Math 421.1—Complex Analysis—Fall 2010

Class number: 73992
Meets MWF 10:10–11:00 a.m. in LGRT 219
(except: F Sept 10 and M Sept 13 in Math & Stat Resource Center, LGRT 110)
Course web site:


Prof. Murray EisenbergProf. Murray Eisenberg
LGRT 1335 G
Office hours: W & F 2:00–3:00 p.m., LGRT 1335G
+ M 2:00–3:00 p.m., Resource Center, LGRT 110
+ other times (ask!)
Phone: 5-2859
E-mail: murray<at>
(see these e-mail guidelines)
Mailbox: LGRT 1623D (do not leave papers at my office!)

There is no TA for this course.

Required textbook

Mathews and Howell, Complex Analysis for Mathematics and Engineering, Fifth (5th) Edition, 2006. ISBN 13-digit: 9780763737481; ISBN 10-digit: 0763737488. At Textbook Annex: used $103.95, new $137.95.

Copies are also available, e.g., from, at a variety of prices. Then:

  • be sure you are getting the current, fifth (5th), edition; and
  • keep in mind that reading assignments will begin the very first day of class, and homework assignments of problems from the text may be due as early as the second week of classes.

Copies of the text are not available at the UMass Library.

Required software

  1. Mathematica, version 7.0 or later, Wolfram Research Inc. This numerics-symbolics-graphics software is available at all public OIT labs, on both Macs and Windows PCs. It includes complete documentation. I will help you learn how to use it. If you want to run Mathematica on your own Windows, Mac OS-X, or Linux computer, you can buy one of several versions of Mathematica for Students. These are available directly from the publisher, Wolfram Research. You have several options:
    • 6-month (“semester”) license: $44.95
    • 1-year license: $69.95
    • regular student license: $139.95

    Despite the low prices, Mathematica for Students is the full product and not crippled in any way—although you see “Mathematica for Students” on print-outs, and for tech support beyond installation issues you must rely on on-line forums. (By contrast, a retail “home” edition costs $295, and a normal “professional” version costs more than $1,000!)

    Like all versions, Mathematica for Students requires that you obtain a license number and password from Wolfram Research that will be tied to the hardware of a single computer. This means that you cannot share your copy for use on another computer.

  2. David Park’s Presentations add-on for Mathematica. This is free! I will provide you with a copy strictly for your own, personal use, courtesy of the Department of Mathematics and Statistics and by special arrangement with the author, David Park. You will be able to install Presentations on your own computer and use it with your licensed copy of Mathematica for Students).Presentations, along with Mathematica, will already be installed at the Math/Stat Resource Center (where we shall hold some hands-on sessions). Presentations will not be installed already on OIT public lab computers. However, you will be able to use your copy from a USB flash drive with Mathematica, which is already installed at OIT public labs.
  3. The WeBWorK on-line homework system. This is free! It will be used for some problems on some homework sets. Its URL is


  • Mathematical: Multivariable calculus (Math 233)
  • Computer: you do not need previous experience with Mathematica


  • take the final exam at the officially scheduled time
  • take the two mid-semester exams during regular class meetings; one or both of these may be given at a computer lab so you can have use of Mathematica
  • hand in the frequent homework sets some of which will involve computer work with Mathematica or in the on-line WeBWorK system.


The principal source cited below is the text. Not everything of importance will be covered in lecture: you will need to learn some things from readings, doing Mathematica work, and solving homework problems.

How far we progress in the topics list depends on the pace we maintain.

  • Complex numbers: their algebra, geometry, and topology [Chap. 1]
  • Complex functions [Chap. 2]
  • Analytic and harmonic functions [Chap. 3]
  • Complex sequences and series with application to Julia and Mandelbrot sets; and power series [Chap. 4]
  • Elementary functions (exponential, log, trig) [Chap. 5, selections from Sections 10.2–10.4]
  • Contour integrals [Chap. 6]
  • Taylor and Laurent series [Chap. 7]
  • Residues, with applications to computing real integrals [Chap. 8]
  • Conformal mapping [selection from Chap. 10]
  • Application of harmonic functions to problems in science and engineering [from Chap. 11]

Course aims

  1. Learn basic concepts—including precise definitions and statements of key theorems—about complex numbers and complex functions.
  2. Understand the reasoning behind key results and methods.
  3. Develop the ability to do calculations involving complex numbers and complex functions.
  4. Learn a few instances of how complex variables are applied in science, engineering, and mathematics.
  5. Learn enough about Mathematica to use it effectively in dealing with complex variables and to be able learn more about Mathematica.

