Cauchy--Goursat theorem

Coming from complex integration.
Theorem
If $f$ is a holomorphic function on a simply connected open set $U$ and $\gamma$ is a closed loop then

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Proof
It uses a fundamental property of the Polya vector field. The Polya vector field has null divergence an null rotational when the function $f$ is holomorphic (using Cauchy-Riemann equations) , and on the other hand we have:

with $W$ the work and $F$ the flux.
Source: this video.
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Visualization

Coming from complex integration#Visualization. For the function $f(z)=1/z$ see @Needham1997Visual page 391. Also, it is shown why