I need to compute the two-sided (bilateral) Laplace transform of a numerically given function $F$,$$I(t) = \int_{-\infty}^{+\infty} {dx} \, e^{-x} \, F(x + t) ~,$$where $F(x)$ has some sharp features, e.g., at $ x = \{ 0, x_p, \cdots \} $, has finite support, and vanishes sufficiently fast for $ x < 0 $ to prevent a divergence of the integral.Due to the sharp features, I have to divide the integration region to some subregions, for instance,\begin{align}I(t) &= \int_{-\infty}^{-\Delta} {dx} \, e^{-x} \, F(x + t) + \int_{-\Delta}^{+\Delta} {dx} \, e^{-x} \, F(x + t) \\&\quad + \int_{+\Delta}^{x_p - \Delta} {dx} \, e^{-x} \, F(x + t) + \int_{x_p - \Delta}^{x_p + \Delta} {dx} \, e^{-x} \, F(x + t) \\&\quad + \int_{x_p + \Delta}^{+\infty} {dx} \, e^{-x} \, F(x + t) ~,\end{align}where $\Delta > 0$.I'd like to know the best and most precise way to perform such an integral numerically (quadrature methods). Is there something like Gauss-Laguerre quadrature for such an integration?I have also considered a transformation of the integration variable,$$u = e^{-x} ~,$$to absorb the exponential factor,$$I(t) = \int_{0^+ = \, u(x \rightarrow +\infty)}^{+\infty = \, u(x \rightarrow -\infty)} {du} \, F(-\ln(u) + t) ~,$$and performing the transformed integration by a Tanh-Sinh quadrature; yet I am not sure if this is the best method.As an example, one can take$$f(x,\omega) = e^{-\alpha (x - \omega)} \, \Theta(x - \omega)$$where $\Theta$ denotes the Heaviside step function and $\alpha > 0$. Hence, the width of the peak at $\omega$ can be varied by $\alpha$.The two-sided Laplace transform of $f$ will be$$\mathcal{L}[f](\omega) := \int_{-\infty}^{+\infty} dx \, e^{-x} \, f(x, \omega) = \frac{ e^{-\omega} }{1 + \alpha} ~.$$
Numerical computation of two-sided (bilateral) Laplace transform
quadrature