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Unformatted text preview: (b) If u,v C 2 ( R ), ie. all second order partial derivatives of u,v exist and are continuous on R , then u xx + u yy = 0 and v xx + v yy = 0 on R (in this case u and v are called harmonic functions ). 5. Let C be a region. Let f = u + iv be analytic on . (a) Determine all f for which g = u 2 + iv 2 (ie. g ( z ) := [ u ( x,y )] 2 + i [ v ( x,y )] 2 ) is also analytic on . (b) Show that if there exist , C such that u + v is constant on , then f must be constant on . (c) Show that if for all z , either f ( z ) = 0 or f ( z ) = 0, then f must be constant on . Date : September 19, 2007 (Version 1.1); posted: September 17, 2007; due: September 24, 2007. 1...
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This note was uploaded on 08/01/2008 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.
- Fall '07