Math 43 (Spring 2012): Functions of a single complex variable

Quick links on this page: News | Introduction | Class coordinates | Instructor information | Exam Schedule | Grading | Homework Policy | Textbook

Quick links to Math 43 pages: Homework Assignments | Syllabus and schedule | Exam page | Final exam page

News

• 5/27/2012: There will be a special office hours on Tuesday, 5/29, 3pm - 4pm. Also, final exam information is available at the final exam page.
• 4/9/2012: Class on Wednesday, 4/11 is cancelled, and we will have a replacement class during the X-hour on Thursday, 4/12. Midterm/quiz resubmissions are still due on Wednesday afternoon, though.
• 4/6/2012: Information about the exam resubmission policy is available at the exam page. Eventually solutions to old exams will be available there as well.
• 4/3/2012: Homework will usually be due on Mondays instead of Fridays for the rest of the term. Also, for the first quiz/midterm, all the content through Monday's (4/2/2012) class is on the test. You can bring one sheet of notes that you prepare yourself to help you during the test.
• 3/14/2012: First version of this webpage goes up. If you are taking this class, please read this webpage carefully, especially the sections on grading and homework.

Introduction

Math 43 is an introduction to the theory of functions of a single complex variable. In other words, Math 43 is about calculus on single variable functions which accept and take values in complex numbers instead of real numbers. We will discuss differentiation and integration of complex functions, and explore the many consequences that complex differentiability implies. A major goal of this class is to discuss the Cauchy integral theorem and its corollaries. We will also discuss product factorizations of everywhere complex differentiable functions (entire functions), and if time permits, we will examine applications to the gamma and (Riemann) zeta functions, and a proof of the Prime Number Theorem.

Although the official ORC pre-requisite for this class is only multivariable calculus, some knowledge of real analysis will be handy. We will not assume any real analysis background, but will quickly discuss and review relevant ideas that subject when necessary.

Class Coordinates

Instructor information

Examination Schedule

There will be four 45 minute midterms, administered during the X-hour, and a final exam. A limited amount of notes will probably be permitted during each exam, but no computational aids will be allowed on exams.

• Midterm 1: Thursday, April 5, 2012, 12:00pm - 12:50pm, our classroom
• Midterm 2: Thursday, April 19, 2012, 12:00pm - 12:50pm, our classroom
• Midterm 3: Thursday, May 3, 2012, 12:00pm - 12:50pm, our classroom
• Midterm 4: Thursday, May 17, 2012, 12:00pm - 12:50pm, our classroom
• Final Exam: Friday, June 1, 2012, 8:00am - 11:00am, 007 Kemeny (our usual classroom)

If you are unable to be at any of these exams, please contact me as soon as possible so we can setup alternate test-taking arrangements.

Grading

Your grade in this class will be determined by homework and exams.

Weekly written assignments will consist of questions which usually cover the material from the previous week. Questions will typically involve computing a certain quantity or proving a particular statement. In either case, answers must be justified using clear and reasonably complete arguments. Written homework will be due on Monday afternoons.

There will be four 45 minute midterm exams, administered during the X-hour, and a final exam. They will probably be limited open note exams.

Each of the above contributes to your final grade in the following fashion:

• Homework: 40%
• Midterms: 7.5% each, 30% total
• Final exam: 30%

Homework Policy

Written homework assignments will be posted at the homework page of this website, and will be usually due about a week after they are posted. Late assignments will only be accepted when granted an extension, which must be requested from the instructor several days in advance. In general, extensions will only be granted for health-related reasons or family emergencies. Exceptions may be made for school-related travel.

The homework collaboration policy for this class is more or less in line with other Dartmouth math classes. You are allowed to collaborate with others on homework, but must write your own solutions. A good rule of thumb is that you should never be copying phrases or sentences from anyone else or any source. You may use theorems, lemmas, etc. that we have covered from the textbook, but in general you should not use theorems, lemmas, etc. from sections of the book we have not covered or from external sources. Also, please write down the people you collaborated with and outside sources (namely, anything besides the required textbook) you consulted on your homework assignments.

Textbook

The required book for this class is Complex Analysis, by Elias Stein and Rami Shakarchi, ISBN 978-0691113852. This relatively new book is one part of a series of texts on real and complex analysis that grew out of teaching the analysis sequence for undergraduates at Princeton University. As this sequence is one of the best, if not the best, analysis sequence for math majors in the country, the quality of the exposition and choice of topics in this book (and the other books in this series) are fantastic. Elias Stein is one of the leading world authorities in analysis (in particular, harmonic analysis), and has had decades of experience teaching top undergraduate and graduate students. Another plus of this book is that it contains a lot of content, yet is reasonably priced.

There are many books about complex analysis. The following are a few books which might be worth consulting:

• Complex Analysis, 3rd Edition, by Lars Ahlfors, ISBN 978-0070006577.
  This book is considered a classic text in the field. It was also written by a top mathematician (Ahlfors received the Fields Medal), but the price of the book is incredibly high. Nevertheless, it might be worthwhile to check this book out in the library.
• Real and Complex Analysis, 3rd Edition, by Walter Rudin, ISBN 978-0070542341.
  This book is considered one of the standard texts in analysis. We are not using it since most of the book is devoted to real analysis, and the level of sophisticiation is slightly higher than what we are aiming for in this class. Nevertheless, it might be worth taking a look at this book after taking this class and a class in real analysis.
• Complex Analysis, 3rd Edition, by Joseph Bak and Donald Newman, ISBN 978-1441972873.
  This book is probably not as well known as the others listed above, but I used it as an undergraduate and found it very well-written. The book is aimed squarely at undergraduate students (unlike the previous two books listed above), and is quite enjoyable to read.