MATH 331: Complex Analysis
James Cook's Complex Analysis Homepage
Course Materials for Fall 2015:
For a day by day overview of our course see:
Fall 2015 Complex Analysis Course Planner
Complex Analysis Lectures of Fall 2015 this is the playlist for our course on my You Tube channel, enjoy.
In this semester we use Gamelin's Complex Analysis as the required text. For the most part, his approach aligns closely with out aims, but, I have some examples to add in many sections. Also, I have added a fair number of historically significant quotes and pointers to rigorous treatments of various topics as found in Reinhold Remmert's Complex Function Theory. You should read Gamelin before my notes for most sections. Enjoy:
Guide to Gamelin's Complex Analysis (updated November 11, 2014)
I am fairly happy with these notes, however, some of the sections towards the end could use a few more proofs. If you would like the LaTeX source file for these notes feel free to ask me for the file. In Fall 2015 semester I think we'll cover Chapter IV on hyperbolic geometry and we may drop some of the material on infinite products depending on the interest of the class.
Useful Materials and Links:
Course Notes from Spring 2013:
- Wolfram Alpha, a good way to check your answers (and learn more math)
- Infinite Series and the Residue Theorem by Noah A. Hughes (a student who worked with my brother at ASU)
- Gamma Function some notes from Gamelin Chapter XIV Section 1.
- Homeworks assigned in Fall 2015 semester:
I collect here the documents which comprise complex analysis of the Spring 2013 semester. We follow, roughly, Chapters 1-3 of Frietag and select topics of Chapter 4. However, it's more accurate to say we covered most of Churchill or Saff and Snider. Marsden, Gamelin, Alhfors, Albowitz and Fokas also contributed to our thinking. I hope to produce a more coherent set of notes with greater analytical depth when I next teach this course. If I teach the course in 2014 then I will likely use Gamelin for the text.
- Complex numbers and basic properties. Pages 1-6.
- Complex exponential notation and Euler's formula. Pages 7-10.
- Complex exponential, roots of unity, logs. Pages 8-21.
- Sequences, series, some proofs. Pages 21-28.
- basic topological concepts, continuity. Pages 29-34.
- limit laws, discontinuity. Pages 29-38.
- complex differentiation by limits. Pages 39-45.
- complex differentiation and its relation to real differentiability. Cauchy Riemann eqns. Pages 46-61.
- harmonic functions, complex mappings as transformations. Pages 62-78.
- contour integral, Cauchy's Integral Theorem, Cauchy-Goursat, other properties of the complex integration theory (see older notes for bounding theorem proof). Pages 79-105.
- theory of power series, introductory remarks on analytic continuation and Laurent series. E65-E73.
- discussion of meromorphic functions.
- methods of contour integration (has E100 to E117).
- methods of contour integration (this came with a handout)
- analytic continuation, argument principle and Rouche's Theorem, Mobius transformation revealed, existence of Laurent expansion.
- lecture by Dr. Ethan Smith on Analytic Number theory
- final lecture on Mittag Leffler and Weierstrauss' Product theorem (did not get to full story of products in lecture)
Solutions to Homework of 2013 and 2014:
- Mission 1 solution (2014)
- Mission 2 solution (2014)
- Mission 3 solution (2014)
- Mission 4 solution (2014)
- Mission 5 solution (2014)
- Mission 6 solution (2014)
- Mission 7 solution (2014)
- Mission 8 solution (2014)
- Mission 9 solution (2014)
- Problem Set 1 solution (2013)
- Problem Set 2 solution (2013)
- Problem Set 3 solution (2013)
- Problem Set 4 solution (2013)
- Problem Set 5 solution (2013)
- Problem Set 6 solution (2013)
- Problem Set 7 solution (2013)
Tests and Solutions from 2013:
- Test 1 solution (2013)
- Test 2 solution (2013)
- Test 3 takehome (no solution, just the problems)(2013)
- Test 1 Solution, Spring 2010.
- Test 3 Solution, Spring 2010.
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Last Modified: 8-15-2015