math185-hw8sol

math185-hw8sol - MATH 185 COMPLEX ANALYSIS FALL 2007/08...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 8 SOLUTIONS For a ∈ C , r > 0, we write D * ( a,r ) := { z ∈ C | < | z- a | < r } . We write C × = C \{ } . You may use without proof any results that had been proved in the lectures. 1. (a) Find a function with a non-isolated singularity at 0. Solution. The denominator of the function f ( z ) = 1 sin 1 z has zeroes at z n = 1 nπ for all n ∈ Z \{ } . So f has isolated singularities at each z n . Since z = 0 is also a singularity of f and z n → 0, it is a non-isolated singularity. (b) Suppose f has a non-isolated singularity at a ∈ C satisfying the following: (i) ( a n ) ∞ n =1 is a sequence of poles of f that converges to a ; (ii) f is analytic on Ω := C \{ a n | n = 0 , 1 , 2 ,... } . Show that f ( D * ( a ,ε ) ∩ Ω) is dense in C for every ε > 0. Solution. See Exercise 10 on pp. 113–114 of the textbook and its solution on pp. 274. 2. Let Ω ⊆ C be a region. Let a ∈ Ω and f : Ω \{ a } → C be a function with an isolated singularity at a . (a) Prove the converse of Casorati-Weierstraß’s theorem, ie. show that if f ( D * ( a,ε )) is dense in C for every ε > 0 (as long as D * ( a,ε ) ⊆ Ω), then f has an essential singularity at a . Solution. If f has a removable singularity at a , then g ( z ) := ( f ( z ) if z 6 = a, lim z → a f ( z ) if z = a, defines an analytic fucntion on D ( a,ε ) and is thus bounded on D ( a,δ ) for all 0 ≤ δ < ε ....
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This note was uploaded on 08/01/2008 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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math185-hw8sol - MATH 185 COMPLEX ANALYSIS FALL 2007/08...

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