Complex Analysis

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.

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

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

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

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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