math185-hw8sol - MATH 185 COMPLEX ANALYSIS FALL 2007\/08 PROBLEM SET 8 SOLUTIONS For a C r > 0 we write D(a r:={z C | 0 < |z a| < r We write C = C{0 You

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

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 8 SOLUTIONS 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. Solution. The denominator of the function f ( z ) = 1 sin 1 z has zeroes at z n = 1 for all n Z \{ 0 } . 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 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. 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 . So f is bounded on D * ( a, δ ) (since f = g on D * ( a, δ )) and so f ( D * ( a, δ )) is not dense in C .
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  • Fall '07
  • Lim
  • Math, lim, Essential singularity, removable singularity

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