SYLLABUS AND HOMEWORK

Mathematics 6350

Functions of a Complex Variable 1

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Homework is due in class and must be stapled, with your name on it, to receive credit.


Date Topics covered Notes
Monday August 24
Introduction:
Complex numbers, polar coordinates, Riemann sphere, complex numbers as matrices, conformal linear maps.

Wednesday August 26
Topology 1:
Topological spaces, continuous maps, metric spaces.
For review, you may want to read Chapter 6 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996.  (Electronic version available from the library.)
Friday August 28
Topology 2:
Constructing topological spaces.

Monday August 31
Topology 3:
Sequences in metric spaces, compact spaces.

Wednesday September 2
Topology 4:
Connectedness, path connectedness.

Friday September 4
Differentiation 1:
Differentiable maps, basic properties, and complex differentiable maps.
Homework 1

For review, you may want to read Chapter 8 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996.  (Electronic version available from the library.)
Monday September 7
NO CLASS
LABOR DAY
Wednesday September 9
Differentiation 2:
Derivations, tangent spaces, and differentials.

Friday September 11
Differentiation 3:
Inverse and implicit function theorems.
Homework 2
Monday September 14
Differentiation 4:
Complex differentials revisited.

Wednesday September 16
Analytic functions 1:
Review of uniform convergence.

Friday September 18
Analytic functions 2:
Definitions and basic properties of real and complex analytic functions.
Homework 3

For review, you may want to read Chapters 2-3 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996.  (Electronic version available from the library.)
Monday September 21
Analytic functions 3:
Real and complex analytic functions, continued.

Wednesday September 23
Integration 1:
Overview of integration, manifolds with boundary, and differential forms.


Friday September 25
Integration 2:
Review of path integrals, and path integrals for vector valued functions.
Homework 4

We will be covering topics out of Chapters 11-14 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996.  (Electronic version available from the library.)
Monday September 28
Integration 3:
Complex path integrals.

Wednesday September 30
Integration 4:
Cauchy integral formula and first applications.

Friday October 2
Manifolds 1:
Manifolds, manifolds with boundary.
Homework 5

We will be covering topics out of Chapters 11-14 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996.  (Electronic version available from the library.)
Monday October 5
Manifolds 2:
Tangent bundles to manifolds.

Wednesday October 7
Manifolds 3:
Tangent bundles to manifolds, continued.

Friday October 9
Manifolds 4:
Linear algebra.
Homework 6

We will be covering topics out of Chapters 11-14 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996.  (Electronic version available from the library.)
Monday October 12
Manifolds 5:
Differential forms and de Rham's theorem.

Wednesday October 14
Manifolds 6:
Integration on manifolds.

Friday October 16
Manifolds 7:
Stokes' theorem.
Homework 7
Monday October 19
Applications of Stokes' theorem 1:
Integral formulas in the complex plane.

Wednesday October 21
Applications of Stokes' theorem 2:
Integral formulas continued.

Friday October 23
Consequences of Cauchy's integral formula 1:
Zeros of holomorphic functions,  and singularities of holomorphic functions.
Homework 8
Monday October 26
Consequences of Cauchy's integral formula 2:
Local structure of holomorphic maps, the open mapping theorem, and the maximum modulus principle.

Wednesday October 28
Consequences of Cauchy's integral formula 3:
Uniform convergence on compact sets.

Friday October 30
Laurent series 1:
Meromorphic functions, and introduction to Laurent series.
Homework 9
Monday November  2
Residue theory 1:
Residue theorem, and applications.

Wednesday November 4
Residue theory 2:
Residue theorem, and applications.

Friday November 6 Meromorphic functions:
Definitions, and basic properties.
Homework 10
Monday November 9
Series and products 1:
Motivation, and the Mittag-Lefler problem.

Wednesday November 11
Series and products 2:
Mittag-Lefler continued.

Friday November 13
Series and products 3:
Canonical products
Homework 11
Monday November 16
Series and products 4:
Canonical products continued, the Weierstrass Factorization Theorem.

Wednesday November 18
Series and products 5:
Weierstrass Factorization Theorem continued.

Friday November 20
Series and products 6:
The  interpolation problem.
Homework 12
Monday November 23 - Friday November 27
NO CLASS THANKSGIVING BREAK
Monday November 30
Introduction to the gamma function

Wednesday December 2
The gamma function continued

Friday December 4

Homework 13
Monday December 6


Wednesday December 9


Friday December 11

Homework 14
Sunday December 13
FINAL EXAM 4:30-7:00 PM MUEN E432
FINAL EXAM



I strongly encourage everyone to use LaTex for typing their homework.  If you have a mac, one possible easy way to get started is with texshop.  If you are using linux, there are a number of other possible ways to go, using emacs, ghostview, etc.  If you are using windows, you're on your own, but I'm sure there's something online.  Here is a sample homework file to use: (the .tex file, the .bib file, and the .pdf file).  It is probably easiest to just look at the .tex file, and start to experiment.  There is also the (Not so) short introduction to latex, which will answer most of your questions (although typing your question into your favorite search engine will probably work well, too).  You may also want to try https://cloud.sagemath.com/ or https://www.sharelatex.com/ for a cloud version.