I am curious to know what kind of applications the Laplace transform has. Yes, I know people will reference Wikipedia, and other online sites which discuss the Laplace transform at length. However, all the applications are very one-dimensional. For example, even looking at Wikipedia most the "applications" are towards solving differential equations.
Furthermore, I have been searching for many books, engineering books, physics books, math books, ect., which contain much material on Laplace transforms. All of those books use the Laplace transform only as a means to solve differential equations. I never see any other applications.
To add further to my question, I heard it said, each time the Laplace transform is introduced, of how valuable it is to electrical engineering. In fact, I said so myself, but looking at books, I again only find the applications of the transform to solving differential equations. Nothing really beyond that.
This is what I mean by "one-dimensional applications". Yes, the Laplace transform has "applications", but it really seems that the only application is solving differential equations and nothing beyond that.
Though, that is not entirely true, there is one more application of the Laplace transform which is not usually mentioned. And that is the moment generating function from probability theory. After all that is the original motivation of Laplace to create that transform in the first place. Unfortuantely, moment generating functions are not of superior importance to probability theory (to the best of my knowledge), and so the the only "big" applications of this transform appears to be only to the solution of differential equations (both ordinary and partial).
Contrast this with the Fourier transform. The Fourier transform can be used to also solve differential equations, in fact, more so. The Fourier transform can be used for sampling, imaging, processing, ect. And even in probability theory the Fourier transform is the characteristic function which is far more fundamental than the moment generating function.
The Fourier transform is certaintly a huge powerful tool with vast applications all across mathematics, physics, and engineering. There are books, across all fields, all devoted to the different applications of this transform.
But does the Laplace transform have any other "applications" to it other than solving differential equations? If you say that it does, then please provide a book reference which has an entire chapter, or large part of the book, discussing a non-differential equation application to which the Laplace transform is of fundamental importance?