For the purpose of my research on stochastic processes on graphs I need to compute the behaviour of a correlation function at time $t=0$, $C(t=0)$. I was able to compute its bilateral Laplace transform, which is (simplifying all the parameters $$ C(z) \propto \left[\frac{\sqrt{(z+\lambda)^2-a^2}}{z} - \frac{\sqrt{(z-\lambda)^2-a^2}}{z} - 2 \right] $$ where $a,\lambda>0$.
Is there a way to compute $C(t=0)$ without computing the inverse bilateral Laplace transform $$ C(t)=\frac{1}{2\pi j}\int_{\gamma -j \infty}^{\gamma + j \infty}e^{zt}C(z) $$ If this is not possible:
- What is the ROC of $C(z)$?
- What contour should I take to compute the complex integral?