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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)
È 2iz ˘
F(z) = a + (b - a)ImÍ
Îz - 1˚
solves the Dirichlet problem.
Exercises Set 7
p. 907 #9, 11, 18
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
u(x, y) = Ù
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
Ó0 if a ≤ x ≤ b
otherwise . For th...
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