### Abstract:

The most important in the theory of harmonic functions is that of finding a harmonic function with given boundary values; it is known as the Dirichlet problem. The Dirichlet problem consists in determining all regionsG such that for any continuous function U;→ℝ there is a continu(ous function→ℝ such that
)=(Z) for inand is harmonic on To study the Dirichlet problem we are concerned with two main questions. Does a solution exists, and if so, is it uniquely determined by the given boundary values? To solve the boundary value problem the major tool is to develop the Poisson integral formula which is integral representation of harmonic functions. We are, in fact, able to show the Poisson integral of harmonic functions on a disk and upper half plane. The theory of harmonic function on the upper half planedevelop by transforming the theory of harmonic function of a unit disc on to upper half plane by conformal mapping.
The purpose of the project is to study the integral representation of harmonic functions in a disc and upper half plane and then compiled as reading material. It means that for a harmonic function on a disc≔ {∈ℂ:||<} has a Poisson integral formula =12and for the upper half plane using Mbius transformation which maps to upper half plane . Then we have to develop the integral representation of harmonic functions on the upper half plane For a harmonic function∈ℂ:on the upper half plane ≔ {>0} has a Poisson integral formula and denoted by.
Key words: Harmonic function, Poisson integral, Dirichlet problem