Eigenfunctions of Laplace Transform

1. Aug 25, 2014

Whovian

Firstly, if this is an inappropriate forum for this thread, feel free to move it. This is a calculus-y equation related to differential equations, but I don't believe it's strictly a differential equation.

The question I'm asking is which functions $f:\left[0,\infty\right)\rightarrow\mathbb{R}$ and real constants $\lambda$ have the property that $\int_0^\infty\left(f\left(t\right)\cdot e^{-s\cdot t}\right)\cdot\mathrm{d}t=f\left(s\right)$ for all $s$ in some open interval.

The question was left somewhat open-ended in this old thread, but since it was from 6 years ago, I felt reviving it would be somewhat unnecessary.

Induction on $n$ gives us the apparently trivial condition that $\int_0^\infty\left(\left(-t\right)^n\cdot e^{-s\cdot t}\cdot f\left(t\right)\right)\cdot\mathrm{d}t=\lambda\cdot f^{\left(n\right)}\left(s\right)$; the left hand side seems to be screaming Caputo fractional derivative, so perhaps this is of some use. That's basically all I've got.

2. Sep 1, 2014

Greg Bernhardt

One eigenfunction (if I recall correctly) is $\frac{1}{\sqrt{t}}$; I recall working this out in a complex analysis homework assignment but don't recall the eigenvalue. Others may exist - try looking in large tables of Laplace Transforms and you may find others.