Confusion in proof of Inverse Laplace Transform For the proof of inverse Laplace transform, we change the integral from $\omega$ to $s$. I want to know the reason why we need to change the integral?
To summarize the discussion:


The usual substitution is $s = \sigma + j \omega$ where $\sigma$ is the real part of the $s$ variable and $\omega$ is the imaginary part.
The equation in the image is for the Fourier transform, not the Laplace transform. The Fourier transform can be thought of as the Laplace transform evaluated on the imaginary axis ($\sigma = 0$).
The differential $ds$, when looking at real and imaginary parts distinctly, becomes $d\sigma + j d\omega$.
Any differential is an infinitesimal (very small) change in that variable. $dx$ is a small change in $x$.