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If we have a function f defined in the complex plane, and a curve C in the complex plane, then the integral of f along C is called a contour integral and is written ∫_C f(z)dz .
Such an integral can be defined as follows: dissect C into n segments with endpoints x₀...x_n. Pick a point z_i in each segment. The integral is the limit, if it exists, of the sum ∑f(z _i ) ( x_i − x_i−1 ) , as n approaches infinity and the length of the longest segment goes to zero.