Proof of Lemma 7.2. Constructible Primitives of Holomorphic Functions along Paths. Integration of Holomorphic Functions over Arbitrary Paths. Homotopy. Simply-Connected Domains. The Riemann Mapping Theorem

Abstract

We will now prove Goursat’s Lemma stated at the end of Lecture 7. Proof (Lemma 7.2). Fix a triangle $T \subset D$. Using the mid-points of the sides, we split T into four subtriangles $T^{\prime},\ T^{\prime\prime},\ T^{\prime\prime\prime},\ T^{\prime\prime\prime\prime}$ (see Fig. 8.1).