Inverse Laplace transform related to modified Bessel function of the second kind

Question

To solve a fluid diffusion problem, I need to calculate an inverse Laplace transform. The integral, according to the Inverse Laplace theorem, has the form: \begin{equation} \label{i_laplace_1} p(r,t) = \frac{p_{c}}{2\pi i}\lim_{n \rightarrow \infty}\int_{1-i\beta_{n}}^{1+i\beta_{n}}\frac{e^{st}}{s}\frac{K_{0}(\sqrt{\frac{s}{\kappa}} r)}{K_{0}(\sqrt{\frac{s}{\kappa}} a)} d s \end{equation} where $p_{c},r,a$ can be seen as constants. I choose a contour as in attached Figure 1 to avoid the branch point $z=0$ during the computation. The integral is then instead calculated on the contour including $C_{R},C_{\rho},\Gamma_{1},\Gamma_{2}$. On $C_{\rho}$, in several textbooks and papers I saw an approximation as followed: \begin{equation} \label{i_laplace_form} \lim_{\rho \rightarrow 0}\int_{C_{\rho}} \frac{e^{st}}{s}\frac{K_{0}(\sqrt{\frac{s}{\kappa}} r)}{K_{0}(\sqrt{\frac{s}{\kappa}} a)} ds \approx \lim_{\rho \rightarrow 0}\int_{C_{\rho}} \frac{e^{st}}{s} d s = 2\pi i \end{equation} However, I still do not fully understand how good is such an approximation and whether it is mathematically rigorous. Can you give me some insightful discussion?