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Poisson integral formula 26 we saw that once we know

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Unformatted text preview: point i. (B) Mapping the unit disk into the right-half plane Here, we choose -1Æ0, iÆi and 1ÆÏ Looking at (*), we get (w )(i - Ï) (z + 1)(i - 1) = (w - Ï)(i) (z - 1)(i + 1) To evaluate the left-hand side, we treat it as a limit: lim i - z = 1, zÆÏw - z so we get w (z + 1)(i - 1) i(z + 1) = = i (z - 1)(i + 1) z-1 giving z+1 1+z w== z-1 1-z (C) Mapping a moon-shaped region into the top-half plane Using a map into the top-half plane, solve the following Dirichlet problem: Solution The easiest is to map the given region into a horizontal strip 0 ≤ y ≤ 1 by sending the inner circle to the x-axis and the outer circle to the line y = 1. this means sending the point 1 to Ï. Let us therefore take 1ÆÏ, 0Æ0, and -1Æi.‡ Using (*), we get ‡ For some inexplicable reason, the textbook does something more complicated, requiring a lot more algebra to deal with 25 (w - Ï)(0 - i) (z - 1)(0 + 1) = (w - i)(0 - Ï) (z + 1)(0 - 1) Taking limits and simplifying gives -i z-1 = w-i -(z + 1) Solving for w gives 2iz w= . z-1 We can also check that ±iÆi±1. We now solve the Dir...
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