Dr. Tobias Mai

Potential Theory in the Complex Plane

(Summer Semester 2020)


News

Lecture

Monday, 14 - 16    (online, via the platform Zoom)
The link will be sent to the registered participants on the day of the lecture.


Potential theory is concerned with the study of harmonic functions, namely solutions f of Laplace's equation Δ f ≡ 0. The origins of this field lie in mathematical physics of the 19th century when it was noticed that harmonic functions play an important role in modeling gravitation and electrostatics.
Conceptually, potential theory is very similar to complex analysis where holomorphic functions are the objects of interest. In fact, harmonic and holomorphic functions have equally strong properties, however, the techniques used to study them are often substantially different and each of those theories exhibits its own phenomenas.
There are nonetheless close connections between these important classes of functions, which become visible in the particular case of the complex plane. On the one hand, this makes appear some results about harmonic functions as reflections of corresponding statements about holomorphic functions. On the other hand, these relations shed a new light on complex analysis and give access to new powerful tools, which had some great impact on developments in that area. Among others, this applies to questions about orthogonal polynomials and polynomial approximations of holomorphic functions.

In this lecture, which can be seen as a continuation of the core course Funktionentheorie, we want to give an introduction to potential theory with an eye towards its connections to and applications in complex analysis.

Lecture announcement

For further information, please contact Tobias Mai.

Lecture notes

Lecture Notes    (final version)
Lecture Notes (additional material)

A typed version of the lecture notes, kindly provided by Friedrich Günther,
is available here (final version).

Tutorials

The tutorials (exercise sessions) take place every second week on

Monday, 13:00 - 14:30    (online, via the platform Zoom)

The link will be sent to the registered participants on the day of the event.
The tutorials are followed by the lecture which begins on those days at 14:30.
In every other week, the lecture starts at the ordinary time 14:15.

There will be no further problem session.

Assignment 1 A    (for the tutorial on Monday, May 18)
Assignment 1 B    (for the tutorial on Monday, May 18)
    Solutions for Assignment 1 A & B
Assignment 2 A    (for the tutorial on Monday, June 8)
Assignment 2 B    (for the tutorial on Monday, June 8)
    Solutions for Assignment 2 A & B
Assignment 3 A    (for the tutorial on Monday, June 22)
Assignment 3 B    (for the tutorial on Monday, June 22)
    Solutions for Assignment 3 A & B
Assignment 4 A    (for the tutorial on Monday, July 6)
Assignment 4 B    (for the tutorial on Monday, July 6)
    Solutions for Assignment 4 A & B
    Application of Exercise 2, Assignment 4 A

You do not have to hand in written solutions. We will discuss the problems in
the tutorials, but you should be prepared then to present your solution.

References

The lecture notes of the lecture Funktionentheorie given in the Summer term 2017 can be found here.

More references will be provided during the course of the lecture.


updated: 22 September 2020   Tobias Mai