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math185-hw5

math185-hw5 - f 00 ± 1 n ² f ± 1 n ² = 0 for all n ∈...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 5 Notations: D (0 , 1) = { z C | | z | < 1 } ; ∂D (0 , 1) = { z C | | z | = 1 } ; N = { 1 , 2 , 3 ,... } ; f g denotes the composition of f and g and is defined by f g ( z ) = f ( g ( z )). 1. Let f : C C be an entire function. Let a R be an arbitrary constant. (a) Show that if Re f ( z ) a for all z C , then f is constant. (b) Show that if Re f ( z ) a for all z C , then f is constant. (c) Show that if [Re f ( z )] 2 [Im f ( z )] 2 for all z C , then f is constant. (d) Show that if [Re f ( z )] 2 [Im f ( z )] 2 for all z C , then f is constant. (e) Suppose h is another entire functions and suppose there exists an a R , a > 0, such that Re f ( z ) a Re h ( z ) for all z C . Show that there exist α,β C such that f ( z ) = αh ( z ) + β for all z C . [Hint: if f and g are both entire, then so are f g and g f ; find an appropriate g so that you may apply Liouville’s theorem.] 2. Find all entire functions f that satisfy
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Unformatted text preview: f 00 ± 1 n ² + f ± 1 n ² = 0 for all n ∈ N . 3. Let f be a continous function on Γ = ∂D (0 , 1) (note that f is not necessarily analytic). Define a function F : D (0 , 1) → C by F ( z ) = Z Γ f ( w ) w-z dw for all z ∈ D (0 , 1). Show that F is analytic in D (0 , 1). 4. If f is analytic on D (0 , 1) and its derivative satisfies | f ( z ) | ≤ 1 1- | z | for all z ∈ D (0 , 1). Let the power series expansion of f be f ( z ) = ∞ X n =0 a n z n . Show that | a n | < e for all n ∈ N , where e is the base of natural logarithms, ie. e = exp(1). Date : October 18, 2007 (Version 1.0); posted: October 18, 2007; due: October 24, 2007. 1...
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