb. 8): Changes of variables, the fundamental theorems of calculus.
Week 4 (Feb. 11- Feb. 15): Integration of vector-valued functions, rectifiable curves, review of  integration.
Week 5 (Feb. 18 - Feb. 22): Analytic functions: power series, uniform convergence and differentiability, Abel's theorem.
Week 6 (Feb. 25 - Feb. 29): Taylor's theorem, uniqueness of power series expansions.
Week 7 (Mar. 3 - Mar. 7): The exponential and logarithmic functions, the trigonometric functions.
Week 8 (Mar. 10 - Mar. 14): Fourier series (main definitions), the Bessel identities, Parseval's theorem.
Week 9 (Mar. 24 - Mar. 28): The Diriclet kernel, pointwise convergence of Fourier series, approximations using trigonometric polynomials.
Week 10 (Mar. 31 -  Apr. 4): Regularization of continuous functions, algebraic completeness of the complex field, the Gamma function.
Week 11 (Apr. 7 - Apr. 11): Properties of the Gamma function, the Beta function, linear transformations.
Week 12 (Apr. 14 - Apr. 18): Normed spaces, invertible linear transformations.
Week 13 (Apr. 21 - Apr. 25): Differentiation: the total derivative, partial derivatives, the chain rule, the mean value theorem.
Week 14 (Apr. 28 - May 2): The contraction principle,  the inverse function theorem,  the implicit function theorem.