Mathquery

Cauchy's Integral Theorem: Statement :       The theorem is usually formulated for closed paths as follows :        Let U be an open subset of C which is simply connected . Let f: U-->C be a holomorphic  function , and let γ be a rectifiable path in U whose start point is equal to its end point . Then                         ∮ f(z) dz = 0                         γ Proof :       Let us assume that the   partial derivatives  of a holomorphic function are continuous , the Cauchy Integral Theorem can be proved as direct sequence of    Green's Theorem and the fact that the real and imaginary parts of f = u + i v must satisfy the Cauchy - Riemann equations in the region bounded by γ , and moreover in the open neighbourhood U of this region.               Cauchy provided this proof , but it was later proved by Goursat without requiring techniques from vector calculus , or the continuity of partial derivatives .             We can brea