Let $f$ be analytic on $\mathbb{H}$ and continuous and bounded on $\overline{\mathbb{H}}.$ Let $R$ be a closed rectangle in $\overline{\mathbb{H}}$ such that one side of $R$ falls on the real line $\mathbb{R}.$
- Is it still true that $$\int_R f(z)dz=0?$$
- Can we make a domain $\mathcal{D}\supset \mathbb{H},$ which also contains $\mathbb{R}$ such that $f$ is analytic on $\mathcal{D}?$
NB: $\mathbb{H}=\{z\in\mathbb{C}: \Im{z}>0\}$