The idea behind numerical ILT and how to compute it

My question on Mathematics is still not getting an answer. And since, this topic is related on computational, i hope this site matches with my question. So please if you understand this topic explain it to me.

I’ve been reading so much articles about this topic on Mathematics and Computer Sciences journal and i would like to know the idea behind this topic. What i got from my understanding is :

We have this inversion formula:
$$f(t) = frac{1}{2pi i}int_{sigma – iinfty}^{sigma + iinfty} e^{st}F(s), mathrm ds$$
Where $F(s)$ is the Laplace transform of $f(t)$

By using u-substitution (i forgot the detail) we have to change the bound becomes the improper one (from $-infty$ to $infty$, and notice there is no imaginary $i$ and $sigma$ also) and we must somehow split the integrand into real part and the imaginary part. Then the approximation of the Bromwich integral is given by reducing the obtained integral to the real part only (if i’m not mistakenly understood).

Here is the final form of the substitution that i’m talking about taken from article which written by Mr. Joseph Abate

$$f(t) = frac{2e^{sigma t}}{pi} int_{0}^{infty} operatorname{Re}left(F(sigma + iu)right)cos(ut),Bbb du$$

Well, my main question is just in which part should i approximate the integral by for example trapezoidal rule? I mean when we evaluating the bromwich integral with singularities, we usually use Residual theorem to calculate it and then proving that the big arc goes to $0$ as $Rtoinfty$ (my main discussion here is the function has no branch cut i.e. the inverse Laplace transform of meromorphic function).

So, is the approximation for calculating the integral along vertical line or the integral along the whole path including the big arc and the vertical line of the Bromwich contour?

And how does it make sense when i still can’t understand where i go from residual theorem or even a countour with branch points to the approximation as my main discussion? Is the approximation talking about the whole approximation of the integral given by my first equation above instead of the application of residual theorem or what? And i read there’s a method called Talbot’s Method and just rephrased the integral bound to be $-pi$ to $pi$. But again which part is called “numerical inverse laplace transform” if the method itself can be evaluated using standard integral. Or should i change the integral using numerical integral or is there something different that i thought?

I’m really new into this topic and i want to start writing my own article about this topic. Hope you can help me to understand. And by the way i’m just an undergraduate student, so please give the best explanation that easy to understand.

Thanks in advance!