hw8-201141 - f is constant on B(0 1 then f is constant b...

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Homework 8. (1) Prove that if z = x + iy and f ( z ) = p ( | xy | ), then the real part and imaginary part of f satisfy the Cauchy-Riemann equations at z = 0, but f is not differentiable at z = 0. (2) Let G C be an open and connected set and let f : G C be a holomorphic function such that f 0 ( z ) = 0 for all z G . Prove that f is constant on G . (Hint: let S = { z G : f ( z ) = f ( z 0 ) } for some fixed z 0 G and show that S is open and closed. You can use from undergraduate analysis that if a real valued differentiable function has zero derivative on an interval, then that function is constant on the interval.) (3) Let f be holomorphic on the unit disk B (0; 1). a. Prove that if Re
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Unformatted text preview: f is constant on B (0; 1), then f is constant. b. Prove that if e f is constant on B (0; 1), then f is constant. (4) Let G ⊂ C be open and let f be holomorphic on G . Let G * = { z : ¯ z ∈ G } and define f * ( z ) = f (¯ z ) for all z ∈ G * . Prove that f * is holomorphic on G * and express f * ( z ) in terms of f . (5) (Quals ’04) Let G ⊂ C be an open and connected set and let f : G → C be a holomorphic function such that | f ( z ) | = C for all z ∈ G . Prove that f is constant on G . 1...
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