# 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

### Staff: Admin

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?

3. Sep 2, 2014

### jasonRF

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.

jason

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