\to 0}
\frac{d_0^2}{4\pi^2 c_1^2 \omega}e^{- \pi \omega \lambda}\times 
\nonumber \\
&&\int_{0-{\mathrm{i}}\epsilon}^{\infty-{\mathrm{i}}\epsilon} \omega'
\omega_+^{-\mathrm{i}\lambda\varepsilon-2}
\int_{\mathrm{i}\omega_+ x_0}^{\mathrm{i}\omega_+ x_0+\infty} e^{-y} 
y^{-\mathrm{i}\omega\lambda} \; {\mathrm{d}}y \; 
\int_{-{\mathrm{i}}\omega_+^* x_0}^{-\mathrm{i}\omega_+^* x_0+\infty} 
e^{-\tilde{y}} 
\tilde{y}^{{\mathrm{i}}\omega \lambda} \; 
{\mathrm{d}}\tilde{y} \;{\mathrm{d}}\omega' \; .
\label{beforesub}\end{aligned}$$

Let us now take the difference to the $x_0 \to 0$ limit (\[x0lim\]), $$\begin{aligned}
&&\int_0^{\infty} \vert\beta^1_{\omega'\omega}\vert^2 \; \mathrm{d}\omega' -
\lim_{x_0 \to 0}\int_0^{\infty} \vert\beta^1_{\omega'\omega}\vert^2 \; 
{\mathrm{d}}\omega' =  
\lim_{\epsilon \to 0 \atop \varepsilon \to 0} 
\frac{d_0^2}{4\pi^2 c_1^2 \omega} e^{-\pi\omega\lambda}\times \nonumber \\
&&\int_{0-\mathrm{i}\epsilon}^{\infty-\mathrm{i}\epsilon} \omega' 
\omega_+^{-\mathrm{i}\lambda\varepsilon - 2} 
\left[\left|\int_{\mathrm{i}\omega_+ x_0}^{\mathrm{i}\omega_+ x_0+\infty} 
y^{-\mathrm{i}\omega\lambda} e^{- y} \mbox{d}y \right|^2\; - 
\vert \Gamma(1-\mathrm{i}\omega\lambda)\vert^2\right] \; \mbox{d}\omega' \; .\end{aligned}$$ Note that the expression inside the square bracket is well defined for all $\omega'$. We now replace the integration variable $\omega'$ by $\omega_+$, using its definition from above, and obtain (we omit $\epsilon$ and $\varepsilon$ in the discussion of the following step) $$\begin{aligned}
&& 
\frac{1}{4\pi^2\omega} e^{-\pi\omega\lambda}
\left\{\int_{\omega}^{\infty} \omega_+^{- 1} 
\left[\left|\int_{\mathrm{i}\omega_+ x_0}^{\mathrm{i}\omega_+x_0+\infty} 
y^{-\mathrm{i}\omega\lambda} e^{-y} \mbox{d}y \; \right|^2 - 
\vert\Gamma(1-\mathrm{i}\omega\lambda) \vert^2 \right] \mbox{d}\omega_+ 
\right.\\ 
&& \left. - \omega \int_{\omega }^{\infty} \omega_+^{- 2} 
\left[\left|\int_{\mathrm{i}\omega_+ x_0}^{\mathrm{i}\omega_+ x_0 +\infty} 
y^{-\mathrm{i}\omega\lambda} e^{- y} \mbox{d}y \; \right|^2 
- \vert \Gamma(1-\mathrm{i}\omega\lambda)\vert^2 \right] 
\mbox{d}\omega_+ \right\} \ .\end{aligned}$$ The second term is finite (see Appendix) and, since we expect a divergent result, we can neglect it. The first 