---
tag: 16W9PP8ET0
kind: statement
role: theorem
name: Abelian theorems for integral transforms
msc: [44A10, "40E05"]
uses: [05XT5AVJBK, D6XWK1NRY1, BYNV84PF86]
about: [0V49JEZW06, D6XWK1NRY1]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_and_Tauberian_theorems
---

## Statement

Let $f \colon [0, \infty) \to \C$ be locally integrable, and let $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ be its [[Laplace transform|05XT5AVJBK]].
Let $c \in \C$.

1. (Behaviour at infinity.) If $\frac{1}{T} \int_0^T f(t)\, dt \to c$ as $T \to \infty$, then $F(s)$ converges for every real $s > 0$, and $s F(s) \to c$ as $s \to 0^+$.
In particular this holds when $f(t) \to c$ as $t \to \infty$.
2. (Behaviour at zero.) If $\int_0^\infty \abs{f(t)} e^{-s_0 t}\, dt < \infty$ for some real $s_0$, and $f(t) \to c$ as $t \to 0^+$, then $s F(s) \to c$ as $s \to +\infty$ through real values.

## 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|BYNV84PF86]].
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|NV54MWKT6T]].

## Explanations

### L1 human

These results say that the Laplace transform remembers the limits of $f$.
The long-run value of $f$ shows up in $sF(s)$ for small $s$, and the starting value of $f$ shows up in $sF(s)$ for large $s$.
This is the integral analogue of Abel's theorem on power series, which is where the name "Abelian" comes from.

### L2 human

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.