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By Ryan Julian


Abstract. Fourier analysis began as an attempt to approximate periodic functions with infinite summations of trigonometric polynomials. For certain functions, these sums, known as Fourier series, converge exactly to the original function. Using this property, along with some other properties of trigonometric polynomials, in particular that they are sums of holomorphic and antiholomorphic 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 plane

Year: 2011
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