Cauchy's Integral Theorem

Cauchy's Integral Theorem states that if a function is complex-differentiable (holomorphic) within and on a closed loop (contour), then the integral of that function around the loop is zero. In practical terms, this means that moving along such a path, the total accumulated value of the function's changes cancels out, resulting in no net change. This theorem highlights that for well-behaved functions in a region, their integral depends only on the start and end points, which, for closed loops, are the same point, leading to a zero overall integral.