Harmonic Function

Harmonic functions are closely connected to analytic functions. Since thernreal and imaginary parts of analytic functions are harmonic functions.rnThe theories of harmonic functions have many interesting features. Amongrnthese interesting features are mean value property; maximum principle andrnthese properties with Poisson integral enables to solve Dirichlet problem.rnThe Poisson integral formula shows that if u(z) is harmonic in a disk andrncontinuous on the closed disk, then its value at any interior point is com-rnpletely determined by its value on the boundary circle.rnThese facts suggest the following two questions:rn(a) Given a real valued bounded piecewise continuous function u(ei_) on thernunit circle, do we obtain a harmonic function v(z) through the Poisson inte-rngral;rnv(z) =rn1rn2_rnZ 2_rn0rnu(ei_)Pr(_ À€€ t)d_; z = reit?rn(b) If so, do the boundary values of v(z) agree with u(ei_)?rnThe answer is a_rmative and the unique solution is given by;rnv(z) =rn(rnpu; ifz 2 D(0; 1)rnu(z); ifz 2 @D(0; 1)rnThe above two questions leads to the problem of _nding a function that isrnharmonic in a region and has pre assigned values on the boundary which isrnknown as the Dirichlet problem.

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