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I'm learning about eigenvalues and eigenvectors (finally starting to get them). This might be a silly question, but what is/are the eigenvector(s) of the Laplace transform? I mean, what $\vec{x}_{i}$'s and $\lambda_{i}$'s satisfy \begin{align}\mathcal{L}\left\{\vec{x}_{i}\right\}&=\lambda_{i}\vec{x}_{i}.\end{align} I'm just trying to extrapolate a bit from the fact that \begin{align}D_{t}e^{\lambda t}&=\lambda e^{\lambda t},\end{align} but I cannot think of any function that remains unchanged under the transformation.

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    $\begingroup$ Eigenvalues only make sense for operators that act on a single space. The Laplace transform goes from t space to s space, so eigenvalues won't have any real meaning. The same is true for a linear map between two different finite dimensional spaces. $\endgroup$
    – Jules
    Apr 8, 2018 at 14:21

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The Laplace transform of $t^p$ is proportional to $\frac{1}{s^{p+1}}$ for $p>-1$. Take $p=-1/2$.

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    $\begingroup$ You mean $-\frac12$, right? $\endgroup$
    – sbares
    Dec 21, 2015 at 0:13
  • $\begingroup$ Is this the only one? $\endgroup$
    – bjd2385
    Dec 21, 2015 at 0:23
  • $\begingroup$ SBareS - thanks. @jm324354 - dunno, this is the only one I know off the top of my head. $\endgroup$
    – Batman
    Dec 21, 2015 at 0:43

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