Cauchy's integral theorem is a fundamental result in complex analysis, stating that if a function is holomorphic (complex differentiable) within and on a simple closed curve, then the integral of that function over the curve is zero. This means that the value of the integral does not depend on the specific path taken within the region, as long as the path does not cross any singularities. The theorem is named after the French mathematician Augustin-Louis Cauchy, who made significant contributions to the field of complex analysis. Cauchy's integral theorem lays the groundwork for many other important results, including Cauchy's integral formula, which provides a way to evaluate integrals of holomorphic functions.