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Introduction to Comlpex Analysis

Overview

Lectures: MWF 3-4pm, Room 71 Evans

Instructor: Jan Reimann

Office: 705 Evans Hall

Office hours: Tu 2-3pm, We 4-6pm

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 Cauchy-Riemann 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, 5-8pm 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 Prat-Waldron. His office hours are: Monday, 9am-10am and 11am to 3pm and Tuesday 9am-2pm 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 Cauchy-Riemann differential equations § I.4, I.5
01/29 Differentiability and linear functions,
the Jacobian of a complex differentiable function, the Cauchy-Riemann 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 anti-derivative
§ II.1
02/07 sufficient conditions for the existence of an anti-derivative
triangular paths
§ II.2
02/09 The Cauchy integral theorem for triangular paths,
star-shaped domains
§ II.2
02/12 The Cauchy integral theorem for star-shaped 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 self-mappings 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; non-essential singularities; poles and orders. III.4, pp. 136-139
04/04 Computations with orders; mapping properties around poles; essential singularities; mapping properties near essential singularities; the Casorati-Weierstrass theorem; the big theorem of Picard. III.4, pp. 139-142
04/06 Proof of the Casorati-Weierstrass theorem; Laurent decomposition - introduction III.4, pp. 141-142, pp. 145-146
04/09 Laurent decomposition - proof of uniqueness; Cauchy integral theorem for annuli. III.5, pp. 146-148
04/11 Laurent decomposition - proof of existence; Laurent series; examples III.5, pp.148-150, p.152
04/13 Laurent series - examples; classification of singularities through Laurent series. III.5, pp. 152-153, 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. 166-169
04/20 second part of the proof of the residue theorem; generalized Cauchy integral formula; Computation of residues for non-essential singularities; examples III.6, pp. 169-171
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.181-184
04/27 Using the Residue Theorem to compute integrals (part II) III.7, pp.186-187
04/30 Consequences of the Residue Theorem: the argument principle, Hurwitz' Theorem III.7, pp.174-176
05/02 Rouche's Theorem; applications III.7, pp. 177-179
05/04 Review of the material for the final