Mathematics 365: Complex Variables With Applications
The final exam is Monday, May 4, 12:00 - 2:00 p.m. in Sweeney 1120. Make-ups must be arranged in advance.
Spring 2015, 3 credits
Professor Stephen J. Willson
Office: Carver 411
TR at 10:00-10:50, W at 10:00- 10:50 and 1:10-2:00 in Carver 411. Please also feel free to make an appointment if the office hours are inconvenient for you.
Class meetings: TR at 12:40 - 2:00 in Sweeney 1120.
The first class is Tuesday, January 13, 2015, and the last class is Thursday, April 30, 2015.
There are no classes Spring Break (the week
of March 16 - 20).
Prerequisites: Math 265.
Catalog description: Functions of a complex variable, including differentiation, integration and series expansions, residues, evaluation of integrals, conformal mapping. Nonmajor graduate credit.
Dennis Zill and Patrick Shanahan. Complex Analysis: A First Course with Applications, 3rd edition, Jones & Bartlett Learning ISBN 978-1-4496-9461-6.
Grading: Grades will be based on points accumulated. Points will be awarded for
I anticipate a total of around 404 points. Grades may include a plus or a minus. Grade cutoffs will be announced for each exam.
- three hour exams (approximately 50 points each),
- a comprehensive final exam (100 points)
- approximately 10 graded homework sets with total normalized to approximately 100 points
- a graded problem (lecture challenge) each day there is not an exam (2 points each for a total of approximately 54 points)
Homework: There will be approximately ten graded homework sets, typically due on Thursday (except during the weeks of hour exams, and the first and last weeks of class). There will also be some ungraded suggested problems. Most exam questions will be similar to homework problems. Current homework assignments are available by clicking
Current course grades:
Current course grades and all recorded scores are available on Blackboard. If a score has been incorrectly recorded, you should contact the instructor.
- Exam, Thursday, February 12.
- Exam, Thursday, March 12.
- Exam, Thursday, April 23.
- Final exam, the week of May 4-8.
Proposed order of topics:
- Complex numbers and the complex plane (1.1-1.5) (2 weeks)
- Complex functions and mappings (2.1-2.4) (2 weeks)
- Analytic functions (3.1-3.5) (2.5 weeks)
- Elementary functions (4.1-4.3, 4.5) (2 weeks)
- Integration in the complex plane (5.1-5.5) (3 weeks)
- Series and residues (6.1-6.6) (2.5 weeks)
- Conformal mapping (7.1-7.2) (1 week)
Drop Deadline: The last day to drop this course is Friday, March 27, 2015.
Sickness policy: If you are sick the day of an exam, you should contact the instructor by phone on the day of the exam to arrange a timely make-up.
Calculators: A graphing calculator is sometimes useful for this course. I recommend the TI-89, but the TI-83 Plus, TI-84 Plus, TI-85, TI-86, TI-92, and TI Voyage 200 are also acceptable. For help in using your calculator, see your manual or else use
web calculator help.
Please address any special needs or special accommodations with me at the beginning of the semester or as soon as you become aware of your needs. Those seeking accommodations based on disabilities should obtain a Student Academic Accommodation Request (SAAR) form from the Disability Resources (DR) office (515-294-6624). DR is located on the main floor of the Student Services Building, Room 1076.
The Mathematics Department has standard policies on such matters as make-up exams, final exams, student behavior in class, harassment, discrimination, and religious accommodation. You should read these at
Mathematics Department Class Policies .
Objectives for Math 365
The following are the kinds of techniques one should learn from Math 365:
Complex numbers and the complex plane (1.1 - 1.5) (2 weeks)
- Add, subtract, multiply, and divide complex numbers
- Represent points in the plane by complex numbers
- Represent complex numbers in polar form
- Find powers and roots of complex numbers
- Find sets of points in the complex plane solving simple equations
Complex functions and mappings (2.1-2.4) (2 weeks)
- Describe complex functions by their real and imaginary parts
- Compute images of simple sets under complex functions
- Draw parametric curves in the complex plane
- Visualize images of sets under linear mappings
- Visualize images of sets under power functions and roots
- Visualize inverse functions
Analytic functions (3.1-3.5) (2.5 weeks)
- Understand limits in the complex plane
- Understand continuity in the complex plane
- Compute derivatives of complex functions
- Use the Cauchy-Riemann equations to test for analytic functions
- Recognize harmonic functions and find their harmonic conjugates
Elementary functions (4.1-4.3, 4.5) (2 weeks)
- Use the complex exponential function
- Use complex logarithm functions, especially the principal value
- Use complex powers
- Use complex trigonometric and hyperbolic functions
- Solve very simple Dirichlet problems for heat flow or electrical potential
Integration in the complex plane (5.1-5.5) (3 weeks)
- Understand the definition of line integrals (contour integrals) in the complex plane
- Evaluate contour integrals
- Find upper bounds for contour integrals
- Understand the Cauchy-Goursat theorem for closed contour integrals
- Use Cauchy's integral formulas to evaluate closed contour integrals
- Understand consequences of Cauchy's integral formulas
Series and residues (6.1-6.6) (2.5 weeks)
- Understand convergence of sequences and series in the complex plane
- Compute Taylor series in the complex plane
- Compute Laurent series about an isolated singularity
- Understand zeros and poles of complex functions
- Compute residues of a complex function at a pole
- Use Cauchy's Residue Theorem to compute closed contour integrals and related real improper integrals
Conformal mapping (7.1-7.2) (1 week)
- Understand the nature of conformal mappings
- Tell where a complex function is conformal
- Understand angle magnification at a critical point
- Use linear fractional transformations
Information about the instructor: For more information about the instructor, click here.
Last updated January 13, 2015.