916 1 3 express ux 0 as a sum of step functions and

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Unformatted text preview: is simple kind of function, we take Èz - b˘ 1 1 ˙ u(x, y) = [Arg(z - b) - Arg(z - a)] = ArgÍ (I) Îz - a˚ π π (Note that, since we are on the upper half plane, all arguments are between 0 and π, and the above equality holds because none of the angles in question cross zero.) The reason this works is: (1) u is the imaginary part of an analytic function, so that u does satisfy Laplace’s equation. (2) For z on the real axis outside of [a, b], (z-b)/(z-a) is a real positive number, and so its argument is zero. (3) For z on the real axis between a and b, (z-b)/(z-a) is a real negative number, and so its argument is π. Now u(x, y) = 1 [Arg(z - b) - Arg(z - a)] π b 1Û d = Ù [Arg(z - t)] dt πıdt a Now we use a little trick: Arg(z – t) z–t z y t x Èy˘ -1 ˙ Arg(z - t) = tan Í Îx - t˚ Therefore, b Û 1Ù d -1È y ˘ ˙ u(x, y) = ı tan Í Îx - t˚ dt π dt a 27 b 1Û y =Ù dt 2 πı (x-t) + y2 a Ï yÛ f(t) Ù = dt , π ı (x-t)2 + y2 -Ï because of the definition of f(t). Therefore, the for...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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