ps13 - A 2/z has a relation with the Milne-Thompson Circle...

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MATH 406: Problem Set 13 due Wednesday, April 27, 2005 1. (a) Without using any facts about electromagnetism (except that the electrostatic potential is harmonic), guess the functional dependence of the potentional φ for a purely radial electric field, which is due to an isolated electric charge (such as an electron). Define your reasoning clearly, and specify the associated analytic function in C . (b) Using superposition, find the total electric potential due to this charge in a spa- tially constant field E 0 . 2. (a) Show that the Zhukovsky mapping w = z + (
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Unformatted text preview: A 2 /z ) has a relation with the Milne-Thompson Circle Theorem for a circle of radius R = A . (b) (based on Fisher 4.2.15) Show that the Zhukovsky map transforms flow past a circle with R < A to flow past an ellipse with axes a and b < a (see Fisher p.282). (c) Describe what happens when A → R . Why is this different from your answer to part (a)? HINT: it is. 3. Find the Fourier transform ˆ u ( ω ) for the following functions of time t : a) u ( t ) = 5 1 + t 2 b) u ( t ) = 1 20 + 8 t + t 2 (F 5.1.5) Version 1.0...
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