Corra Compares Contour Integrals, A proof of the Cauchy-Goursat theorem

This articles assumes familiarity with the concepts of complex functions, their continuity, differentiability and contour integration at an early undergraduate level.  It is aimed at a student taking an introductory course on complex analysis. It was my entry for SoMEπ, held in 2024, and was ranked 21 of 44 non video entries.

Corrections:

• The energy function defined in (2) on page 1 should be a function from the complex plane to the complex plane, and not from the integers to the integers. Thanks anonymous SoME reviewer for pointing out the mistake.

• Also on page 1, (3) initially had an error which implied the contour integral is a real number. The contour integral is a complex number and the error is now fixed. Thanks anonymous SoME reviewer for pointing this out.

• The extension of the proof to arbitrary simple closed contours does not work as presented here as I have not shown that despite the contribution of each partial square becoming smaller, the sum of all the contributions is still not significant. Thanks anonymous Redditor for highlighting this error.

Found a mistake? Let me know!

Acknowledgements:

I was first exposed to this theorem in my complex analysis course. The proof for triangles is based on the discussions in the classes.

The extension to arbitrary simple closed contours is my original attempt.

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