# Light Commentary on Laplace Transforms

Ever since I learned about the Laplace transform method in undergraduate engineering, I’ve always been somewhat mystified by it. The Laplace transform of a function $f(t)$ is defined by:

$L[f(t)] = \int_{0}^{\infty}e^{-st}f(t)dt$

The Wikipedia page has all of the information on what the Laplace transform is used for and why it is useful. Briefly, a Laplace transform allows for easy solution of $n^{th}$-order linear non-homogeneous differential equations, as well as some partial differential equations. However, the Wiki is somewhat light on the history and origins of the transform.

My question is: what motivated Laplace to dream up such a strange integral, and apply it to solving differential equations?

This source says it was first used by Laplace to prove the central limit theorem.

Arthur Mattuck has given a pretty good lecture connecting the Laplace Transform with the power series, but I still don’t know if that’s what Laplace was thinking when he made his discovery. Unfortunately, this source probably has all the information I’d like in it, but it is paywalled.