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In the context of the complex-analytic structure within the open unit disk, that was established in a previous paper, here we establish a simple generalization of the Cauchy-Goursat theorem of complex analytic functions. We do this first for the case of inner analytic functions, and then generalize the result to all analytic functions. We thus show that the Cauchy-Goursat theorem holds even if the complex function has isolated singularities located on the integration contour, so long as these are all integrable ones. 

singularity classification, Fourier coefficients, Taylor series, analytic functions, conformal transformations, Cauchy-Goursat theorem.