I thought it would be obvious, but I can't find a series representation of the Laplace transform. I'm looking for something analogous to the Fourier series and how it can be used to derive the Fourier transform. I though it would simply be [tex]f(x) = \sum_{s=-\infty}^{\infty}{C_{s} e^{sx}}[/tex](adsbygoogle = window.adsbygoogle || []).push({});

, but that was under the assumption that the Laplace transform was merely a decomposition of a function into an exponential basis (I just simply noticed the analogies to the Fourier decomposition of a function into its basis imaginary exponential functions.) However, I'm increasingly thinking the interpretation is wrong, there appears to be no mention in any literature of a "Laplace series" other than people sometimes referring to the series of spherical harmonics as the "Laplace series".

I notice that the inverse Laplace transform appears to be a contour integral, and having not taken complex analysis before, I'm further mystified by how I am to interpret the Laplace transform. I had long thought it was a just an exponential decomposition, but I can't justify such a claim because the inverse Laplace transform is not as simple as the inverse Fourier transform where it is completely obvious that the inverse Fourier transform linearly combines basis sine and cosine terms to produce the original function.

Is there a series by which the Laplace transform can be derived? What is the Laplace transform if not a method to represent a function in a particular basis of an infinite dimensional vector space? Maybe the basis representation it produces is something I don't have in mind? Why do conceptual analogies between the Laplace and Fourier transform break down?

Sorry if the subject of this post doesn't seem completely clear. I learned the Laplace transform in ODE before I learned the Fourier transform in PDE. The Fourier transform has a fairly clean infinite dimensional vector space basis interpretation and I've tried to go back and apply it to the Laplace transform thinking it would be an easy analogy, but I am suddenly surprised by the Laplace transform's complexities and am essentially asking for reasons and explanations behind the conceptual differences between Laplace and Fourier transforms coming from the angle of somebody who understands basic Fourier decomposition theory.

Thanks to anyone who tries to bear with my apparent nonsense.. heh

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# Series form of the Laplace transform

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