This concept covers complex line integration and Cauchy's theorem within complex analysis. It establishes that single integrals in the complex plane are inherently path-dependent line integrals, then develops a complex-analytic Fundamental Theorem of Calculus stating that if a continuous function f has an analytic primitive F (F' = f) on a domain, the integral of f along any path depends only on the endpoints. It further proves the ML-inequality (a triangle inequality bound for line integrals), an equivalence between a function having a primitive and its integral vanishing over every closed curve, and Cauchy's theorem, that the integral of an analytic function over the boundary of a domain (extending smoothly to the boundary) is zero.
This concept covers complex line integration and Cauchy's theorem within complex analysis. It establishes that single integrals in the complex plane are inherently path-dependent line integrals, then…