# Integral involving expoential - does it have a name?

• Apr 26th 2010, 11:31 AM
astronomerroyal
Integral involving expoential - does it have a name?
I'm working with this simple little integral,

$\displaystyle I_{l}=\int_{x_{i}}^{x_{f}}x^{l}\exp[-x^{2}]dx$,

and was wondering if it had a name. It seems so basic that I imagine there's a lot of literature written about it - I would like to know more about it.

An observation: If the limits were 0 and infinity the solution would be related to gamma functions.

I have found solutions for integer $\displaystyle l$ (integrate by parts once, then note a recursion relation). Then general solutions can be written down; one for even $\displaystyle l$ and one for odd $\displaystyle l$.

e.g. for odd $\displaystyle l$

$\displaystyle I_{l}=\left[-\frac{1}{2}\exp\left(-x^{2}\right)\left\{ x^{l-1}+\sum_{n=1}^{N}\left(x^{(l-1)-2n}\prod_{m=1}^{n}\frac{(l-1)-2(m-1)}{2}\right)\right\} \right]_{x_{i}}^{x_{f}}$

I think that's correct - I'm just getting the feeling that it must have been solved a million times before...

Any thoughts?

Tom.
• Apr 26th 2010, 09:57 PM
chiph588@
Quote:

Originally Posted by astronomerroyal
I have found solutions for integer $\displaystyle l$

What if $\displaystyle l=0$?
• Apr 27th 2010, 09:39 AM
astronomerroyal
Quote:

Originally Posted by chiph588@
What if $\displaystyle l=0$?

$\displaystyle l=0$? Error functions.

For non-zero, positive integer $\displaystyle l$ the integral reminds one of the moment equation for the gaussian. This is just one of the reasons I believe that huge tomes have already been written on this particular integral. However, I'm not finding the tomes (via google, wikipedia, colleagues etc.).