Question

9. Jordan Contour Theorem for Step Paths. Define a crossing of a closed step path σ :[a, b] → D to be a point σ(t1) in the image of σ, such that σ(t1) = σ(t2)(a ≤ t1 < t2 < b) Say that σ is simple if it has no crossings. Prove the Jordan Contour Theorem for step paths: Let σ be a simple closed step path in C. Then (i) C \ˆσ has precisely two connected components. One (call it I(σ)) is bounded, the other (call it O(σ)) is unbounded. (ii) Either w(σ, z) = 1 for all z ∈ I(σ), or w(σ, z) =−1 for all z ∈ I(σ). (iii) If z ∈ O(σ), then w(σ, z) = 0.

Answer 1

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