homework108c7

homework108c7 - C , suppose that for all holomorphic...

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HOMEWORK 7 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, MAY 26, 2009 (1) Show that there do not exist non-constant harmonic functions u : C R with u ( z ) 0 for all z C . Do there exist non-constant subharmonic func- tions with this property? What about non-constant subharmonic functions u satisfying u ( z ) 0 for all z C ? (2) Compute a Poisson integral formula for the upper half plane H = { z | = ( z ) > 0 } . That is, given f : R R , give a formula of the form h ( z ) = Z R f ( x ) K ( z,x ) dx, such that h ( z ) is harmonic on H and the limit lim z x h ( z ) = f ( x ) for all x R . (3) Given f : U R , where U is some open subset of
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Unformatted text preview: C , suppose that for all holomorphic functions F : D (0 , 1) U the function f F is subharmonic. Prove that f itself is subharmonic. (4) Let f C 2 ( U ) for some open U C . Suppose f is subharmonic. Show that f 0 on U . (Hint: Taylor expansion) Now suppose f > 0 on U , show that f is subharmonic. (Hint: f has no local maxima) Finally suppose f 0 on U , show that f is subharmonic. (Hint: Consider ( f + | z | 2 )). 1...
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This document was uploaded on 12/08/2010.

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