Problem Involving the Improper Integral

[Someone2841]

How would one find an (explicit) \(\displaystyle f(t)\) such that \(\displaystyle \int_{0}^{\infty}f(t,n)e^{-t}dt=f(n)\). Specifically, I am looking for the case where \(\displaystyle f(n)=n^2-1\).

Any help on this would be much appreciated.

 

[tkhunny]

You'll have to give a better clue. Exactly what is it you are doing and what tools are at your disposal? It is difficult to tell what sort of direction would be of any benefit to you?

Related-Looking Issues
1) Moment Generating Function
2) Gamma Function
3) Laplace Transform

It is possible some of these things could be of benefit, but not if your specific direction proscribes them.

 

[Someone2841]

You'll have to give a better clue. Exactly what is it you are doing and what tools are at your disposal? It is difficult to tell what sort of direction would be of any benefit to you?

Related-Looking Issues
1) Moment Generating Function
2) Gamma Function
3) Laplace Transform

It is possible some of these things could be of benefit, but not if your specific direction proscribes them.

Any direction would benefit me as this is just for personal learning. Thanks for pointing me towards some things to look into, especially the moment-generating function. It turns out I was making it more complicated than I should have, as \(\displaystyle \int_{0}^{\infty}f(n)e^{-t}dt=f(n)\) and therefore \(\displaystyle \int_{0}^{\infty}n^2e^{-t}dt=n^2\).