Course Syllabus for

Potential Theory in the Complex Plane
Potentialteori i det komplexa planet

FMA305F, 7.5 credits

Valid from: Autumn 2018
Decided by: Professor Thomas Johansson
Date of establishment: 2018-11-15

General Information

Division: Mathematics
Course type: Third-cycle course
Teaching language: English


To give a postgraduate student in, e.g., complex analysis, harmonic analysis and partial differential equations good knowledge about a number of basic concepts and tools in modern analysis.


Knowledge and Understanding

For a passing grade the doctoral student must

Course Contents

Harmonic functions of two variables: Harmonic and holomorphic functions, the Dirichlet problem on the disc, positive harmonic functions. Subharmonic functions: Upper semicontinuous functions, subharmonic functions, the maximum principle, criteria for subharmoniciy, integrability, convexity, smoothing. Potential theory: Potentials, polar sets, equilibrium measures, upper semicontinuous regularization, minus-infinity sets, removable singularities, the generalized Laplacian, thinness. The Dirichlet problem: Solution of the Dirichlet problem, criteria for regularity, harmonic measure, Green's functions, the Poisson-Jensens formula. Capacity: Capacity as a set function, computation of capacity, estimation of capacity, criteria for thinness, transfinite diameter.

Course Literature

Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, 1995. ISBN 9780521466547.

Instruction Details

Type of instruction: Lectures

Examination Details

Examination format: Seminars given by participants
Grading scale: Failed, pass

Admission Details

Assumed prior knowledge: Analytic functions and basic measure theory.

Further Information

Contacts: Jacob Stordal Christiansen ( and Frank Wikström ( .

Course Occasion Information

Contact and Other Information

Course coordinators:

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