Legesse, Hunduma (PhD)Yeneabat, Shiferaw2018-07-182023-11-042018-07-182023-11-042017-09http://etd.aau.edu.et/handle/123456789/9099In this thesis, using the properties of convergence of Fourier series and some other properties of trigonometric polynomials, in particular that they are sums of holomorphic and anti-holomorphic functions, we are able to solve the Dirichlet problem on the Disc. Then, applying the result for the unit Disc along with a couple of Möbius transformations, we are able to solve the Dirichlet problem on the upper half planeenFourier analysis and the DirichletThesis