# Harmonic Functions on a Disc and Upper Half Plane

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## Date

2010-06-06

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Addis Ababa University

## Abstract

This seminar report is devoted to the study of harmonic functions on disc and upper half plane. These functions are closely connected to analytic functions since the real and imaginary parts of analytic function are harmonic functions. The study of harmonic functions is important in physics and engineering, and there are many results in the theory of harmonic functions that are not connected directly with complex analysis. However, in this seminar we consider that part of the theory of harmonic functions that grows out of the Cauchy Theory and Mo ̈bius transformation. These functions have many interesting properties like mean value property (MVP) and the maximum principle. One of the most important aspects of harmonic functions is that they arise as functions that solve a boundary value problem for analytic functions, known as the Dirichlet problem. An example is the problem of ﬁnding a function continuous in a closed disk D (upper half plane H)that assumes certain known values on the boundary of the disk D (H) and is harmonic in the interior of the disk D (H). An important tool in the solution is the Poisson Integral formula. This paper is organized in to three chapters.
In the first chapter we consider basic properties of real and complex function theory with some topological concept. It includes basic definition and notation, class of analytic function and continuous function and Cauchy theory.
The second chapter deals with harmonic function locally as real part of analytic function and the interesting property of harmonic functions such that mean value property (MVP) and the maximum principle and its integral representation, that develops from the theory of conformal mapping.
The third chapter deals with the solution of Dirichlet problem on disc and upper half plane and positive harmonic functions.

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## Keywords

Harmonic, Functions, Disc, Upper Half Plane