Applied Complex Analysis

Math 715: Applied Complex Analysis

Days & Times 	Room 	Instructor 	Meeting Dates
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MWF 10am-10:50am  PE101   Eric Olson  01/19/2016 to 05/03/2016

Course Information

Instructor:
Eric Olson
email:
ejolson at unr edu
Office:
Monday and Wednesday 1pm DMS 238 and by appointment.
Homepage:
http://fractal.math.unr.edu/~ejolson/715

Texts:
John Conway, Functions of One Complex Variable, Second Edition, Springer, 1978.
Other Complex Analysis Books:
A.I. Markushevic, Theorey of Functions of a Complex Variable, Second Edition, AMS, 2005.

Lars Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill, 1979.

Carrier, Krook and Pearson, Functions of a Complex Variable: Theory and Technique, SIAM, 2005.

M.A. Evgrafov, Analytic Functions, Dover, 1966.

Announcements

[06-May-2016] Final Exam

The final exam will be Friday, May 6 at 10:15am in our usual room PE101. Please study
    Contour integral problems C and D.
    Conformal mapping problems A and C.
    Two additional problems chosen from the submitted problems.
    The proof of the Poisson Formula up to equation (2-24).
Again note that the last homework can be turned in at the final or the following week Monday, May 9.

[05-May-2016] Voting Results

The conformal mapping problems have now been ranked. The results are

A=10, B=13, C=6, D=12, E=17

Also, please note that the last homework due date has been postponed to Monday, May 9.

[02-May-2016] Final Exam Material

Voting results for contour integrals are

A=19, B=24, C=17, D=12, E=24 and F=30

where lower scores indicate higher preference. I have made scans of the sample problem solutions and some recently submitted clarifications. Please let me know if you find any errors. To rank the conformal map problems send an email to ejolson at unr.edu with your rankings in the format

1=?, 2=?, 3=?, 4=? and 5=?

where each the question mark stands for one of the letters A, B, C, D or E. Note that 1 means most preferred and 5 means least preferred. Use each letter only once.

[22-Apr-2016] Maple Worksheet

We discussed conformal transformations in class and demonstrated that analytic functions preserve angles in the complex plane using Maple. If you have access to Maple you may download the live worksheet for further experimentation. A version of the same document in portable document format is also available.

[11-Mar-2016] Maximum Modulus Theorem

Here are my lecture notes of the proof of the maximum modulus theorem.

[18-Mar-2016] Homework 1 and Quiz 1

The day before spring break we will have a quiz. I have created a review sheet to help you study. The first homework assignment will also be due at this time.

Homework

	HW #1  Carrier, Krook and Pearson (solutions)
           pg 29 # 1
           pg 35 # 1
           pg 37 # 8abc
           pg 40 # 1, 2, 3, 6, 7

    HW #2  Carrier, Krook and Pearson (solutions)
           pg 82 # 1, 2
           pg 89 # 8, 10, 14
           pg 98 # 1, 2

Course Description

The main text we shall follow, Functions of One Complex Variable by John Conway, is self-contained and suitable for students who have not had previous experience with complex analysis. Note also that neither Math 713 nor Math 714 are prerequisite: differentiability of a complex function imposes significantly more structure than in the real case and immediately leads to a distinctly different theory that is almost magical in strength and applicability.

Complex analysis is used in many fields of mathematics and science. In pure mathematics, the location in the complex plane of the zeros of the Riemann Zeta function constitutes one of the open millennium prize problems whose solution is worth $1,000,000 from the Clay Mathematics Institute. In quantum physics, Regge theory is the study of the analytic properties of scattering as a function of angular momentum, where the angular momentum is not restricted to be an integer but is allowed to take any complex value. In my field of fluid dynamics estimates on the radius of analyticity of the complexified-in-time Navier-Stokes equations can be used to obtain exponential decay of the energy spectrum of the flow.

We shall start with the complex numbers and show that complex functions have a remarkable property related to differentiability: one-times differentiable implies infinitely many times differentiable. This interplay between differentiability and the structure of the complex numbers then leads to many powerful results. For example, the Cauchy residue theory is a powerful computational tool used in probability, Fourier analysis and many other fields. Liouville's theorem leads to a simple proof of the fundamental theorem of algebra. Complex analysis techniques yield the Cauchy-Kovalevskaya theorem on the existence and uniqueness of solutions to partial differential equations. We seek to understand these results and their applications.

Equal Opportunity Statement

The Mathematics Department is committed to equal opportunity in education for all students, including those with documented physical disabilities or documented learning disabilities. University policy states that it is the responsibility of students with documented disabilities to contact instructors during the first week of each semester to discuss appropriate accommodations to ensure equity in grading, classroom experiences and outside assignments.

Academic Conduct

Bring your student identification to all exams. Work independently on all exams and quizzes. Behaviors inappropriate to test taking may disturb other students and will be considered cheating. Don't talk or pass notes with other students during an exam. Don't read notes or books while taking exams given in the classroom. You may work on the programming assignments in groups of two if desired. Homework may be discussed freely. If you are unclear as to what constitutes cheating, please consult with me.
Last updated: Tue Aug 21 12:11:04 PDT 2012