Final Value Theorem of Laplace Transform/General Result

Theorem

Let $f$ and $g$ be real functions.

Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.

Let $\ds \lim_{t \mathop \to \infty} \dfrac {\map f t} {\map g t} = 1$.

Then:

$\ds \lim_{s \mathop \to 0} \dfrac {\map F s} {\map G s} = 1$

if those limits exist.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $14$. Generalization of final-value theorem: Theorem $1 \text{-} 19$