# calculate $\int_\gamma \frac{dz}{z}$ using $\ln(z)$ function

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.

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 $$\int_\gamma \frac{dz} z = \lim_{t\to2\pi} \ln \gamma(t) - \lim_{t\to0} \ln \gamma(t) .$$