# Injectivity of Laplace Transform/Corollary

Jump to navigation
Jump to search

## Corollary to Injectivity of Laplace Transform

Let $f$, $g$ be functions from $\R_{\ge 0} \to K$ of a real variable $t$, where $K \in \set {\R, \C}$.

Let $f$ and $g$ both admit Laplace transforms.

Suppose that the Laplace transforms $\laptrans f$ and $\laptrans g$ satisfy:

- $\forall t \ge 0: \laptrans {\map f t} = \laptrans {\map g t}$

Let $f$ and $g$ be continuous everywhere on their domains, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\R_{\ge 0}$.

Then $f = g$ everywhere on $\R_{\ge 0}$, except possibly where $f$ or $g$ have discontinuities of the first kind.

This article, or a section of it, needs explaining.In particular: Establish whether it is "finite subinterval" that is needed here, or what we have already defined as "Definition:Finite Subdivision".You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.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 `{{Explain}}` from the code. |

## 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. |