Why isn't integration of dz/z not 0 in a closed contour doesn't this violate antiderivative theorem
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Step-by-step explanation:
Parametrize CR by z=Reiϕ for ϕ∈[0,2π). Then
∮CRdzz−1=iR∫2π0dϕeiϕR−1e−iϕ=i2π
Hope it helps you..
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