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Please I want to calculate $\int_\gamma \frac{dz}{z}$ with $\gamma=a cos(t)+i b sin(t)$ $t\in[0,2\pi]$ using the complex logarithm function, I mean using the primitive of $\frac{dz}{z}$ which is $ln(z)$, but I don't know how to choose the limits of the integral.

Thank you for your help

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  • $\begingroup$ The technique is the same as the one I used in this answer. $\endgroup$ – Git Gud Nov 5 '14 at 9:42
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Hint:

You have to be careful that $\ln z$ is continuous in the interval $t\in(0,2\pi)$ otherwise the fundamental theorem of calculus does not apply. Luckily there is a lot of freedom to choose $\ln z$ in complex analysis. Given the "correct" form of $\ln z$ you then have by the aforementioned theorem $$\int_\gamma \frac{dz} z = \lim_{t\to2\pi} \ln \gamma(t) - \lim_{t\to0} \ln \gamma(t) .$$

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    $\begingroup$ I see, it s all about the branch cut. Thank you :) $\endgroup$ – Hajar Elhammouti Nov 5 '14 at 10:38

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