# general topology – Understanding Path Independence in Complex Analysis, Cauchy Integral Theorem

I don’t understand the path independence in the Cauchy integral theorem which $$0$$ for a closed curve.

For example, consider Cauchy integral theorem on the closed curve $$gamma$$ which is created by gluing $$2$$ non-identical or asymmetrical looking curves $$gamma_1, gamma_2$$. Since, it is on complex plain, then all values of an arbitrary function $$f_1$$ on both $$gamma_1, gamma_2$$ of x-axis is same but the complex component is not, so while integrating on the closed curve $$gamma$$, all the x-components will be cancelled out (since $$gamma_1, gamma_2$$ are clockwise anti-clock-wise curve, respectively), but the y-axis or the complex component will not be a $$0$$!

What am I missing?

I am not saying Cauchy integral theorem is wrong, I am just trying to understand as newbie. I have seen other posts ( like, 1, 2,, 3) on this forum, but didn’t help.