Abelian theorems for integral transforms

Remarks

Both parts compare $f$ with the constant $c$, using $s \int_0^\infty e^{-st}\, dt = 1$ for $s > 0$. For part 2, write $sF(s) - c = s\int_0^\infty (f(t) - c) e^{-st}\, dt$; the part of the integral over $[0, \delta]$ is small because $f$ is close to $c$ there, and the rest tends to $0$ by dominated convergence. For part 1, integrate by parts with $U(T) = \int_0^T f$ to get $sF(s) = s^2 \int_0^\infty U(T) e^{-sT}\, dT$, and use $s^2 \int_0^\infty T e^{-sT}\, dT = 1$. Example: for $f(t) = 1 + e^{-t}$, $F(s) = 1/s + 1/(s + 1)$, so $sF(s) \to 1 = \lim_{t \to \infty} f(t)$ as $s \to 0^+$, and $sF(s) \to 2 = f(0)$ as $s \to \infty$. The converses are false without extra conditions; they are the Tauberian theorems.

Going deeper

For a student

In engineering these are the final value theorem and the initial value theorem. They read off the limits of a solution of a linear differential equation from its transform without inverting it. Part 1 is the Laplace analogue of the fact that Cesàro summability implies Abel summability. Analogous Abelian theorems hold for other transforms, such as the Stieltjes and Mellin transforms: an asymptotic behaviour of the function gives a matching asymptotic behaviour of the transform.