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AMCS 608 Fall 2010
Home page for AMCS 608:
Analytic Techniques for AMCS, I
Fall 2010
Instructor: Charles L. Epstein
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The Course
The main goal of analysis is the solution of equations. We begin by
reviewing the basic results from linear and non-linear finite
dimensional analysis and the solution of "algebraic"
equations, and Stokes theorem. We next turn to the case of 1-variable complex analysis, where certain
aspects of the problem are simplified and develop tools that will be
very useful in the infinite dimensional context. In the
second
semester we turn to the problem of analysis in infinite
dimensional spaces, and develop the framework needed to find solutions
of ordinary and partial differential equations.
The material on finite dimensional analysis is drawn from a variety of
sources: Peter Lax, Linear
Algebra;
Robert Strichartz, The Way of Analysis;
W. Rudin, Principles of
Mathematical Analysis; Jerrold Marsden, Elementary Classical Analysis;
Michael Spivak, Calculus
on Manifolds.
The material on complex analysis is taken from E. Stein and R.
Shakarchi, Complex
Analysis;
Lars Ahlfors, Complex
Analysis;
Zeev Nehari, Conformal
Mapping.
Good references for functional analysis are: Peter D. Lax, Functional Analysis;
Ward Cheney, Analysis
for Applied Mathematics;
Michael Reed and Barry Simon, Methods
of Modern Mathematical Physics, Vol. 1: Functional Analysis.
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 10:30-12:00 in room 4C8 of the
David
Rittenhouse Labs.
- Tentatively my office hour will be Mondays 3:30-5PM.
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 for this course is Eugene So. His office is 4C15 in DRL, his phone
number is 8-5224, and his e-mail is soeugene -at-
sas.upenn.edu. His office hours are Tuesday: 3:00-4:00 and Thursday 12:00-1:00.
Syllabus
- A quick review of analysis in Rn
- Completeness:Cauchy sequences and convergence
- Convergence of series
- Connectedness, Compactness, and contractions
- Continuity, differentiability and approximation of
functions
- Methods from calculus
- Inverse and Implicit Function theorems, Newton's Method
- Stokes Theorem
- Tools from complex analysis
- Analytic functions and Cauchy's theorems
- The residue theorem and the argument principle
- The Cauchy Riemann equations and harmonic functions
- Methods of asymptotic analysis: Laplace's method and
stationary
phase
- Conformal mapping and imcompressible steady flow
- The Dirichlet and Neumann problems for harmonic functions
Announcements
- This class will have its first meeting on Thursday, September 9, 2010.
- Less time will be spent reviewing basic analysis than in previous years. Please be sure you know that contents of this review of elementary analysis, through section B6.
Problem sets
- Problem set 1, due September 21, 2010.
- Problem set 2, due September 28, 2010.
- Problem set 3, due October 5, 2010.
- Problem set 4, due October 14, 2010.
- Problem set 5, due October 26, 2010.
- Problem set 6, due November 2, 2010.
- Problem set 7, due November 9, 2010.
- Problem set 8, due November 16, 2010.
- Problem set 9, due November 23, 2010.
- Problem set 10, due December 13, 2010.
Return to cle's home
page.