Home page for AMCS 608:
Analytic Techniques for AMCS, I
Fall 2009
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.
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;
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 a
take-home midterm and final exams.
- The class meets on TuTh from 12:00-1:30 in room 4N30 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.
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
- 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
The first class will be held on Thursday, September 10, 2009.
Problem sets
- Problem set 1, due September 22, 2009.
- Problem set 2, due September 29, 2009.
- Problem set 3, due October 6, 2009.
- Problem set 4, due October 13, 2009.
- Problem set 5, due October 20, 2009.
- Problem set 6, due October 27, 2009.
- Problem set 7, due November 3, 2009.
- Problem set 8, due November 10, 2009.
- Problem set 9, due November 17, 2009.
- Problem set 10, due November 24, 2009.
Return to cle's home
page.