So frist, define $L(z) = \log(r)+i\theta $ is the holomorphic branch of $\log(z)$ on the cut-plane $\mathbb{C} \setminus (-\infty,0]$ such that $L(1)=0$
Let$[1,i]$ denote the line segment from 1 to $i$ in $\mathbb{C}\setminus (-\infty,0]$.
(i) Determine $\int_{[1,i]}L(z)\,dz$.
[you may assume, without proof, an appropriate version of the fundamental theorem of calculus.]
(ii) Let $H= \{z\in\mathbb{C}:\operatorname{Im}z > 0 \} $. What is the image L(H) of the upper half-plane under L?
Hi, I am kind of confused about integration, should I substitute $L(z)$ by $\log r + i \theta$.
and, could you please give me some hints about part ii as well ? thanks!
the calculus result hinted at is the fact that the antiderivative of $L(z)$ is $zL(z)-z$ on a simply-connected open region excluding the cut you have made. hence the answer is just a matter of substitution and subtraction as with a real definite integral.
for the second question, describe the upper half plane in terms of polar co-ordinates, and see what are the corresponding values for the real and imaginary parts of the $L$-branch of the logarithm