Instructor: Jon Simone

Email: jsimone@umass.edu

Class Meetings: MWF 1:25pm-2:15pm in LGRT 147 on Zoom

Office: LGRT 1332

Office Hours: Monday 3-4pm
                        Wednesday 11:30am-12:30pm, 3-4pm
                        and by appointment

Textbook: Complex Variables and Applications (9th edition) by Brown and Churchill


The course is an introduction to functions of a complex variable. We will cover most topics in Chapters 1-7 of the textbook and if time allows, we will explore additional topics in Chapters 8-10. Please read this page for class policies.


Homework:

There will be weekly written homework assignments (posted in the schedule below). These will be due at 4pm every Wednesday (with the first assignment due on 1/29). Assignments are to be put in my mailbox in LGRT 1623D. Since I am teaching another course, be sure to put your assignment in the envelope labelled Math 421. Completed assignments are to be posted to Moodle by 5pm on the day that they are due.

These assignments will consist mainly of problems from the textbook. Thus it is important for you to have access to the correct edition of the text.

These assignments will be graded on logical flow and clarity as well as mathematical correctness. That is to say, please write neatly and concisely. Using a pencil is alway a good idea. Late homework will not be accepted. The lowest homework grade will be dropped.

You may work on your own or collaborate with others on these assignments. If you choose to collaborate, be sure to write your solutions in your own words---solutions should not be simply copied from collaborators.

I will typically post homework at least one week in advance of the due date. It is a good idea to begin working on these assignments well before the due date. This will give you plenty of time think about the problems, collaborate, and come to office hours if you have questions.

Late homework will not be accepted. The lowest grade will be dropped.


Exams:

There will be three in-class exams and a final exam. Past exams can be found here.

Exam 1: Friday, February 14

Exam 2: Friday, March 6

Exam 3: Friday, April 10

Final Exam: Monday, May 4th 10:30am-12:30pm in LGRT 147


Grading:

All of your grades will be posted on Moodle. The course grade is broken down as follows.

Homework:25%
Exam 1:15%
Exam 2:15%
Exam 3:15%
Final:30%


Disability Services Accommodations:

The University of Massachusetts Amherst is committed to making reasonable, effective and appropriate accommodations to meet the needs of students with disabilities and help create a barrier-free campus. If you have a disability and require accommodations, please register with Disability Services (161 Whitmore Administration building; phone 413-545-0892) to have an accommodation letter sent to your faculty. Information on services and materials for registering are also available on their website.


Schedule (to be regularly updated and subject to change throughout the semester):

