I am revising complex analysis and am stuck on a question.
I have defined a branch of $Log$ on $\Bbb{C}\setminus(-\infty,0]$, that is $$Log(z)=log(|z|)+iargz, argz \in[-\pi,\pi)$$
But how do I find this integral: $$\int_{[1,i]}Log(z) dz$$ I'm guessing I want to use the Fundamental Theorem of Calculus, as I was just asked to state that, but I don't know how, what is the antiderivative of Log?
You can do integration by parts (similar as in real analysis), either for determining an antiderivative: $$ \int 1 \cdot \operatorname{Log}(z) \, dz = z \cdot \operatorname{Log}(z) \, dz - \int z \frac 1z \, dz = z \cdot \operatorname{Log}(z) - z + C $$ or for computing an integral along the curve: $$ \int_\gamma 1 \cdot \operatorname{Log}(z) \, dz = \bigl[ z \cdot \operatorname{Log}(z)\bigr ]_{z=\gamma(0)}^{z=\gamma(1)} - \int_\gamma z \cdot \frac 1z \, dz = \bigl[ z \cdot \operatorname{Log}(z) - z\bigr ]_{z=\gamma(0)}^{z=\gamma(1)} $$