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Notice that and are conjugate harmonic functions in

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Unformatted text preview: mula works for this simple function. BUT: (1) Potential functions are additive, as are integrals (2) Any function can be approximated arbitrarily closely by a linear combination of steps functions. Therefore, it works for all integrable functions. QED. Ï Ôx if -1 ≤ x ≤ 1 Example Solve Dirichlet's problem on H with u(x, 0) = Ì0 otherwise Ô Ó 1 Û t Solution We get u(x, y) = Ù ı (x-t)2 + y2 dt -1 Substituting s = x - t transforms this to x+1 Û x-s u(x, y) = Ù 2 ı s + y2 ds x-1 1 1 Èy 22 -1Êx-tˆ˘ = Í ln[(x-t) +y ] - xtan Á ˜˙ Ë y ¯˚ π Î2 -1 ÈÊ ˆ˘ 22 y Í Á(x-1) +y ˜˙ x È -1Êx+1ˆ -1Êx-1ˆ˘ Í Á ˜ Á ˜˙ = ÍlnÁ 2 2˜˙ + π Îtan Ë y ¯ - tan Ë y ¯˚ 2π Î Ë(x+1) +y ¯˚ Note We can also use this method for regions other than H. Take a conformal map onto H from another region, see what it does on the boundary, solve it on H as above, and then compose it with the conformal map to get the solution (see the homework) Exercise Set 8 p. 916 #1, 3 (Express u(x, 0) as a sum of step functions and use (I) on each one (rather then do...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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