Math 7339 Spring 2026
(Dec 30, 2025 Version)
[Advanced Analysis/ Potential Theory, 3 Credit hours, CRN31434]
Lectures: Tuesday, Thursday in person in College of Computing 53. Lectures run from 9:30 to 10:45. Lectures will be recorded, and recordings placed in Media Gallery. Students may also watch lectures remotely through Zoom. A link will be sent before classes. Handwritten lectures will be posted in “Files” in Canvas.
Instructor Information
Doron Lubinsky,
lubinsky@math.gatech.edu,
Office hours: Tuesday, Thursday 11:00-12:00 via Zoom or in person in Skiles 237A
General Information
Description
Potential theory is a tool that useful in many areas, including complex analysis, partial differential equations, approximation theory, numerical analysis, and integral equations. We'll study potential theory mainly in the complex plane, but occasionally in higher dimensions. We'll examine energy integrals, logarithmic and other potentials, solutions to certain p.d.e's, and some applications to polynomials and complex analysis. The topics to be covered are:
(1) Cauchy principal value integrals and Hadamard finite part integrals
(2) The Hilbert transform and the Sokotkii-Plemelj formulae
(3) Harmonic functions in the plane
(4) Subharmonic functions in the plane
(5) The maximum principle and Phragmen-Lindelof principle
(6) Potentials, polar sets, and equilibrium measures
(7) The Dirichlet problem, Green’s functions, and Poisson-Jensen formula
(8) Logarithmic capacity and transfinite diameter
(9) Bernstein-Walsh inequality and theorem on rates of polynomial approximation
· Computing Cauchy-Principal Value and Finite Part Integrals
· Applying the Sokotkii-Plemelj formulae
· Applying the maximum principle for subharmonic functions
· Measuring thinness using polar sets and capacity
· Comparing logarithmic capacity and other measures of thinness of sets
· Understanding of subharmonic function theory and potential theory
Required Background
You need to have completed a course on measure theory (such as Math6337) and also have some basic complex analysis skills: such as the Cauchy-Riemann equations, Cauchy's integral formula, the maximum-modulus principle.
Course Requirements & Grading
There will be regular Homework, a midterm test, and a final exam
The items are weighted as follows:
Homework: 45%
Midterm test 25%
Final exam: 30%
No extra credit.
Description of Graded Components
Homework
There will be regular graded homework, throughout the course. Homeworks will be posted on Gradescope, and students must upload their solutions to Gradescope. Tentative dates for homeworks:
Homework 1: Thursday 29 January at 11pm (2 days later than original)
Homework 2: Tuesday 10 February at 11pm
Homework 3: Tuesday 24 February at 11pm
Homework 4: Tuesday 31 March at 11pm
Homework 5: Tuesday 14 April at 11pm
Homework 6: Tuesday 28 April at 11pm
Midterm Test
TENTATIVE date: Tuesday 10 March
It is emphasized that this date may be changed due to unforeseen circumstances.
Final Exam
The final exam will be comprehensive.
The final exam is on Monday May 4 from 8:00am-10:50am, in the classroom.
Makeup Test
This will only be given where there is a medical certificate provided, or approved university absence.
Grading Scale
You can be guaranteed at least the following grades if your percentage lies in the specified range:
A 90-100%
B 80-89%
C 70-79%
D 60-69%
F 0-59%
There will also be a curve that might allow e.g. an A for a grade slightly lower than 90%. This is decided at the end of the course based on the distribution of final percentages in the class.
See http://registrar.gatech.edu/info/grading-system for more information about the grading system at Georgia Tech.
Course Materials
Course Text
Potential Theory in the Complex Plane, by Tom Ransford, Cambridge University Press, 1995.
Additional Materials/Resources:
Chapter 16 of Analytic Function Theory, Vol. 2, by E. Hille, Chelsea, New York, 1987
Volume 3 of Applied and Computational Complex Analysis, by P. Henrici, Wiley Interscience.
There will be a weekly announcement posted in Canvas.
Piazza Link: https://piazza.com/gatech/spring2026/math7339a/homeurs