solutionshw12-201141 - Solutions Homework 12(1(Quals 1995...

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Solutions Homework 12 (1) (Quals 1995) Let f be an entire function on C and assume that | f ( z ) | ≤ A | z | k + B for some constants A, B , integer k and all z C . Prove that f is a polynomial. Solution: Let R > 0 and γ R = Re it , 0 t 2 π . From Cauchy’s integral formula we get for | z | < R f ( n ) (0) = n ! 2 πi Z γ R f ( z ) z n +1 dz, and thus | f ( n ) (0) | ≤ 2 πR n ! 2 π AR k + B R n +1 0 , as R → ∞ for all n > k . Thus f ( n ) (0) = 0 for all n > k . Applying this to the power series expandsion of f around z = 0 we see that f ( z ) = k n =0 c n z n . (2) (Quals 1995) Let G C be a connected open set and let h f n i be a sequence of holomorphic functions on G , which converges uniformly on every compact subset of G to a function f . Prove that f is holomorphic on G . Solution: Let a G and r > 0 such that B ( a ; r ) G . Then h f n i converges uniformly to f on B ( a ; r ) and thus f is continuous on B ( a ; r ). Let Δ B ( a ; r ). Then by Cauchy’s Theorem for starlike open sets we know that R Δ f n dz = 0 for all n . From the uniform convergence it follows that also R
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