The Generalization of Cauchy Integral Formula

The Cauchy Integral Formula is a fundamental result in complex analysis, providing a powerful method for evaluating analytic functions within a closed contour. This paper explores the Cauchy Integral Formula in its basic and higher-order forms. The basic Cauchy Integral Formula allows for the computation of a function’s value at a point inside a closed contour by an integral over the contour. The higher-order form extends this to compute the derivatives of analytic functions. The Cauchy-Goursat Theorem will be discussed, which generalizes the Cauchy Integral Formula by addressing functions with singularities inside the contour. Additionally, the paper covers the generalized Cauchy Integral Formula involving contours that enclose infinity, providing insights into its application for evaluating functions at such singularities. Practical examples illustrate the theoretical concepts, demonstrating the application of these formulas in various scenarios. This study underscores the significance of these integral formulas in complex analysis and their utility in solving complex function problems.