Cauchy's integral formula

Cauchy's integral formula is a fundamental concept in complex analysis that relates the value of a holomorphic (complex differentiable) function inside a circular region to an integral around its boundary. Essentially, it states that if you know how the function behaves along a closed loop, you can determine its value at any point inside that loop by integrating the function divided by the difference between the variable point and the point of interest. This formula highlights the deep connection between a function's values inside a region and its boundary behavior, emphasizing how complex functions are tightly constrained by their values on the boundary.