math185f09-hw7 - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 7 1. Let α,β C and | α | < r . (a) Show that for all n = 0 , 1 , 2 ,..., ± β n n ! ² 2 = 1 2 πi Z ∂D ( α,r ) β n e βz n ! z n +1 dz. (b) Show that X n =0 ± β n n ! ² 2 = 1 2 π Z 2 π 0 e 2 β cos θ dθ. [ Hint : Consider power series expansion of e β/z and apply (a) on z - 1 e β ( z +1 /z ) .] 2. 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.] 3. Let f : C C be an entire function. (a) Suppose there exists α,β C ×
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math185f09-hw7 - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

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