math185-hw8 - MATH 185 COMPLEX ANALYSIS FALL 2007\/08 PROBLEM SET 8 For a C r > 0 we write D(a r:={z C | 0 < |z a| < r We write C = C{0 You may use

math185-hw8 - MATH 185 COMPLEX ANALYSIS FALL 2007/08...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 8 For a C , r > 0, we write D * ( a, r ) := { z C | 0 < | z - a | < r } . We write C × = C \{ 0 } . 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. (b) Suppose f has a non-isolated singularity at a 0 C satisfying the following: (i) ( a n ) n =1 is a sequence of poles of f that converges to a 0 ; (ii) f is analytic on Ω := C \{ a n | n = 0 , 1 , 2 , . . . } . Show that f ( D * ( a 0 , ε ) Ω) is dense in C for every ε > 0. 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 . (b) Show that if f has a pole or an essential singularity at a , then e f has an essential singularity
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Unformatted text preview: at a . 3. Let Ω ⊆ C be a region. Let a ∈ Ω and f : Ω \{ a } → C be a function with an isolated singularity at a . Suppose for some m ∈ N and ε > 0, Re f ( z ) ≤ -m log | z-a | for all z ∈ D * ( a,ε ). Show that a is a removable singularity of f . 4. Let f : C × → C be analytic on C × with a pole of order 1 at 0. Show that if f ( z ) ∈ R for all | z | = 1, then for some α ∈ C × and β ∈ R , f ( z ) = αz + α 1 z + β for all z ∈ C × . 5. Let f : D * (0 , 1) → C be analytic. Show that if | f ( z ) | ≤ log 1 | z | for all z ∈ D * (0 , 1), then f ≡ 0. Date : November 15, 2007 (Version 1.0); posted: November 15, 2007; due: November 21, 2007. 1...
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  • Fall '07
  • Lim
  • Math, Essential singularity, singularity, isolated singularity

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