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I could use some help in calculating $$\int_{\gamma} \bar{z} \; dz,$$ where $\gamma$ may or may not be a closed curve. Of course, if $\gamma$ is known then this process can be done quite directly (eg. Evaluate $\int \bar z dz$), though that is not the case here.

For instance, if $\gamma$ is indeed a closed curve then I can show the above integral is purely imaginary, but still don't know how to explicitly calculate it.

Thanks!

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