If f(z) is analytic in some simply connected region R, then ∮_γ f(z) d z = 0 for any closed contour γ completely contained in R. Writing z as z congruent x + i y and f(z) as f(z) congruent u + i v then gives ∮_γ f(z) d z | = | integral_γ(u + i v)(d x + i d y) | = | integral_γ u d x - v d y + i integral_γ v d x + u d y.
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