380_hw_9_sp10[1]

380_hw_9_sp10[1] - U ( x; 0) = 2 for x > and U ( x; 0) =...

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Math 380 Assignment 9 due 4/14 Section 4.6 # 5, 6, 17 Section 5.1 # 11 Section 4.7 # 11 Additional problems A. the integral.) B. De±ne the function u ( x; y ) on the upper half-plane as follows: u ( x; y ) = 8 < : 1 Arg ( x + iy ) when y > 0 0 when y = 0 and x & 0 1 when y = 0 and x < 0 : 1. Verify that u is harmonic for y > 0 : 2. Show that when x 6 = 0 , lim y ! 0 + u ( x; y ) = u ( x; 0) : 3. Use u to ±nd a function U ( x; y ) that is harmonic in the upper half-plane and has boundary values
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Unformatted text preview: U ( x; 0) = 2 for x > and U ( x; 0) = ± 1 for x < : Hint: you can take U ( x; y ) = a + bu ( x; y ) with appropriate constants a and b: C. Let D 1 and D 2 be complex domains. Suppose f : D 1 ! D 2 is analytic and & : D 2 ! R is harmonic. De±ne g by g ( x; y ) = & ( u ( x; y ) ; v ( x; y )) where f ( x + iy ) = u ( x; y ) + iv ( x; y ) : Verify that g is harmonic on D 1 : 1...
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This note was uploaded on 09/16/2011 for the course MATH 380 taught by Professor Staff during the Spring '11 term at S.F. State.

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