Is there a function that equals its Laplace transform?

[Bipolarity]

Google seems to provide not much information on this. In essence, I am asking about the eigenfunctions of the Laplace transform when λ=1? Anyone have any insights on this rather unusual problem?

BiP

 

[Simon Bridge]

You want: $$F(s) = \int_0^\infty f(t)e^{-st}dt = f(s)$$ ... for a function to be it's own Laplace transform.

Been discussed before:
https://www.physicsforums.com/archive/index.php/t-180250.html
I don't think there is any F(s)=f(s) ... but I'd be hard pressed to prove it.

 

[JJacquelin]

You want: $$F(s) = \int_0^\infty f(t)e^{-st}dt = f(s)$$ ... for a function to be it's own Laplace transform.
Been discussed before:
https://www.physicsforums.com/archive/index.php/t-180250.html
I don't think there is any F(s)=f(s) ... but I'd be hard pressed to prove it.

I agree in case of real function.
In case of complex function, there are many solutions. For example, one of them :

 

[Simon Bridge]

Yike - well there you have it :)