P2) a) Find the Green's function for the half-plane {(x, y) | y > 0}. Justify...

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P2) a) Find the Green's function for the half-plane {(x, y) | y > 0}. Justify your answer. b) Use part a) to solve the Dirichlet problem for the Laplace's equation in the half-plane, with boundary values u(x, 0) = h(x). c) Use part b) to find the solution of the Dirichlet problem for h(x) = 1 for all x. P2) a) Find the Green's function for the half-plane {(x, y) | y > 0}. Justify your answer. b) Use part a) to solve the Dirichlet problem for the Laplace's equation in the half-plane, with boundary values u(x, 0) = h(x). c) Use part b) to find the solution of the Dirichlet problem for h(x) = 1 for all x.