AMCS/MATH 608 Fall 2014

AMCS/MATH 608: Analysis I

Fall 2014
Instructor: Charles L. Epstein

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The Course

This year the Analysis I-II sequence is being redesigned to cover Complex Analysis, Real Analysis (Measure Theory), and Functional Analysis. The first semester of analysis provides an introduction, at the graduate level, to functions of one complex variable and the first half of real analysis.

For the first ten weeks, we cover the basic Cauchy theory, we will focus on more analytic features of the subject, highlighting its connection to PDE and Fourier analysis. As references for complex analysis we use E. Stein and R. Shakarchi, Complex Analysis; Lars Ahlfors, Complex Analysis; Zeev Nehari, Conformal Mapping. More advanced material on the d-bar equation can be found in T. Napier and M. Ramachandran, An Introduction to Riemann Surfaces.

For weeks 11-14 we begin our discussion of Real Analysis, which will be followed by 4 more weeks in the second semester. In the first semester we will cover Measure Theory and Lebesgue Integration in Euclidean space. References for Real Analysis are Real Analysis by E. Stein and R. Shakarchi, Real Analysis by Gerald Folland, and Real Analysis by H. Royden.

A problem set will be assigned every week on Tuesday, due the following week on Tuesday. I very much prefer that students do the problem sets alone. We may have midterm and final exams.

The class meets on TuTh from 1:30 to 3:00 in room 4C8 of the David Rittenhouse Labs.
Tentatively my office hour will be Mondays 3:30-5:00PM. Contact me by e-mail for an appointment if you can not come during this time.
My office in the Math Department is 4E7 DRL, tel. 8-8476.
email: cle@math.upenn.edu. Send e-mail if you have a question or need to contact me.
The TA is Soum Nayak. His office is 3N2B in DRL, and e-mail is nsoum [@] math [.] upenn [.] edu. His office hours are on Wednesdays from 2:00 to 4:00.

Syllabus

A quick review of analysis in Rⁿ

Forms and Stokes theorem in the plane (Informal introduction to forms in 1 and 2 dimensions. Good online notes look at parts on "Exterior Calculus.")
Inverse and Implicit Function theorems
Newton's Method

Complex analysis

Analytic functions and Cauchy's theorems
The residue theorem and the argument principle
The Cauchy Riemann equations and harmonic functions
The maximum principle and the three circle theorem
The d-bar equation
The Fourier transform, the Hilbert transform
Conformal mapping and incompressible steady flow
The Dirichlet and Neumann problems for harmonic functions

Real Analysis

Measurable sets
Measurable functions
The Lebesgue integral
L^1-functions
Fubini's theorem

Announcements

This class will have its first meeting on Tuesday, September 2, 2014.
Please be sure you know the contents of this review of elementary analysis and this Informal introduction to forms in 1 and 2 dimensions. .
Class is canceled for Thursday, Oct. 9 due to Fall break.
The textbook we will follow for the Real Analysis segment of the course is Real Analysis by E. Stein and R. Shakarchi. It is available from Amazon or directly from Princeton Press.

Problem sets

Problem set 0, These problems in advanced caclulus do not have to be handed in, but should be done this week, so you are ready for our discussion of contour integration.
Problem set 1, due September 16, 2014.
Problem set 2, due September 23, 2014.
Problem set 3, due September 30, 2014.
Problem set 4, due October 7, 2014.
Problem set 5, due October 21, 2014.
Problem set 6, due October 28, 2014.
Problem set 7, due November 11, 2014.
Problem set 8, due November 18, 2014.
Problem set 9, due November 25, 2014.
Problem set 10, due December 9, 2014.
Problem set 11, due December 16, 2014.

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