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**Unformatted text preview: **(b) If u,v ∈ C 2 (Ω R ), ie. all second order partial derivatives of u,v exist and are continuous on Ω R , then u xx + u yy = 0 and v xx + v yy = 0 on Ω R (in this case u and v are called harmonic functions ). 5. Let Ω ⊆ C be a region. Let f = u + iv be analytic on Ω. (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 Ω. (b) Show that if there exist α,β ∈ C × such that αu + βv is constant on Ω, then f must be constant on Ω. (c) Show that if for all z ∈ Ω, either f ( z ) = 0 or f ( z ) = 0, then f must be constant on Ω. Date : September 19, 2007 (Version 1.1); posted: September 17, 2007; due: September 24, 2007. 1...

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- Fall '07
- Lim
- Math, Calculus, Derivative, fixed positive integer