I have to solve $\int_L \frac{1}{z} dz$ with L going from $1 \rightarrow1+i \rightarrow-1+i\rightarrow-1-i\rightarrow2-i\rightarrow2$
I tried to solve this by using the parametric form of all of the 5 parts ($1 \rightarrow1+i$, $1+i \rightarrow-1+i$ and so on). And next I calculated all the 5 different integrals with respect to dt.
The solution I became was false. The book gives a hint by saying that I have to split the figure in 2 parts: The part above the real axis and the part under the real axis.
This has to do with the next picture:
When reading in my textbook, they give 2 options. You can take $[0,2\pi)$ or $[-\pi, \pi)$
When L crosses the real axis there would be a problem and the one time you have to take $[0,2\pi)$ and the other time $[-\pi, \pi)$. Can someone explain why?
If there were a sixth part from $2$ to $1$ we would have a loop around the origin. For such a loop the value of the integral is $2\pi i$, as we all know. We therefore have the equation $$\int_L{dz\over z}+\int_2^1{dz\over z}=2\pi i\ ,$$ which leads to $$\int_L{dz\over z}=\log 2+2\pi i\ .$$