Let $U$ be an open neighbourhood of $\overline{\mathbb{H}}=\{z\in\mathbb{C}:\Im(z)\ge0\}$ and $g:U\rightarrow\mathbb{C}$ meromorphic with a finite number of poles in $\mathbb{H}=\{z\in\mathbb{C}:\Im(z)>0\}$ and one pole $a\in\mathbb{R}$ with order one. Assume that $\lim_{z\rightarrow\infty} g(z)=0$ and put $f(z)=g(z)e^{iz}$.
I want to show:
$$\lim_{\rho\to0}\int_{\gamma_{\rho}}f(z)dz=-\pi i Res(f,a) ,$$ where $\gamma_{\rho}:[0,\pi]\to \mathbb{C}, \gamma_{\rho}(t)=a+\rho e^{i(\pi-t)}$.
If we have this result: How does this help to show $$\int_{-\infty}^{a}f(x)dx+\int_a^{\infty}f(x)dx=\pi iRes(f,a)+2\pi i \sum_{z\in\mathbb{H}}Res(f,z)?$$
It would be great if somebody has an indea how we can apply the Residue theorem.