Browsing by Author "Erjabo, Abebe"

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    Harmonic Function
    (Addis Ababa University, 2014-02-02) Erjabo, Abebe; Goa, Mengistu (PhD)
    Harmonic functions are closely connected to analytic functions. Since the real and imaginary parts of analytic functions are harmonic functions. The theories of harmonic functions have many interesting features. Among these interesting features are mean value property; maximum principle and these properties with Poisson integral enables to solve Dirichlet problem. The Poisson integral formula shows that if u(z) is harmonic in a disk and continuous on the closed disk, then its value at any interior point is com- pletely determined by its value on the boundary circle. These facts suggest the following two questions: (a) Given a real valued bounded piecewise continuous function u(ei_) on the unit circle, do we obtain a harmonic function v(z) through the Poisson inte- gral; v(z) = 1 2_ Z 2_ 0 u(ei_)Pr(_ 􀀀 t)d_; z = reit? (b) If so, do the boundary values of v(z) agree with u(ei_)? The answer is a_rmative and the unique solution is given by; v(z) = ( pu; ifz 2 D(0; 1) u(z); ifz 2 @D(0; 1) The above two questions leads to the problem of _nding a function that is harmonic in a region and has pre assigned values on the boundary which is known as the Dirichlet problem.