Mathematica is used as a tool for doing numeric and symbolic calculations and for visualizing complex functions.


Scores below are on a 100-point scale.  Let

F = final exam score
X = average of mid-semester exam scores
= average of best 80% of homework set scores

Then your course score S is the weighted average defined, in Mathematica notation, by:

    S = Total[{0.25,0.50,0.25}*{F,X,H}]

However, if F > X, then F will be given greater weight, and X lesser.

Course scores 86–100 earn a course grade of A; 76–85 at least B; 60–75 at least C; 50–59 at least D.  Intermediate scores may earn course grades of A−, B+, B−, C+,  C−, D+.

How to succeed in Math 421

  • Read assigned material—before the class where it is discussed.
  • Keep up with new material every day: many topics build upon preceding ones.
  • Come to every class.
  • Don’t leave homework until the last minute.
  • On exams and written homework: show all relevant work and organize it unambiguously; indicate clearly what you are doing, and why.

Classroom etiquette

To avoid distracting you, your classmates, and your professor, before each class begins please check that your cellphone and other noisy devices are turned off.

Ordinarily, you may not use a laptop or similar type of computer during class. This is to avoid distracting other students (and me), and to help you avoid the temptation of diverting your attention elsewhere. (Exception: anybody certified by Disability Services as needing such assistive technology.) On occasions where Mathematica is being used, you may use a computer for purposes of this class—but not for checking e-mail, keeping up with Facebook,  and other extraneous purposes.

If you arrive late to class, please sit in the first seat you can find so as not to disturb others, and do not come up to the front of the room to pick up returned papers.


These cover both theory and calculation. You need to know definitions and statements of theorems. Some of the exams may be given at a lab where you will have access to Mathematica. In any case, you should show the setup leading to the computations.

Final exam

Covers the entire semester’s work, with some emphasis on material since second mid-semester exam. You must take it at the scheduled time except in the case of an official final exam conflict.

Final exam conflicts: Obtain from the Registrar’s Office a certified copy of your entire final exam schedule that indicates the conflict. On that form write your e-mail and phone contact information as well as the name and contact information of the instructor in the course creating the final conflict. No later than two weeks before the Math 421final exam, give me: (i) that form, and (ii) a copy of the syllabus,web page, or note from the instructor of the other course that states a final exam is actually being given there.

Mid-semester exams

Each exam is announced at least a week in advance.  If class on the exam date is canceled for any reason, the exam will be given at the next class.

There are no make-ups for mid-semester exams. Any excuse for missing an exam due to unavoidable cause such as serious illness must be in writing. By University policy, I will excuse missing an exam due to certain University-sanctioned events or a day of religious observance (about which you appropriately notified me). The normal arrangement in the case of a valid excuse is that your grade will be based upon the rest of your work.

Special accommodation on exams

Should you require special accommodation for exams, you must arrange directly with Disability Services for alternate proctoring at a time that includes the scheduled period for our exam. At least one week before the first exam for which you are entitled to special accommodation, give me the requisite written notice from Disability Services.

Homework sets

Each homework set has a due date that is strictly enforced except for unavoidable cause affecting the whole class. Normally the due date for written (and Mathematica) work is a class day, and then the work is due at the start of class. If class on the due date is canceled for any reason, the next class meeting becomes the due date. On-line WeBWorK answers are due at the time prescribed—often shortly before class begins.

The due dates help you keep up-to-date. Shortly after the due date, solutions to written work will be posted and answers to WeBWorK problems will be revealed. Hence for logistical as well as pedagogical reasons, late homework will not be accepted.

However, only the best 80% of your homework set scores are counted. This is to allow for all reasons that homework is  not submitted on time, including illness, emergencies, officially excused absence from campus, absence for religious observance, computer crashes, etc.

