Complex Analysis

  1. Given the following local descriptions of an isolated singularity determine which ones are possible. If such a local form is possible given an explicit example, otherwise prove that no such example exists. 

  2. a)  f:Ω\{w}is holomorphic with an order 4 pole at z=0 but res0f=0.

  3. b)  f : Ω \ {w} is holomorphic with an order 2 pole and    limz→w(z − w)3/2f(z) = 0

  4. c)  Both f and g are holomorphic on Ω \ {w} with poles of order 2 and 3 respectively but f + g has a removable singularity at        z = w

  5. d)  f is holomorpic on Ω\{w1} with an order 1 pole at w1 and g is holomorphic on Ω\{w2} with an order 2 pole at w2, (w1 = w2) but fg has removable singularities at w1 and w2. 

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