Math 185
Section 3  Spring 2007Overview
Lectures: MWF 34pm, Room 71 Evans
Instructor: Jan Reimann
Office: 705 Evans Hall
Office hours: Tu 23pm, We 46pm
Email:
Personal Website: http://math.berkeley.edu/~reimann/
Course Description
The course is a standard introduction to complex analysis. The main results are more than 150 years old, and the presentation has been polished over decades. The basic theory now appears as an ensemble of beautiful theorems.
The main topics are: The field of complex numbers, complex derivatives, the CauchyRiemann differential equations; line integrals, the Cauchy integral theorem, the Cauchy integral formulas; power series of analytic functions, singularities, Laurent series, the residue theorem, applications of the residue theorem.
If time permits, we will deal with additional topics such as the Gamma function or the Riemann mapping theorem.
Literature
The textbook for the course will be E. Freitag and R.Busam, Complex analysis, Springer, 2005. We will follow it more or less closely. The first three chapters cover the core topics, maybe we will add material from Chapter IV. The book extends far beyond the scope of this class. Whoever likes the material, can go on to read about elliptic functions, modular forms, Dirichlet series, and analytic numbner theory.
Exams
There will be two midterms: the first on Friday, February 16 (in class), the second on Wednesday, March 21 (also in class).
Information on the first midterm (02/16): This will be a closed book exam(sorry, no cheat sheets!). Bring your blue books. There is a sample midterm available. (Solutions to the sample midterm)
Information on the second midterm (03/21): This will be a closed book exam(no cheat sheets!). Bring your blue books. There is a sample midterm available. (Solutions to the sample midterm)
The final exam is held on Tuesday May 15, 58pm in 9 Lewis. Again, closed book exam, bring blue books, no cheat sheets.
Grading
The final grade will be determined as follows: 20% homework, 20% each midterm, 40% final exam.
Homework
A note on academic honesty: Collaboration among
students to solve homework assignments is welcome. This is a good way
to learn mathematics. So is the consultation of other sources such as
other textbooks. However, every student should hand in an own set
of solutions, and if you use other people's work or ideas you
should indicate the source in your solutions.
(In any
case, complete and correct homework receives full credit.)
Date  File  Due  Solutions 

01/24  Homework 1 (pdf)  01/31  Solutions (pdf) 
01/31  Homework 2 (pdf)  02/07  Solutions (pdf) 
02/07  Homework 3 (pdf)  02/14  
02/16  Homework 4 (pdf)  02/21  Solutions (pdf) 
02/21  Homework 5 (pdf)  02/28  Solutions (pdf) 
02/28  Homework 6 (pdf)  03/07  Solutions (pdf) 
03/07  Homework 7 (pdf)  03/14  Solutions (pdf) 
03/14  Homework 8 (pdf)  03/21  
04/04  Homework 9 (pdf)  04/11  Solutions (pdf) 
04/11  Homework 10 (pdf)  04/18  Solutions (pdf) 
04/20  Homework 11 (pdf)  04/27  Solutions (pdf) 
04/30  Bonus Homework (pdf)  05/07 
Graduate Student Instructor
There is a GSI for this course, Arturo PratWaldron. His office hours are: Monday, 9am10am and 11am to 3pm and Tuesday 9am2pm in 891 Evans.
Brief Summary of Lectures
Date  Material Covered  Suggested Textbook Reading 

01/17  The field of complex numbers  § I.1 
01/19  Visualization of algebraic opperations, roots of unity, the exponential function 
§ I.2 
01/24  Euler's formula, principal branch of the logarithm, continuity 
§ I.2, I.3 
01/26  Complex differentiability, the CauchyRiemann differential equations  § I.4, I.5 
01/29  Differentiability and linear functions, the Jacobian of a complex differentiable function, the CauchyRiemann differential equations a sufficient criterion for complex differentiability. 
§ I.5 
01/31  Analytic functions, harmonic functions.  § I.5 
02/02  Construction of conjugate harmonic functions Path integrals  § I.5, II.1 
02/05  Properties of path integrals necessary conditions for the existence of an antiderivative  § II.1 
02/07  sufficient conditions for the existence of an antiderivative
triangular paths  § II.2 
02/09  The Cauchy integral theorem for triangular paths,
starshaped domains  § II.2 
02/12  The Cauchy integral theorem for starshaped domains,
the principle branch of the logarithm as a primitive of 1/z, elementary domains.  § II.2 
02/14  The Cauchy integral formula.  § II.3 
02/21  The generalized Cauchy integral formula.
Liouville's theorem 
§ II.3 
02/23  Example: Computing integrals with the Cauchy integral formula;
Morera's theorem; Liouville's theorem; the fundamental theorem of algreba. 
§ II.3 
02/26  Uniform convergence; local uniform convergence; uniform convergence and path integrals; uniform convergence and analyticity;
series of functions, normal convergence, stability properties of series of normally convergent series of analytic functions; the Riemann zeta function. 
§ III.1, III.2 
02/28  The Riemann zeta function; power series; radius of convergence; power series representation of analytic functions  § III.1, III.2 
03/02  Applications of power series representation.
Some topology: discrete sets in subsets of the complex plane. 
§ III.2, III.3 
03/05  The identity theorem for analytic functions  § III.3 
03/07  Algebraic interpretation of the identity theorem;
the oppen mapping theorem, roots of analytic functions, the implicit function theorem. 
§ III.3 
03/09  Conformal mappings; geometric interpretation and properties. The mapping z > z^2.  § I.5 
03/12  Properties of conformal mappings;
Moebius transformations, examples; The maximum modulus principle. 

03/14  The minimum modulus principle; Schwarz' Lemma;
globally conformal mappings of the unit disk to itself. 
III.3 
03/16  Conformal selfmappings of the unit disk;
harmonic functions on elementary domains; the Dirichlet problem 

03/19  The Dirichlet problem;
Review of material for 2nd midterm. 

04/02  Isolated singularities of analytic functions; the Riemann removability condition; nonessential singularities; poles and orders.  III.4, pp. 136139 
04/04  Computations with orders; mapping properties around poles; essential singularities; mapping properties near essential singularities; the CasoratiWeierstrass theorem; the big theorem of Picard.  III.4, pp. 139142 
04/06  Proof of the CasoratiWeierstrass theorem; Laurent decomposition  introduction  III.4, pp. 141142, pp. 145146 
04/09  Laurent decomposition  proof of uniqueness; Cauchy integral theorem for annuli.  III.5, pp. 146148 
04/11  Laurent decomposition  proof of existence; Laurent series; examples  III.5, pp.148150, p.152 
04/13  Laurent series  examples; classification of singularities through Laurent series.  III.5, pp. 152153, p. 150 
04/16  Winding numbers  III.6, pp. 165  168 
04/18  Computation of winding numbers; Residues, the residue theorem, first part of the proof  III.6, pp. 166169 
04/20  second part of the proof of the residue theorem; generalized Cauchy integral formula; Computation of residues for nonessential singularities; examples  III.6, pp. 169171 
04/23  Computation of residues (cont'd); residues of essential singularities.  III.7, p. 171 
04/25  Using the Residue Theorem to compute integrals (part I).  III.7, pp.181184 
04/27  Using the Residue Theorem to compute integrals (part II)  III.7, pp.186187 
04/30  Consequences of the Residue Theorem: the argument principle, Hurwitz' Theorem  III.7, pp.174176 
05/02  Rouche's Theorem; applications  III.7, pp. 177179 
05/04  Review of the material for the final 