MATH 3851
Complex Variables
Winter 2018

This is the homepage for MATH 3851 (Complex Variables). This page will be updated throughout the term with important information for our course, including homework assignments, review materials, solutions to assignments, and more. CHECK IT FREQUENTLY!


  • Solutions for Assignment 7 have been posted.
  • A practice final exam and topics list for the final exam have been posted below. I placed estimates of time per problem as an aid, but I can't promise they'll be completely accurate for each of you.
  • Assignment 8 has been posted. This assignment does NOT need to be turned in. The final problem is a bit of a challenge problem, using the ideas we started working with Thursday. We'll finish up those ideas on Tuesday, but they won't be on the final exam.
Course Information

Instructor: Ronnie Pavlov
Office: Knudson Hall 204
Phone: (303)-871-4001
Office hours: Monday 11-12, Thursday 4-5, Friday 1-2, or by appointment (try to give 24 hours notice for an appointment!)


Text: Complex Variables and Applications, 9th Edition, by Brown and Churchill.

This book is available at the DU Bookstore.

We will cover Chapters 1-5 of the textbook, and as much of Chapter 6 as we can.

Course summary

The purpose of this course is to develop the theory of calculus for functions whose input and output are both complex numbers. It turns out that differentiability is a much stronger property for such functions, and as a result there are many beautiful theorems that one can prove. For example, the integral around any closed loop in the complex plane of any function which is differentiable inside the loop is 0! We will also be able to use this theory to prove some surprising results, such as the Fundamental Theorem of Algebra, which states that any polynomial with complex coefficients has a complex root.

We can also prove some results which, on the surface, seem to have nothing at all to do with complex numbers. For instance, suppose we want to find the integral over the real line of the function 1/(1 + x^10). This can be done with only a few lines by using complex analysis, even though the indefinite integral of this function takes many pages to write down. (try plugging it into Mathematica/Wolfram Alpha sometime!)

The only prerequisites for this course are our calculus sequence, Math 2200 (Proofs), and Math 2080 (Multivariable Calculus.) Though our emphasis will still be mostly on computation and applications, this course will be more theoretical than a Calculus course, and I will require you to be able to understand and write simple "proofs."

The most important advice I can give you for this course is to honestly evaluate your own progress. Mathematics, perhaps more than any other subject, allows for constant easy self-evaluation; either you know how to complete exercises from the section on your own, without outside help, or you do not. If you are having trouble, come see me! I am always happy to discuss any aspect of the class which is causing trouble, either informally after class or during office hours. A difference of even a few days in seeking help can make a huge difference, so BE PROACTIVE!

Grading scheme

Your term grade will consist of weekly homework assignments, a midterm exam, and a final exam, broken down in the following way:

40% Final exam
30% Midterm
30% Homework


Late assignments (without an accepted reason) will have a percentage subtracted according to the following policy:

3-5 days late: -50%
>5 days late: not accepted


You will have one midterm exam on February 13th, during class time in our classroom. Your final exam will be on Thursday, March 15th, also during class time in our class room.
Important Documents

None yet!

Course Policies

Honor Code: Students in this course are expected to abide by the University of Denver’s Honor Code and the procedures put forth by the Office of Citizenship and Community Standards. Academic dishonesty - including, but not limited to, plagiarism and cheating - is in violation of the code and will result in a failing grade for the assignment or for the course. As student members of a community committed to academic integrity and honesty, it is your responsibility to become familiar with the DU Honor Code and its procedures: see

Students with Disabilities: If you qualify for academic accommodations because of a disability or medical issue, please submit a faculty letter to me from Disability Services Program (DSP) in a timely manner so that your needs may be addressed. DSP determines accommodations based on documented disabilities/medical issues. DSP is located on the 4th floor of Ruffatto Hall, 1999 E Evans Ave, 303-871-2278. Information is also available online at; see the Handbook for Students with Disabilities.

Religious Accommodations: University policy grants students excused absences from class or other organized activities for observance of religious holy days, unless the accommodation would create an undue hardship. Faculty are asked to be responsive to requests when students contact them in advance to request such an excused absence. Students are responsible for completing assignments given during their absence, but should be given an opportunity to make up work missed because of religious observance. Once a student has registered for a class, the student is expected to examine the course syllabus for potential conflicts with holy days and to notify the instructor by the end of the first week of classes of any conflicts that may require an absence (including any required additional preparation/travel time). The student is also expected to remind the faculty member in advance of the missed class, and to make arrangements in advance (with the faculty member) to make up any missed work or in-class material within a reasonable amount of time.