Class Meeting Topics Textbook Sections Assignments
1/22 Wed Complex numbers, modulus, conjugation Sections 1-6
1/24 Fri Polar and exponential form, Euler formula Sections 7-9
1/27 Mon nth roots, polynomial functions Sections 10, 11, 13, 14
1/29 Wed Rational functions, stereographic projection, multiple-valued functions Sections 13, 17 Homework 1 Due:
p.4 # 2(a), 4
p.13 # 2, 5, 8, 9
p.16 # 1, 10(a), 15
p.23 # 1, 2, 5(a), 5(c), 10
p.30 # 1, 4, 8
HW1 Solutions
1/31 Fri The exponential function, logarithmic functions Sections 30-32, 34-36
2/3 Mon Power functions, trig and hyperbolic trig functions Sections 37-39
Interactive Tools:
Geometry of multiplication, the reciprocal function, the exponential function, the logarithmic function, the sine function
Geometry of Stereographic Projection
Search around Geogebra for more toys.
2/5 Wed Limits and continuity Sections 15-18 Homework 2 Due:
p.43 # 1(c), 1(d), 4, 8
p.89 # 1, 6, 10(a)
p.95 # 1, 2, 3, 5, 8
p.103 # 1(a), 5
p.111 # 2(a), 7(a)
Additional Problem
HW2 Solutions
2/7 Fri Derivatives Sections 19-20
2/10 Mon Cauchy-Riemann equations Sections 21-23
2/12 Wed Review Homework 3 Due:
p.54 # 2(a), 5, 11
p.61 # 1, 2(b), 2(d), 8(a)
p.70 # 1(a), 1(c), 8
HW3 Solutions
2/14 Fri Exam 1 (Covers lectures 1/22-2/7) Exam 1 Solutions
2/18 Tue Polar version of the Cauchy Riemann equations Sections 24
2/19 Wed Analytic functions and branch cuts Sections 25, 33
2/21 Fri Analytic functions continued Sections 25, 26
Interactive Tools:
Riemann Surfaces: graphs of the real and imaginary parts of logz and z^(1/2)
2/24 Mon Definite integrals, parametrized curves Sections 41-43
2/26 Wed Contour Integrals Sections 44, 45 Homework 4 Due:
p.70 # 1(d), 3, 4(a)
p.76 # 1(a), 2(a), 7
p.95 # 4
p.107 # 1
Additional Problems
HW4 Solutions
2/28 Fri Antiderivatives, Fundamental Theorem of Contour Integrals Sections 48-49
3/2 Mon Cauchy-Goursat Theorem, simply connected domains Sections 50, 52
3/4 Wed Multiply connected domains and some review Section 53 Homework 5 Due:
p.123 # 5
p.132 # 2, 3, 4, 5, 13 (use basic definition of contour integrals for problems in this section)
p. 147 # 1, 2, 3
HW5 Solutions
3/6 Fri Exam 2 (Covers lectures 2/10-2/28) Exam 2 Solutions
3/9 Mon Cauchy Integral Formula, integral modulus bounds Sections 54, 47
3/11 Wed Proof of Cauchy Integral Formula, generalization to nonsimple closed curves Sections 47, 54
3/13 Fri Applications: Cauchy's Inequality, Harmonic functions Section 57, 27 Homework 6 Due:
p.159 # 1(a)-(c), 2(a)-(b), 7
p.170 # 1, 2, 3, 4, 5
p.139 # 1(a), 2, 4
HW6 Solutions
3/16-3/20 SPRING BREAK
3/23 Mon Harmonic conjugates, Liouville's Theorem, Fundamental Theorem of Algebra Sections 115, 58
3/25 Wed Maximum Modulus Principle Section 59
3/27 Fri Using contour integrals to evaluate real integrals Sections 85, 92 Homework 7 Due:
p.171 # 10
p.177 # 1, 2, 8
p.357 # 1(a), 3
Additional Problems
HW7 Solutions
3/30 Mon Calc II review: sequences and series
4/1 Wed Complex sequences and series, divergence test Sections 60, 61 Homework 8 Due
Homework 8
HW8 Solutions
4/3 Fri Absolute convergence, Dirichlet test, power series Section 61, 69
4/6 Mon Power series continued, Taylor series Sections 62, 64, 70-72
4/8 Wed Taylor series continued, proof of Taylor's Theorem using the Cauchy Integral Formula Section 63 Homework 9 Due
Homework 9
HW9 Solutions
4/10 Fri Moodle Exam 3 (Covers lectures 3/2-4/6) Exam 3 Solutions
4/13 Mon Real Taylor's Theorem VS Complex Taylor's Theorem, Laurent Series Sections 65-66, 68
4/15 Wed Laurent series continued, singular points Sections 68, 74, 78, 79
4/17 Fri Singular points continued, residues Sections 84, 75 Homework 10 Due
Homework 10
HW10 Solutions
4/22 Wed Cauchy Residue Theorem, residue at infinity Sections 76, 77
4/24 Fri Residue at infinity continued, residues at poles Sections 77, 80
4/27 Mon Residues at poles continued Sections 80, 81
4/29 Wed Zeros of analytic functions, review Sections 82, 83 Homework 11 Due:
Homework 11
HW11 Solutions
5/4 Mon FINAL EXAM (on Moodle) Practice Problems
Solutions
Final Exam Solutions