Home page for Math 509, Advanced Analysis
Spring 2018
Instructor: Charles L. Epstein
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The Course
In this course we continue our study of the foundations of analysis.
The goal
of this course is to further develop your instincts as a mathematical
analyst.
This
entails understanding the basic concepts and techniques of analysis, as
well as
developing intuition and proficiency in the construction of rigorous
proofs. We will complete several topics from the Fall. After a
quick discussion of differential calculus
in several variables, we turn to Riemann integration in the plane
and prove Green's theorem. We will then introduce
basic concepts of functions
of a complex variable, and finish up with a study of Fourier
series in 1variables and applications. For the
first part of the
course we will continue to use Strichartz (the
textbook from the fall). Marsden's book Elementary Classical Analysis is
also a good reference for the material on differential and integral calculus
of several variable. We will use other sources for the complex
analysis unit. Fourier Series and Complex Analysis by
E.M. Stein and R. Shakarchi are excellent references, but more
advanced and detailed than our treatment of these subjects will
be. Strichartz also has a good treatment of Fourier series.
A problem set will be assigned every week, on Tuesdays. We may have a midterm and final exam.
 The class meets from 3:00 to 4:30 on Tuesdays and Thursday in DRL A1.
 Each student is required to register for and attend an evening lab session.

My office hour is 3:005:00 on Monday. Send email if you have a question or would like to come see me at some other time.

email: cle@math.upenn.edu.
Send email or call if you need to see me at some other time.

My office is DRL 4E7, my telephone number is: 88476.
 The TA is Siyu Heng. His email is siyuheng@sas.upenn.edu and his office is 1N1 in DRL.
Syllabus for the course
 Approximation versus Interpolation
 ArzelaAscoli Theorem
 Topology in R^{n}
 Differential Calculus in Several Variables
 Riemann Integration in the Plane
 Fubini's Theorem
 Change of Variable Formula
 Integration by Parts and Green's Theorem in the Plane
 Introduction to Complex Analysis
 Definitions of Analytic Functions
 Contour and Complex Integration
 Cauchy's Theorem and Formula and their Consequences
 Power Series
 Local Properties of Analytic Functions
 Residue Calculus and Definite Integrals
 Elementary Conformal Mapping
 Fourier Series
 Basic properties
 Convergence of Fourier Series
 Applications of Fourier Series
Lecture Notes
Announcements
 We will have a midterm exam on February 27, 2018
 There will be a review session on Thursday, February 22 at 4:30 in DRL A1.
 I will hold extra office hours on Friday, February 23 from 10:0012:00 in my office.
Problem Sets
 Problem set 1 due January 23, 2018.
 Problem set 2 due January 30, 2018.
 Problem set 3 due February 6, 2018.
 Problem set 4 due February 13, 2018.
 Problem set 5 due February 20, 2018.
 Problem set 6 due March 13, 2018.
 Problem set 7 due March 20, 2018.
 Problem set 8 due March 27, 2018.
 Problem set 9 due April 3, 2018.
 Problem set 10 due April 10, 2018.
 Problem set 11 due April 17, 2018.
 Problem set 12 due April 26, 2018.