For each homework set:

  • Use 8.5-by-11″ paper and write with a pen or dark pencil.
  • There should be no “frizzies” along an edge due to removing pages from a notebook (but punched holes are OK).
  • Leave at least a 1″ right margin on every page, so I can write there.
  • Do not do different problems side-by-side in two or more columns on a page.
  • Arrange your solutions in the order in which the problems were assigned and number them in the same order as assigned (1, 2, 3, etc.).
  • Intermix computer printouts with written work on each question—do not collect the printouts for all questions at the end or beginning. Better yet: include written work on the computer printouts themselves or include appropriate text cells.
  • At the top right of every page, write your name. Below that write “Math 421” and the number of the homework set.
  • If possible, staple the pages together in their upper left corner only; don’t use any cover or folder.

Some problems will require you to use Mathematica. You may use Mathematica in any problem except where explicitly indicated otherwise, or where using Mathematica would render the problem largely pointless.  (If in doubt, include written work.) Include printouts showing the input along with the corresponding output.

Since I want to return papers as soon as possible, some problems you turn in will not be graded at all, while others will merely get a “checkoff” as to whether you made an honest attempt to solve them. Of course, all papers will be treated alike in this regard.


Access to WeBWork is free! The URL for WeBWorK for our class is:

Or click the link for Math 421 on the generic UMass URL for WeBWorK:

WeBWorK login:
  • Username: the part of your official UMass e-mail address before @
  • Password: initially, your UMass student ID number
    (change it as soon as you log on!)

For example, if your official UMass e-mail address is, then your Username would be xyersel.

For additional help in using WeBWorK, see:

Written work

Don’t scatter your work helter-skelter, willy-nilly about the page. Rather, arrange it in an organized fashion so that the logical flow is clear. Indicate clearly what you are doing and why.

Collaboration and plagiarism

For homework, you may work in a group of two or three. For on-paper homework, you must turn in a separate paper of your own with your own write-up and your own Mathematica work, on which you name any collaborators. Note that most WeBWorK problems are “parameterized”, so that different students get somewhat different versions.

On some homework assignment(s), you may be required to work in a group of two or three. In that case, I will provide additional information about the nature of the collaboration expected and the kind of product to be submitted.

You must explicitly cite any sources you use other than the textbook, your own Math 421 class notes, handouts in class, materials available on this web site, or help from me. And you may not use solutions to the homework problems that you find in print or on-line.

Representing somebody else’s work as your own is plagiarism, for which there can be severe penalties under University policy.

Graded paper return

I will return all exam and homework papers in class as soon as they have been graded. If you miss class when a paper is returned, you will need to see me at office hours, or otherwise, in order to retrieve your paper. Unfortunately, the U.S. Family Educational Rights and Privacy Act prohibits me from leaving unclaimed papers outside my office.


You are expected to attend regularly and are responsible for anything you miss!

If illness or other emergency forces you to leave campus, it is best to notify the Dean of Students (545-2684), who will notify all your instructors.

In accordance with University regulations, within the first calendar week of classes you should notify me in writing of expected absences for religious observance.

Drops, Withdrawals, and Incompletes

The last day to drop is Monday, September 20; to withdraw with a W, Thursday, October 21 (Mid-Semester Date).

An Incomplete is possible only if (1) you had a compelling personal reason (e.g., serious illness), (2) your work has clearly been passing, and (3) there’s a good chance you’ll complete the course with a passing grade within the allotted time. Thus, failing work is no reason in itself for an Incomplete.

Copyright information

Many of the materials created for this course are the intellectual property of the instructor. This includes, but is not limited to, the syllabus, lectures, printed handouts, and pages and files on the course web site whose intellectual ownership is not otherwise indicated. Except to the extent not protected by copyright law, any use, distribution, or sale of such materials  in any format—printed or electronic—requires the permission of the instructor. Please be aware that it is a violation of University policy to reproduce, for distribution or sale, class lectures or class notes, unless copyright has been explicitly waived by the faculty member.

Copyright © 2010 Murray Eisenberg.  All rights reserved.