College of the Holy Cross, Spring 2022

Math 305 (Complex Analysis) Problem Sets

Professor Hwang.

Last modified: April 11, 2022


Please bookmark this page and check it weekly for new assignments.

Problem Sets and Portfolio Questions

Most Fridays, exercises will be assigned from that week's daily activity sheets. Written solutions are due following Friday. Late papers are not accepted for any reason. These problem sets are worth 10% of the course grade. Let me know if an illness or other emergency prevents you from completing an assignment.

One primary goal of the course is to give you practice writing mathematics with feedback. Some of the problems, the portfolio candidates, will be marked with a topic from the ten listed below. These write-ups (one question of your choice from each topic) are worth 40% of the course grade.

Portfolio write-ups may only be turned in on Mondays. Starting February 7, you must submit at least one portfolio write-up each Monday unless you have positive scores on seven or more topics. You may submit up to four portfolio write-ups on any given Monday.

You may resubmit a portfolio write-up multiple times in order to raise your score on that topic. (In no event will a resubmission lower your score for a topic.) To resubmit, you must re-write the entire problem, not make corrections to an existing draft.

You are warmly invited to ask me for help writing up portfolio questions, though I'll generally provide strategies rather than details. Write up and turn in each portfolio question separately, as a stand-alone document.

Each portfolio draft will be graded on the following scale:

Exceeds expectations
(E, 100%), a write-up I could photocopy and distribute to the class.
Mastered
(M, 85%), a write-up that substantially answers the question, but is slightly lacking in organization, clarity of exposition, and/or neatness.
In progress
(I, 70%), a write-up that mostly answers the question, and/or that requires effort to read, for reasons including logical disorganization or illegibility.
Not yet
(N, 0%) otherwise.

You may write drafts by hand on paper, use technology such as an iPad, or type them in the web-based LaTeX platform Overleaf. Submissions must include physical pieces of paper; I am unable to accept emailed scans, for example.

If you use Overleaf for portfolio questions, neatness of source code is part of the evaluation criteria. The goal is to develop your ability to write flexible, easy-to-read LaTeX. I will happily provide more detailed guidance as needed.

The topics and prospective questions (pick one for each topic by the end of the semester) are:

  1. Complex numbers (Topic 1)
    • Exercise 1.10, p. 18
    • Exercise 1.24, p. 20
    • Exercise 3.3, Activity Sheet 3 (prove your answer is correct)
    • Exercise 23.5, Activity Sheet 23
  2. Limits and continuity (Topic 2)
    • Exercise 2.3, p. 38
    • Exercise 6.4, Activity Sheet 6
    • Exercise 7.2, Activity Sheet 7
  3. Holomorphic functions (Topic 3)
    • Exercise 2.22, p. 41
    • Exercise 8.5, Activity Sheet 8
    • Exercise 9.5, Activity Sheet 9
  4. Projective transformations (Topic 4)
    • Exercises 12.1 and 12.2, Activity Sheet 12
    • Exercise 3.9, p. 63
    • Exercise 3.23, p. 65 and Exercise 12.5, Activity Sheet 12
  5. Exponential and circular functions (Topic 5)
    • Exercise 3.36, p. 67
    • Prove $\sin$ is injective on the strip where $|\operatorname{Re} z| < \frac{\pi}{2}$.
  6. Multi-valued functions (Topic 6)
    • Exercise 3.51, p. 69
    • Exercise 15.5, Activity Sheet 15
    • Exercise 18.3, Activity Sheet 18
  7. Complex integration (Topic 7)
    • Exercises 19.3, 4, Activity Sheet 19
    • Exercises 20.2, 4, 5, Activity Sheet 20
    • Exercises 21.2, 3, 4, Activity Sheet 21
  8. Harmonic functions (Topic 8)
    • Exercise 24.4, Activity Sheet 24
    • Exercise 6.9, p. 119
    • Exercise 6.12, p. 119
  9. Power series (Topic 9)
    • Exercise 28.5, Activity Sheet 28
    • Exercise 29.4, Activity Sheet 29
    • Exercise 32.4, Activity Sheet 32
  10. Isolated singularities, residues (Topic 10)
    • Exercise 33.2, Activity Sheet 33
    • Exercise 34.1, Activity Sheet 34
    • Exercise 35.4, Activity Sheet 35

Academic Integrity: Accuracy and honesty, including specific, easy-to-locate citations and proper attribution, are essential skills for producing useful written work. The guidelines below are intended to help you develop habits of good scholarship, including honest collaboration, detailed citation, clarity and completeness of written expression.

You are welcome to ask me any questions at all. One of my main roles in the course is to be your guide. The most productive questions will arise after you have made a serious attempt at a question on your own, and will generally be specific and focused.

Because the portfolio questions are a primary indicator of your individual work, they are subject to special rules: You may discuss question statements with each other, but not methods of solution. On written work submitted for grading, the wording, calculations, and structure of the argument must be yours alone. Any instances that violate this policy will result in a score of zero for that topic, and (as mandated by College policy) in my bringing a formal charge of academic dishonesty.

For non-portfolio exercises, you may collaborate with classmates, but must practice honest scholarship. When you turn in your exercises, acknowledge by name any classmates or others you worked with by writing “I worked with...” at the top of your assignment. If you use an external resource, give an appropriate bibliographic citation, such as the author of the resource, the complete URL, and the date on which you retrieved the content; or a book title, author, publisher, date of publication, and page number(s). My goals are to help you develop good habits of documenting your work. This allows others to verify your work more easily.

Read the College policy for definitions of plagiarism, cheating, and collusion. In this course, using wording from a classmate (or anyone else, such as a user on a math question and answer web site) on a portfolio question is cheating. The same act is, in addition, plagiarism if you do not mention the other person's name. Sharing your own portfolio work with a classmate is collusion.

To summarize, these policies are spelled out in order to clarify their scope and purpose. I want each of you to learn mathematical writing. You can only do this by actually writing, in your own words. In “real” writing, you have multiple drafts. Similarly, I expect you'll sometimes turn in E write-ups on a first try, but other times will turn in work that could use improvement in some respect. For each topic, you can resubmit write-ups until your work is as good as you like.


This Week's Problems

Monday, April 25

Activity sheet 28 3, 4
Activity sheet 29 2
Activity sheet 30 2, 3
Activity sheet 31 2, 3

Friday, April 8

Activity sheet 24 2
Activity sheet 25 4, 5
Activity sheet 26 1, 3

Friday, April 1

Activity sheet 19 2
Activity sheet 22 2, 3
Activity sheet 23 1, 3

Friday, March 25 The second midterm will be given in class.


Friday, March 18

Activity sheet 16 4, 5
Activity sheet 17 1, 2, 3, 4
Activity sheet 18 2, 5

Friday, March 4

Activity sheet 13 1, 3, 4, 5
Activity sheet 14 2, 5
Activity sheet 15 1, 2

Friday, February 25

Activity sheet 11 1, 2, 3
Activity sheet 12 3

Friday, February 18 The first midterm will be given in class.

Activity sheet 8 4
Activity sheet 9 4
Activity sheet 10 4

Friday, February 11

Activity sheet 5 1, 2
Activity sheet 6 1, 2, 3, 5
Activity sheet 7 3, 5

Friday, February 4

Activity sheet 1 5, 6
Activity sheet 2 1–5
Activity sheet 3 2, 4, 5
Activity sheet 4 3(a), 3(d), 5