Proof of the Cauchy-Goursat Theorem

This repository presents a full proof of the Cauchy-Goursat Theorem, a foundational result in complex analysis. Authored by Marouf Haider and Chaya Aimen, the document breaks down key lemmas and builds toward a full proof of the theorem, making it accessible for learners and enthusiasts.

Summary

• The Cauchy-Goursat theorem states that for any holomorphic function defined in a simply connected domain, the complex integral over any closed contour within that domain is zero.
• The proof avoids reliance on the continuity of the derivative and instead uses geometric contour constructions and properties of holomorphic functions.

Structure

1. Lemma 1.2 – Characterizes the equivalence between having a primitive and vanishing closed integrals.
2. Lemma 1.4 to 1.6 – Addresses integrals along:
   • Triangular contours
   • Polygonal contours
   • Arbitrary simply closed curves within open disks
3. Theorem 1.3 – Assembles the lemmas to establish the full Cauchy-Goursat theorem.

File Contents

• Cauchy_Goursat_Theorem.pdf – Contains the complete proof and discussion.
• README.md – You're looking at it.

Keywords

Complex Analysis, Cauchy-Goursat, Holomorphic Functions, Contour Integration, Path Independence

Contact

• Marouf Haider – haider.marouf@nhsm.edu.dz
• Chaya Aimen – aimen.chaya@nhsm.edu.dz

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Feel free to open issues for suggestions or comments. Contributions are welcome.