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homework108c8 - that ∑ n ≥ | α n-β n |< ∞ Show...

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HOMEWORK 8 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, JUNE 2, 2009 (1) (a) Give a sequence { a n } such that n 0 a n converges, but Q n 0 (1+ a n ) does not. (b) Give a sequence { a n } such that Q n 0 (1+ a n ) converges, but n 0 a n does not. (c) Show that if n 0 | a n | 2 converges, then n 0 a n converges if and only if Q n 0 (1 + a n ) converges. Remember the definition of convergence of an infinite product as given in class (or in the book): Q n 0 (1 + a n ) is said to converge if and only if there exists an N 0 such that a n 6 = - 1 for n > N 0 and such that lim N →∞ Q N n = N 0 +1 (1 + a n ) exists and is not equal to zero (or infinity). (2) Recall sinh( z ) = ( e z - e - z ) / 2 and cosh( z ) = ( e z + e - z ) / 2. Give the Hadamard product formula for sinh( z ) and cosh( z ), and, using these, prove the doubling formula for sinh(2 z ). (3) (This problem counts double) Suppose sequences α n and β n are given, such
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Unformatted text preview: that ∑ n ≥ | α n-β n | < ∞ . Show that Q n ≥ z-α n z-β n converges normally on C \ { β n } (i.e. the complement of the closure of the set of β n ’s). Now let U ⊂ C be open, and take a sequence { c j } ∈ ∂U , which is dense in ∂U . Let { a j } be the sequence a j = c j-b √ j-1 c 2 , i.e. c 1 ,c 1 ,c 2 ,c 3 ,c 1 ,c 2 ,c 3 ,c 4 ,c 5 ,c 1 ,... and assume b j ∈ U are such that ∑ n ≥ | a n-b n | < ∞ . Show that f ( z ) = Q n ≥ z-b j z-a j is holomorphic on U and no p ∈ ∂U is regular for f , i.e. for any p ∈ ∂U there does not exist any r > 0 and ˆ f : U ∪ D ( p,r ) → C holomorphic, with ˆ f ± ± ± U = f . 1...
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