Now ux y 1 argz b argz a b 1 d argz

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Unformatted text preview: ichlet problem for this. In the horizontal strip in the w-plane, it is the real-valued function given by U(w) = a + (b-a)Im(w) This is the imaginary part of F(w) = a + (b - a)Im(w) So È 2iz ˘ ˙ F(z) = a + (b - a)ImÍ Îz - 1˚ solves the Dirichlet problem. Exercises Set 7 p. 907 #9, 11, 18 Hand-In 1. Express the solution in Example (C) in terms of x and y, verify directly that it satisfies Laplace’s equation , and check that it has the given boundary values given on three different boundary points. 2. Use a conformal mapping to solve the general Dirichlet problem on the annulus: 3. Use a conformal mapping to solve the following Dirichlet problem (see Conformal Mapping #C1 in the book): 8. Poisson Integral Formula 26 We saw that, once we know a potential on the upper half plane, we can find it for any region. So now, the question is to solve Dirichlet's problem on the upper half plane. Theorem (Poisson Integral Formula for H) The (unique) potential u(x, y) on the upper half plane with u(x, 0) = f(x) is Ï yÛ f(t) u(x, y) = Ù dt π ı (x-t)2 + y2 -Ï Sketch of Proof We prove it first for a simple step function of the form f(x) = Ï Ô1 Ì Ô Ó0 if a ≤ x ≤ b otherwise . For th...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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