MATH 2250, Fall 2009. Complex Analysis I.

MWF 1:00--1:50 pm,   Kassar 105 

Instructor: Prof. Sergei Treil,
office: Kassar 216,   ph.: x1122
e-mail: treil<at>math.brown.edu
Web: http://www.mth.brown.edu/~treil/

Office hours: MWF 2:00 pn-3:00 pm and by appointment.

Syllabus. (pdf format)

Assignments:

• 9/9: will be discussed at the next class. 
• 9/14: p. 108, # 1, 4, 7,  p. 120 # 1, 2..
• 9/16: p. 123 #1, 2,  3.
• 9/18: p. 130,  #2, 3,  5
• 9/21: show that exp(log z) = z in any connected simply connected domain containing 1 and not containing 0 (log z is defined as an integral). 
• 9/28: p. 148 # 5, p. 154 # 1, 2, p. 130 # 5, p. 161 # 1 a, b, 3 d, e (to be collected Fri. 10/2).
• 10/02: p. 80, # 1, 2, 3
• 10/09: p. 96 # 1, 2, 7, p. 173 # 1, 2, 3 , handout p. 270 # 2, 3, 4, (to be collected Mon. 10/19)
• 11/11: to be collected Monday 11/16. 
• 11/20: To be collected Monday 11/30
  
  

Handouts:

•  Hyperbolic geometry.  
•  Dirichlet problem

Lecture Notes taken by students (pdf format). 

• 9/9: Introduction: complex numbers, power series, analytic functions,  and operators., Cauchy--Riemann equations.
• 9/11: Cauchy--Gousat theorem (differentiability implies analiticity): Roadmap of the proof. Integral along the path. Closed and exact forms. 
• 9/14: Cauchy theorem for rectangle. Cauchy theorem for the disc (integral over a closed contour in the disc is 0).  Homotopy invariance of integrals.  
• 9/16: Caucy formula for the disc. Taylor series. Corolary: Liouville's theorem. Laurent series.      
• 9/18: Laurent series (proof). Two uniqueness theorems for analytic functions. Removable sigularities.  
• 9/21: Poles and essential singularities. Logarithm in complex domain. Argument principle.  
• 9/23: Rouche theorem. Corollaries: main theorem of algebra, maximum principle, open map theorem.  
• 9/25: Local behavior of analytic functions. General form of Cauch theorem: homology.    
• 9/28: Homology and index. General form of Cauchy formula.  
• 9/30: Cauchy formula for unbounded domains. Analytic capacity.    
• 10/02: Conformal mappings. Linear fractional transformations. Cross ratio.  
• 10/05: LFTs map circles to circles. Symmetric points. All LFT mapping the unit disc onto itself.    
• 10/07: Schwarz lemma. Conformal maps from the unit disct onto itself. Symmetry principle.  
• 10/09: Conformally invariant metric. Hyperbolic geometry.    
• 10/14: Riemann surface of a function.  
• 10/16: Geometric function theory. Koebe 1/4 theorem, Bieberbach conjecture. Proof of Area Principle.  
• 10/19: Proof of baby Bieberbach and Koebe 1/4 theorem. Statement of Riemann mapping theorem. Normal families, an introduction.  
• 10/21: Normal families. Metric for the uniform convergence on compact subsets topology. Proof of the Riemann mapping theorem: step 1---the set in the optimization problem is non-empty. 
• 10/23: Proof of the Riemann mapping theorem. Hurwitz theorem (isolated zeroes cannot just appear).
• 10/26: Harmonic functions. Conformal invariance. Poisson formula via Fourier series.  
• 10/28: Poisson formula. Mean value propery for harmonic functions.    
• 10/30: Harmonic functions via mean value property. Harnack's inequality.  
• 11/02: Subharmonic functions. Classical definition and definition via domination by harmonic functions. Conformal invariance of subharmonic functions.
• 11/04. Green's formulas. Subharmonicity and Laplacians.  
• 11/06: Perron process.  
• 11/09: Barrier. Solvability of Dirichlet problem
• 11/11: Harmonic measure and Green's function.    
• 11/13: Properties of Harmonic measure. Riemann mappng theorem via Green's function. Uniformization.    
• 11/16: Approximation by polynomials and rational functions. Runge theorem. Mergelyan's theorem (w/o proof). Swiss cheese.  
• 11/18: Zeroes of analytic functions. Infinite products. Blaschke cndition.   
• 11/20: Zeroes of bounded analitic function in the disc satisfy Blaschke condition. Jensen formulas.  Canonical products and Weierstrass theorem about zeroes of analytic functions.