Where does the laplace Kernel come from?

Question

Where does $e^{-st} $ kernel come from in a laplace transform?

Answer 1

Euler investigated integrals like $n! = \int_0^1 \left[\log (1/s)\right]^n\,ds$ which is equivalent to the usual definition of the gamma function (offset by $1$), after substution $s=e^x$.

Euler then studied integrals of a more general form $\int_0^{\infty} X(x)e^{ax} \,dx$. Lagrange later studied integrals of the form $\int_0^{\infty} X(x)e^{-ax} a^x dx$. Laplace worked on transforms including what is now known as the Laplace transform, but was the first to use the general properties that these kind of transforms gave (e.g., linearity, invariance, etc).

This could easily be argued an example of Stigler's law of eponymy