An analytic function f(z) whose Laurent series is given by f(z) = sum_(n = - ∞)^∞ a_n (z - z_0)^n, can be integrated term by term using a closed contour γ encircling z_0, integral_γ f(z) d z | = | sum_(n = - ∞)^∞ a_n integral_γ (z - z_0)^n d z | = | sum_(n = - ∞)^(-2) a_n integral_γ (z - z_0)^n d z + a_(-1) integral_γ (d z)/(z - z_0) + sum_(n = 0)^∞ a_n integral_γ (z - z_0)^n d z. The Cauchy integral theorem requires that the first and last terms vanish, so we have integral_γ f(z) d z = a_(-1) integral_γ (d z)/(z - z_0), where a_(-1) is the complex residue.
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