**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 **

**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 ÒAcademic Ethics for UndergraduatesÓ 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.

No Class.

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.

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.

Read 2.3, 2.4, 2.5

Homework 2 due.

Homework 3: Read ÔLevel Curves of Harmonic FunctionsÕ 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.

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): CauchyÕs Integral Theorem and CauchyÕs 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.

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.

Read 5.1

Homework 9 due.

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

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).

Read 6.1, 6.2, 6.3

Homework 11 due.

Practice Problems (will not be collected): Exercises 5.6: 1, 2, 5. Exercises 6.1: 1, 3. Exercises 6.2: 2, 3, 4, 5.

Final on Thursday in class.

Read 6.4, 6.7