MATH 2250, Fall 2010. Complex Analysis I.

MWF 11:00--11:50 am,   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 1:00 pn-2:00 pm and by appointment.

Syllabus (pdf format).

Assignments:

• 09/01: will be discussed at the next class. 
• 09/03: will be discussed next class.
• 09/08: will be discussed next class.
• 09/10: p. 120 # 1, 2, p 123 # 1
• 09/15: p. 120 # 3, p. 123 # 5,  3.
• 09/17: will be collected Mon. 9/20.  
• 09/20: p. 130, # 2, 3, 5, 6, p. 161 # 1.
• 09/22: will be discussed next class.
• 09/27: p. 78 #1,  p. 80 #1, 3, 4.
• 09/29: p. 83, # 2, 4, 7
• 10/01: p. 80 # 2,  p. 96 # 2,
• 10/04: will be discussed next class.
• 10/06: will be collected Fri. Oct. 8.
• 10/08: will be discussed next class.  
• 10/13: will be discussed next class.  
• 10/15: will be discussed next class.
• 10/18: will be discussed next class.
• 10/22: will be discussed next class.
• 10/27: will be discussed next class.
• 10/29: will be discussed next class.
• 11/05: will be discussed next class.
• 11/08: will be discussed next class.
• 11/10: will be discussed next class.
• 11/12: Find a mistake in the lemma in the attached page
• 11/19: Will be collected Mon. 11/29.
• 12/01: Will be discussed next class.

Lecture Notes taken by students (pdf format). 

• 09/01: Introduction: complex numbers, matrix representation, modulud and argument, Euler's formula. (by   Ryan Zelen)
• 09/03: Definition of analytic functions. Review: power series, complex integration.  
• 09/08: Cauchy--Riemann equations,  and  derivatives. Cauchy--Goursat theorem.  
• 09/10: Cauchy theorem for the disc. Integration over arbitrary path. Lebesgue number lemma. 
• 09/13: Homotopy invariance of the integral. Cauchy formula.
• 09/15: Taylor and Laurent series representation.  
• 09/17: Lioville theorem. Uniqueness theorems.  
• 09/20: Classification of zeroes and singularities.  
• 09/22: Residue theorem. Definition of logarithm. Argument principle. 
• 09/24: Rouche Theorem. Corollaries: local structure of analytic maps; open map, maximum principle.
• 09/27: Conformal mappings. Conformal authomorphisms of the plane and  Riemann sphere. 
• 09/29: Schwartz  Lemma. Conformal authomorphisms of the unit disc. Symmetry principle. 
• 10/01: Cross ratios. Preservation of generalised circles under LFT. 
• 10/04: Symmetry with respect to a circle. General form of a conformal authomorphism of a disc.   
• 10/06: General form of Cauchy Theorem. Homology. 
• 10/08: Homology (continued). Index and homology classes.  
• 10/13: Plemelj--Sokhotsy formula.
• 10/15: Convergence of analytic functions.
• 10/18: Normal families. 
• 10/20: Hurwitz Theorem. Riemann mapping theorem.
• 10/22: Riemann mapping theorem (completion of the proof). Harmonic functions.
• 10/25: Harmonic fubctions. Dirichlet problem for the disc and Poisson formula. 
• 10/27: Meand value properties. Maximum and minimum principles for harmonic functions (for functions with a weak mean value property)
• 10/29: Harmonic functions from weak mean value property.  
• 11/01: Conjugate harmonic functions. Scwartz integral  formula.   
•    
• 11/05: Harnak inequality. Subharmonic functions: defnition and strong maximum property.
• 11/08: Green's formulas and subharmonic functions.    
• 11/10: Perron process.     
• 11/12: Berron process. Barrier.
• 11/15: Barrier at a point implies that Perron process gives correct boundary value at that point. Construction of the barrier for points in non-trivial components of the complement.
• 11/17: Harmonic measure.     
• 11/19: Riemann mapping theorem via Green's function. 
  Approximation by rational functions: Runge's theorem, Mergelyan theorem (w/o proof), swiss cheese.
• 11/29: Infinite products. Blaschke products and condition.
• 12/01: Zeroes of bounded analytic functions in the disc. Jensen formula. Weiirstrass theorem about zeroes of entire function.
• 12/03: Analytic continuation and Riemann surface of a function. 
• 12/06: Riemann surface of a function (continued). Monodromy theorem.

Solutions for the  midterm.

• In class part
• Take-home part

Handouts.

• Harmonic functions and Dirichlet problem.