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I have a hard time understanding how one can take the following complex integral. $$\int |z|^2 \, dz.$$ Why is it not path independent.

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Absolute value is not an analytic function, so the integral will not be path independent.

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  • $\begingroup$ But wouldn't the hypotheses of the Fundamental Theorem of Calculus and Cauchy's Theorem apply since the function is continuous, has an anti-derivative, and is differentiable on a domain containing the path? $\endgroup$ – Ptheguy Nov 10 '16 at 6:11
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    $\begingroup$ It does not have an anti-derivative, and is not differentiable (in the complex sense). $\endgroup$ – Robert Israel Nov 10 '16 at 6:34

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