## Mathematics 371:  Complex Analysis Fall 2011Introduction

Professor:        Jeff Johannes                                    Section 1    MWF    12:30-1:20p   Sturges 113
Office:             South 326A
Telephone:       245-5403
Office Hours:  Monday 11:30a - 12:20p, Wednesday 4-5p, Thursday 1-2p, 8-9p, Friday 2:30 - 3:30p, and by appointment or visit
IM:                    JohannesOhrs
Web-page:        http://www.geneseo.edu/~johannes

Textbook
An Introduction to Complex Analysis and Geometry by John P. D'Angelo

Purposes
Extending almost all mathematics to complex numbers.  High school algebra and geometry, calculus, linear algebra, differential equations, abstract algebra, topology, …

Overview
Because of the fact that we will be extending so many areas, this is a perfect opportunity to solidify everything that you have done before, or to get little tastes of things you haven't seen before.  We'll start with some basics of real numbers, then earn the basics of what and why complex numbers are.  After that we will see interplay with geometry, linear algebra, and then head toward topics from calculus:  series, differentiation and end with the richest area of all - integration.

Half of your grade will come from problem sets.  Another tenth will come from a final project and each of two midterm exams.  The final fifth will come from the final exam.

Problem Sets
After we finish each chapter problem sets will be collected.  They will be returned with a letter grade based on the following factors:  number of exercises correctly completed, difficulty of exercises correctly completed, number of exercises completed by classmates, and some subjective determination on my part as to what seems appropriate.  Each problem set will be scaled using a linear function of the number of exercises completed (problems correctly completed by only one student will earn two points).  Submitting no problem set by the day it is due will earn a score of zero.  I strongly recommend consulting with me as you work on these problem sets.  I also recommend working together on them, however I want to carefully emphasise that each must write up their own well-written solutions.  A good rule for this is it is encouraged to speak to each other about the problem, but you should not read each other's solutions.  A violation of this policy will result in a zero for the entire assignment and reporting to the Dean of Students for a violation of academic integrity.

Final Project
Your final project will constitute writing up a detailed explanation (filling in the gaps) of a topic in the text that we will omit (or another topic selected by you and approved by me) , and a completion of a problem set (graded as above) from the exercises in that section.

Exams
The exams will consist of a few straightforward problems designed to emphasise a personal understanding of the basics.

Feedback
Occasionally you will be given anonymous feedback forms.  Please use them to share any thoughts or concerns for how the course is running.  Remember, the sooner you tell me your concerns, the more I can do about them.  I have also created a web-site which accepts anonymous comments.  If we have not yet discussed this in class, please encourage me to create a class code.  This site may also be accessed via our course page on a link entitled anonymous feedback.  Of course, you are always welcome to approach me outside of class to discuss these issues as well.

While working on homework with one another is encouraged, all write-ups of solutions must be your own. You are expected to be able to explain any solution you give me if asked. The Student Academic Dishonesty Policy and Procedures will be followed should incidents of academic dishonesty occur.

Disability Accommodations
SUNY Geneseo will make reasonable accommodations for persons with documented physical, emotional or learning disabilities.  Students should consult with the Director in the Office of Disability Services (Tabitha Buggie-Hunt, 105D Erwin, tbuggieh@geneseo.edu) and their individual faculty regarding any needed accommodations as early as possible in the semester.

Religious Holidays
It is my policy to give students who miss class because of observance of religious holidays the opportunity to make up missed work.  You are responsible for notifying me by September 11 of plans to observe a holiday.

Schedule (loose and subject to variations)

August 29    Introduction
31        Chapter 1 (1-3, 4, 5)
September 2

7
9

12
14      Chapter 2 (1, 2, 6, 7, 8)
16      PS1 due

19
21
23

26
28      Chapter 3 (1, 2, 4, 5) PS2 due
30

October 3    exam (Chapter 1-2)
5
7

12
14

17      Chatper 4 (2.3, 2.4, 2.5, 1, 2)
19
21     PS3 due

24
26
28

31
November 2   Chapter 5 (1, 2, 3, 4, half of 5) PS4 due
4

7    exam (Chapter 3-4)
9
11

14
16
18   Chapter 6 (1, 2, 4, 5)

21   PS5 due

28
30
December 2

5   Chapter 7 (1, 2)
7
9

12      Review, PS6-7 due, Final Project due

Monday, December 19 12N - 3p  Final Exam (half 5-7, half 1-4)

Learning Outcomes
Upon successful completion of this course, a student will be able to
• Express complex numbers in the important equivalent forms - rectangular, polar and exponential.
• Perform the basic arithmetic operations on complex numbers, including powers and roots (using DeMoivre's Theorem).
• Represent complex numbers as points in the complex plane and vice-versa.
• Determine various basic topological properties of sets of complex numbers in the plane.
• Determine convergence of sequencees and series of complex numbers.
• Determine the continuity and differentiability of complex valued functions of a complex variable.
• Determine the analyticity of complex valued functions of a complex variable using the Cauchy-Riemann equations.
• Determine the harmonicity of complex and real valued functions of a complex varaible.
• Perform computations with the elementary compelx functions - exponential, trigonometric, and logarithmic.
• Evaluate contour integrals directly and using the appropriate form of Cauchy's theorems.
• Expand analytic functions in power series of a complex variable.
• Determine the radius and interval of convergence of a complex power series.
• Calculate the residue ofa complex function at a singularity.
• Use the residue theorem to compute definite integrals.