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Theorem: IF f(z) is analytic within and on the Simple closed curve C, Then We have integral_c f(z) dz = 0 Let f(x) be analytic in the disc B'_2 (0) = {z elementof C: 0 < vert z vert < z} and is bounded. That is, vert f(z) vert < M for all z elementof B'_2 (0). Now for 0 < r lessthanorequalto 1, clearly f(x) is analytic within and on C_1. where c_r is circle with rather at 0 and radius r and so f(x) is analytic within and on c_1 = {z elementof C: vert z vert = 1}. Therefore of above theorem integral_c_1 f(z) dz = 0

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