Math 311: Methods of Complex Analysis

Fall 2017

Textbook

Fundamentals of Complex Analysis (with Applications to Engineering and Science), 3rd Edition by E. B. Saff & A. D. Snider. Prentice Hall, 2003.

 

Course Description

This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Material covered includes: functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions.
A working knowledge of the principles of complex analysis is an indispensable part of the formation of any scientist or engineer. The central concepts of "analytic function" and "conformal mapping" are nothing but two faces of the same coin and the interplay between analysis and geometry makes the subject extremely rich in applications. One can use the properties of these functions to easily compute integrals for which standard real-variable methods fail, to generate beautiful fractal figures and to study the ubiquitous "harmonic functions" that appear when dealing with such diverse problems as the steady-state temperature of a plate, non-viscous fluid flow or electrostatic charge distribution. Although fully exploring the richness of applications of the subject is beyond the scope of a first course it is hoped that the ones we present will serve as enticing highlights.
The prerequisite for this course is Calculus III.

 

Instructor

Hang Xu

Email: hxu@math.jhu.edu

Office Location: Krieger Hall Rm 220

Office Hours: Th 1:30-3:30 pm

Grader

Junyan Zhang

Email: jzhan182@jhu.edu

Math Help Room Hours: W 3:00-5:00 pm at Krieger 213

Lecture

TTh 12:00-1:15 at Maryland 104

 

Grading

is based on 300 points distributed as follows:

11 homework problem sets, worth 10 points each, due in the lecture on the following Thursday. The lowest score will be dropped.

Midterm on October 17 in lecture, worth 100 points.

Final, on December 7 in lecture, worth 100 points.

 

Course Policies

No late homework is accepted. Staple your problem sets! Study groups are encouraged, but homework has to be written down independently.

No makeup exam in this course. If you have to miss an exam for a documented, legitimate reason, please inform me as early in the semester as possible.

You are responsible for lecture notes, any course material handed out, and attendance in class. The lectures will be conducted as if you have already read the material and attempted some homework problems. In this manner, you can focus mainly on those parts of the lectures that cover the areas of your reading you found difficulty to understand.

 

If you have any math questions, please feel free to ask me anytime. You can find me after lectures, in office hours, or you can reach me by email.

 

Help Room

Krieger Hall 213. The hours are 9am – 9pm on Monday through Thursday, and 9am – 5pm on Friday. This free service is a very valuable way to get one-on-one help on the current material of a class from other students outside the course. It is staffed by graduate students and advanced undergraduates. Outside of me and the Grader for the course, definitely take your questions to the Help Room. This course is simply an analysis course directed toward particular maps and differential equations. Most graduate students should be able to "see" through the many problems stated in this course. And your attempts to help guide them will be of huge benefit to you also. 

 

Students with disabilities

Students with documented disabilities or other special needs who require accommodation must obtain an accommodation letter from Student Disability Services, 385 Garland, (410)516-4720, studentdisabilityservices@jhu.edu. After that, remind the instructor of the specific needs at two weeks prior to each exam; the instructor must be provided with the official letter stating all the needs from Student Disability Services.

 

JHU ethics statement

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. You may consult the associate dean of student affairs and/or the chairman of the Ethics Board beforehand. See the guide on “Academic Ethics for Undergraduates” and the Ethics Board Web site (http://e-catalog.jhu.edu/undergrad-students/student-life-policies/#UAEB) for more information.

 

Tentative Schedule (will be updated as the course progresses)

Homework will be posted by every Tuesday and due in the lecture on the next Thursday.

Week 1 

No Class.

Week 2 (9/5 & 9/7): Complex Numbers

Read 1.1-1.5 

Homework 1: Exercises 1.1: 4, 6, 8; Exercises 1.2: 7(e), 16; Exercises 1.3: 5(d), 7(h), 12(d); Exercises 1.4: 4, 8; Exercises 1.5: 5(f), 16.

Week 3 (9/12 & 9/14): Planar Sets, Riemann Sphere and Complex Functions

Read 1.6, 1.7, 2.1, 2.2

Homework 1 due in the lecture on 9/14.

Homework 2: Exercises 1.6: 2, 4, 6; Exercises 1.7: 2, 5; Exercises 2.1: 3, 10(a); Exercises 2.2: 9, 11, 18, 21.

Week 4 (9/19 & 9/21): Analytic Functions, Cauchy-Riemann Equations and Harmonic Functions

Read 2.3, 2.4, 2.5

Homework 2 due.

Homework 3: Read ‘Level Curves of Harmonic Functions’ Page 81-83. Exercises 2.3: 4(a)(c), 7(a)(c)(e), 11(b)(f); Exercises 2.4: 3, 4, 8, 12, 13; Exercises 2.5: 3(b)(c)(d), 6, 9, 15.

Week 5 (9/26 & 9/28): Elementary Functions

Read 3.1, 3.2, 3.3

Homework 3 due.

Homework 4:  Exercises 3.1: 5(a), 10, 13(d), 15(a)(b)(c); Exercises 3.2: 5(b)(d)(f), 9(b)(c)(e), 20; Exercises 3.3: 5, 12, 14, 15.

Week 6 (10/3 & 10/5): Elementary Functions

Read 3.4, 3.5

Homework 4 due.

Homework 5: Exercises 3.4: 1, 2, 3, 4; Exercises 3.5: 1(b)(d), 3(b)(c), 6, 7, 15(a).

 

Week 7 (10/10 & 10/12): Contour Integral

Read 4.1, 4.2

Homework 5 due.

Homework 6:  Exercises 4.1: 1(b)(d), 4, 8, 9, 11; Exercises 4.2: 3(b)(d).

 

Week 8 (10/17 & 10/19): Independent of Path

Midterm on Tuesday in class.

Read 4.3

Homework 6 due.

Homework 7: Exercises 4.2: 6, 8, 9, 10, 12, 13, 14.

  

Week 9 (10/24 & 10/26): Cauchy’s Integral Theorem and Cauchy’s Integral Formula

Read 4.4, 4.5

Homework 7 due.

Homework 8: Exercises 4.3: 1(b)(e)(g), 5, 10. Exercises 4.4: 1, 9, 10, 13, 15.

  

Week 10 (10/31 & 11/2): Cauchy’s Integral Formula, Bounds for Analytic Functions, Sequences and Series 

Read 4.6

Homework 8 due.

Homework 9: Exercises 4.4: 17, 18. Exercises 4.5: 3(a)(d)(e)(f), 4, 5, 6, 7, 10.

 

Week 11 (11/7 & 11/9): Bounds for Analytic Functions, Sequences and Series 

Read 5.1

Homework 9 due.

Homework 10: Read ‘Applications to Harmonic Functions’ Page 221-225. Exercises 4.6: 3, 5, 6, 7, 8, 11, 17. Exercises 5.1: 7(a)(c)(e).

 

Week 12 (11/14 & 11/16): Power Series and Laurent Series, Zeros and Singularities

Read 5.2, 5.3, 5.5, 5.6

Homework 10 due.

Homework 11: Exercises 5.1: 11, 12. Exercises 5.2: 1(c)(e), 5(b)(e), 11(a)(c), 13. Exercises 5.3: 2, 3(c)(d), 5, 7. Exercises 5.5: 3, 5, 6, 7(b)(c).

 

Week 13 (11/21 & 11/23) Thanksgiving Vacation

Week 14 (11/28 & 11/30): The Residue Theorem and Improper Integrals

Read 6.1, 6.2, 6.3

Homework 11 due.

Week 15 (12/5 & 12/7): The Argument Principle and Rouche’s Theorem

Final on Thursday in class.

Read 6.4, 6.7