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

    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).

[05-May-2016] Voting Results

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

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

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

[11-Mar-2016] Maximum Modulus Theorem

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

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

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

Academic Conduct