Let $G$ and $H$ be defined by
$G=\mathbb{C}\setminus\{z=x+iy: x\le 0, y=0\}$
$H=\mathbb{C}\setminus\{z=x+iy: x\in \mathbb{Z},x\le 0, y=0\}$
Suppose $f:G\to \mathbb{C} $ and $g:H \to \mathbb{C}$ are analytic functions.Consider the following statements.
I: $\int_{\gamma}f$ is independent of the path $\gamma$ in $G$ joining $-i$ and $i$.
II: $\int_{\gamma}g$ is independent of the path $\gamma$ in $H$ joining $-i$ and $i$.
Then which of the above statements is/are true?
My thoughts:-
For statement I
Let $\gamma_o$ be a fixed path in $G$ joining $i$ to $-i$.
Now if $\gamma$ be any path in $G$ joining $-i$ to $i$ then , let $\Gamma=\gamma+ \gamma_o$ , Then
$\int_{\Gamma}f=0$ (since there are no singularities within $\Gamma$)
$\Rightarrow \int_{\gamma}f+ \int_{\gamma_o}f=0$
$\Rightarrow \int_{\gamma}f=-\int_{\gamma_o}f$, for every $\gamma$ in $G$. So I is true
For statement II
I think it is false because there are paths which contain singularities but I cant write a formal proof.
Please help.Thanks for your time