Question

1. Let D ⊂ C be an open set and let γ be a circle contained in D. Suppose f is holomorphic on D except possibly at a point z0 inside γ. Prove that if f is bounded near z0, then

f(z)dz = 0. γ

2. The function f(z) = e1/z has an essential singularity at z = 0. Verify the truth of Picard’s great theorem for f. In other words, show that for any w ∈ C (with possibly one exception) there is a sequence z1,z2,... with zk → 0 and f(zk) = w for all k.

Answer 1